From 7478090e7190752b5e80004d90a233d88d32a7dc Mon Sep 17 00:00:00 2001 From: Ed_ Date: Sun, 21 Jun 2026 16:32:19 -0400 Subject: [PATCH] conductor(probability_logic): Phase 1 Acquire - transcript.json (3315 segments via yt-dlp VTT fallback) + video.log (84MB mp4 downloaded) Generic reusable drivers added: phase1_acquire.py, phase2_keyframes.py, phase3_ocr.py take slug as arg for batch use across all 12 children. --- .../artifacts/transcript.json | 16583 ++++++++++++++++ .../artifacts/video.log | 10 + .../phase1_acquire.py | 132 + .../phase2_keyframes.py | 37 + .../phase3_ocr.py | 39 + 5 files changed, 16801 insertions(+) create mode 100644 conductor/tracks/video_analysis_probability_logic_20260621/artifacts/transcript.json create mode 100644 conductor/tracks/video_analysis_probability_logic_20260621/artifacts/video.log create mode 100644 scripts/tier2/artifacts/video_analysis_campaign_20260621/phase1_acquire.py create mode 100644 scripts/tier2/artifacts/video_analysis_campaign_20260621/phase2_keyframes.py create mode 100644 scripts/tier2/artifacts/video_analysis_campaign_20260621/phase3_ocr.py diff --git a/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/transcript.json b/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/transcript.json new file mode 100644 index 00000000..f58e9d3e --- /dev/null +++ b/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/transcript.json @@ -0,0 +1,16583 @@ +{ + "video_id": "0yF9TvMeAzM", + "segments": [ + { + "start": 0.0, + "duration": 0.0, + "text": "So,<00:00:01.120> we're<00:00:01.200> going<00:00:01.320> to<00:00:01.400> talk<00:00:01.600> about" + }, + { + "start": 2.03, + "duration": 0.0, + "text": "So, we're going to talk about" + }, + { + "start": 2.04, + "duration": 0.0, + "text": "So, we're going to talk about probability<00:00:02.560> today<00:00:03.160> and<00:00:03.520> we're<00:00:03.600> going<00:00:03.800> to" + }, + { + "start": 3.87, + "duration": 0.0, + "text": "probability today and we're going to" + }, + { + "start": 3.88, + "duration": 0.0, + "text": "probability today and we're going to give<00:00:04.360> a<00:00:04.440> very<00:00:04.584> [snorts]" + }, + { + "start": 5.27, + "duration": 0.0, + "text": "give a very [snorts]" + }, + { + "start": 5.28, + "duration": 0.0, + "text": "give a very [snorts] um" + }, + { + "start": 6.39, + "duration": 0.0, + "text": "um" + }, + { + "start": 6.4, + "duration": 0.0, + "text": "um overlooked<00:00:07.040> and<00:00:07.160> underdeveloped<00:00:07.680> approach" + }, + { + "start": 8.71, + "duration": 0.0, + "text": "overlooked and underdeveloped approach" + }, + { + "start": 8.72, + "duration": 0.0, + "text": "overlooked and underdeveloped approach that<00:00:08.960> sees<00:00:09.280> probability<00:00:10.200> theory<00:00:10.680> as<00:00:10.920> an" + }, + { + "start": 10.99, + "duration": 0.0, + "text": "that sees probability theory as an" + }, + { + "start": 11.0, + "duration": 0.0, + "text": "that sees probability theory as an extension<00:00:11.400> of<00:00:11.520> logic." + }, + { + "start": 12.47, + "duration": 0.0, + "text": "extension of logic." + }, + { + "start": 12.48, + "duration": 0.0, + "text": "extension of logic. Famously,<00:00:13.280> one<00:00:13.440> of<00:00:13.520> the<00:00:14.000> first" + }, + { + "start": 15.19, + "duration": 0.0, + "text": "Famously, one of the first" + }, + { + "start": 15.2, + "duration": 0.0, + "text": "Famously, one of the first scientists<00:00:15.920> and<00:00:16.040> mathematicians<00:00:16.600> to<00:00:16.920> develop" + }, + { + "start": 17.27, + "duration": 0.0, + "text": "scientists and mathematicians to develop" + }, + { + "start": 17.28, + "duration": 0.0, + "text": "scientists and mathematicians to develop this<00:00:17.440> idea<00:00:17.720> was<00:00:17.880> Laplace,<00:00:18.440> who<00:00:18.680> in<00:00:18.840> 1819<00:00:19.360> said," + }, + { + "start": 19.63, + "duration": 0.0, + "text": "this idea was Laplace, who in 1819 said," + }, + { + "start": 19.64, + "duration": 0.0, + "text": "this idea was Laplace, who in 1819 said, \"Probability<00:00:20.120> theory<00:00:20.520> is<00:00:20.680> nothing<00:00:20.960> but" + }, + { + "start": 21.11, + "duration": 0.0, + "text": "\"Probability theory is nothing but" + }, + { + "start": 21.12, + "duration": 0.0, + "text": "\"Probability theory is nothing but common<00:00:21.440> sense<00:00:22.080> reduced<00:00:22.520> to<00:00:22.600> calculation.\"" + }, + { + "start": 24.75, + "duration": 0.0, + "text": "common sense reduced to calculation.\"" + }, + { + "start": 24.76, + "duration": 0.0, + "text": "common sense reduced to calculation.\" And<00:00:24.920> we<00:00:25.000> will<00:00:25.120> see<00:00:25.240> today<00:00:25.520> what<00:00:25.640> that<00:00:25.800> means" + }, + { + "start": 25.99, + "duration": 0.0, + "text": "And we will see today what that means" + }, + { + "start": 26.0, + "duration": 0.0, + "text": "And we will see today what that means exactly." + }, + { + "start": 27.43, + "duration": 0.0, + "text": "exactly." + }, + { + "start": 27.44, + "duration": 0.0, + "text": "exactly. So,<00:00:27.760> first<00:00:27.920> we're<00:00:28.000> going<00:00:28.120> to<00:00:28.200> look<00:00:28.360> at<00:00:28.600> the" + }, + { + "start": 28.67, + "duration": 0.0, + "text": "So, first we're going to look at the" + }, + { + "start": 28.68, + "duration": 0.0, + "text": "So, first we're going to look at the different<00:00:28.880> definitions<00:00:29.280> of<00:00:29.320> probability." + }, + { + "start": 30.03, + "duration": 0.0, + "text": "different definitions of probability." + }, + { + "start": 30.04, + "duration": 0.0, + "text": "different definitions of probability. We're<00:00:30.120> going<00:00:30.240> to<00:00:30.320> talk<00:00:30.520> about<00:00:30.720> some<00:00:30.880> classical" + }, + { + "start": 31.31, + "duration": 0.0, + "text": "We're going to talk about some classical" + }, + { + "start": 31.32, + "duration": 0.0, + "text": "We're going to talk about some classical logic,<00:00:32.160> then<00:00:32.599> some<00:00:32.800> lattice<00:00:33.120> theory<00:00:33.640> because" + }, + { + "start": 33.91, + "duration": 0.0, + "text": "logic, then some lattice theory because" + }, + { + "start": 33.92, + "duration": 0.0, + "text": "logic, then some lattice theory because this<00:00:34.120> is<00:00:34.240> how<00:00:34.480> we're<00:00:34.600> going<00:00:34.800> to" + }, + { + "start": 35.67, + "duration": 0.0, + "text": "this is how we're going to" + }, + { + "start": 35.68, + "duration": 0.0, + "text": "this is how we're going to derive<00:00:36.000> our<00:00:36.120> foundations." + }, + { + "start": 37.51, + "duration": 0.0, + "text": "derive our foundations." + }, + { + "start": 37.52, + "duration": 0.0, + "text": "derive our foundations. We're<00:00:37.680> going<00:00:37.800> to<00:00:38.280> derive<00:00:38.640> the<00:00:38.720> famous<00:00:39.080> sum" + }, + { + "start": 39.27, + "duration": 0.0, + "text": "We're going to derive the famous sum" + }, + { + "start": 39.28, + "duration": 0.0, + "text": "We're going to derive the famous sum rule<00:00:39.760> and<00:00:39.920> the<00:00:40.000> product<00:00:40.360> rules<00:00:40.640> of" + }, + { + "start": 40.71, + "duration": 0.0, + "text": "rule and the product rules of" + }, + { + "start": 40.72, + "duration": 0.0, + "text": "rule and the product rules of probability<00:00:41.440> that<00:00:41.600> you<00:00:41.720> all<00:00:41.880> know.<00:00:42.720> Uh<00:00:43.200> we're" + }, + { + "start": 43.31, + "duration": 0.0, + "text": "probability that you all know. Uh we're" + }, + { + "start": 43.32, + "duration": 0.0, + "text": "probability that you all know. Uh we're going<00:00:43.440> to<00:00:43.520> talk<00:00:43.680> about<00:00:44.720> how<00:00:44.840> this<00:00:45.040> leads<00:00:45.280> to" + }, + { + "start": 45.43, + "duration": 0.0, + "text": "going to talk about how this leads to" + }, + { + "start": 45.44, + "duration": 0.0, + "text": "going to talk about how this leads to Bayesian<00:00:45.720> inference<00:00:46.360> with<00:00:46.840> Bayes'<00:00:47.160> rule<00:00:47.880> and" + }, + { + "start": 48.11, + "duration": 0.0, + "text": "Bayesian inference with Bayes' rule and" + }, + { + "start": 48.12, + "duration": 0.0, + "text": "Bayesian inference with Bayes' rule and then<00:00:48.360> some<00:00:48.680> unique<00:00:48.960> powers<00:00:49.680> of<00:00:50.080> Bayesian" + }, + { + "start": 50.31, + "duration": 0.0, + "text": "then some unique powers of Bayesian" + }, + { + "start": 50.32, + "duration": 0.0, + "text": "then some unique powers of Bayesian inference." + }, + { + "start": 52.59, + "duration": 0.0, + "text": "inference." + }, + { + "start": 52.6, + "duration": 0.0, + "text": "inference. All<00:00:52.760> right.<00:00:53.120> So," + }, + { + "start": 54.87, + "duration": 0.0, + "text": "All right. So," + }, + { + "start": 54.88, + "duration": 0.0, + "text": "All right. So, nowadays<00:00:55.480> there<00:00:55.720> is<00:00:56.680> two<00:00:57.280> big<00:00:58.120> definitions<00:00:58.760> of" + }, + { + "start": 58.87, + "duration": 0.0, + "text": "nowadays there is two big definitions of" + }, + { + "start": 58.88, + "duration": 0.0, + "text": "nowadays there is two big definitions of probability<00:00:59.680> that<00:00:59.880> kind<00:01:00.040> of<00:01:00.360> contend<00:01:00.880> for<00:01:01.880> the" + }, + { + "start": 61.99, + "duration": 0.0, + "text": "probability that kind of contend for the" + }, + { + "start": 62.0, + "duration": 0.0, + "text": "probability that kind of contend for the spot<00:01:02.320> of<00:01:02.400> being<00:01:02.560> correct." + }, + { + "start": 63.79, + "duration": 0.0, + "text": "spot of being correct." + }, + { + "start": 63.8, + "duration": 0.0, + "text": "spot of being correct. And<00:01:04.239> that<00:01:04.400> is<00:01:04.600> the<00:01:05.040> frequentist" + }, + { + "start": 66.47, + "duration": 0.0, + "text": "And that is the frequentist" + }, + { + "start": 66.48, + "duration": 0.0, + "text": "And that is the frequentist uh" + }, + { + "start": 67.31, + "duration": 0.0, + "text": "uh" + }, + { + "start": 67.32, + "duration": 0.0, + "text": "uh interpretation,<00:01:07.880> which<00:01:08.080> sees<00:01:08.320> probability" + }, + { + "start": 68.91, + "duration": 0.0, + "text": "interpretation, which sees probability" + }, + { + "start": 68.92, + "duration": 0.0, + "text": "interpretation, which sees probability as<00:01:09.080> sort<00:01:09.280> of<00:01:09.680> the<00:01:09.800> limit<00:01:10.120> of<00:01:10.240> the<00:01:10.320> frequency<00:01:11.520> uh" + }, + { + "start": 71.75, + "duration": 0.0, + "text": "as sort of the limit of the frequency uh" + }, + { + "start": 71.76, + "duration": 0.0, + "text": "as sort of the limit of the frequency uh of<00:01:11.920> an<00:01:12.000> event<00:01:12.280> happening,<00:01:13.200> and<00:01:13.640> the" + }, + { + "start": 73.75, + "duration": 0.0, + "text": "of an event happening, and the" + }, + { + "start": 73.76, + "duration": 0.0, + "text": "of an event happening, and the plausibility<00:01:14.400> approach,<00:01:14.840> which<00:01:14.960> is<00:01:15.040> the" + }, + { + "start": 75.11, + "duration": 0.0, + "text": "plausibility approach, which is the" + }, + { + "start": 75.12, + "duration": 0.0, + "text": "plausibility approach, which is the Bayesian<00:01:15.400> approach,<00:01:15.840> which<00:01:16.080> sees" + }, + { + "start": 76.75, + "duration": 0.0, + "text": "Bayesian approach, which sees" + }, + { + "start": 76.76, + "duration": 0.0, + "text": "Bayesian approach, which sees probability<00:01:17.360> simply<00:01:18.440> as<00:01:18.640> a<00:01:18.680> quantification" + }, + { + "start": 79.79, + "duration": 0.0, + "text": "probability simply as a quantification" + }, + { + "start": 79.8, + "duration": 0.0, + "text": "probability simply as a quantification of<00:01:20.720> how<00:01:21.400> plausible<00:01:22.600> an<00:01:22.720> event<00:01:23.200> or<00:01:23.320> a" + }, + { + "start": 83.35, + "duration": 0.0, + "text": "of how plausible an event or a" + }, + { + "start": 83.36, + "duration": 0.0, + "text": "of how plausible an event or a proposition<00:01:23.920> is<00:01:24.480> given<00:01:25.200> our<00:01:25.480> state<00:01:25.680> of" + }, + { + "start": 85.79, + "duration": 0.0, + "text": "proposition is given our state of" + }, + { + "start": 85.8, + "duration": 0.0, + "text": "proposition is given our state of knowledge<00:01:26.160> or<00:01:26.480> our<00:01:26.560> state<00:01:26.720> of<00:01:26.840> ignorance," + }, + { + "start": 87.31, + "duration": 0.0, + "text": "knowledge or our state of ignorance," + }, + { + "start": 87.32, + "duration": 0.0, + "text": "knowledge or our state of ignorance, depending<00:01:27.680> on<00:01:27.840> how<00:01:28.040> you<00:01:28.400> look<00:01:28.560> at<00:01:28.640> it." + }, + { + "start": 89.43, + "duration": 0.0, + "text": "depending on how you look at it." + }, + { + "start": 89.44, + "duration": 0.0, + "text": "depending on how you look at it. So,<00:01:30.000> for<00:01:30.200> example,<00:01:31.160> imagine<00:01:32.080> um<00:01:32.640> that<00:01:32.800> we're" + }, + { + "start": 92.91, + "duration": 0.0, + "text": "So, for example, imagine um that we're" + }, + { + "start": 92.92, + "duration": 0.0, + "text": "So, for example, imagine um that we're doing<00:01:33.160> the<00:01:33.240> very<00:01:33.600> simple<00:01:33.840> experiment<00:01:34.200> of" + }, + { + "start": 94.27, + "duration": 0.0, + "text": "doing the very simple experiment of" + }, + { + "start": 94.28, + "duration": 0.0, + "text": "doing the very simple experiment of tossing<00:01:34.600> a<00:01:34.640> coin." + }, + { + "start": 95.67, + "duration": 0.0, + "text": "tossing a coin." + }, + { + "start": 95.68, + "duration": 0.0, + "text": "tossing a coin. And<00:01:36.040> imagine<00:01:36.320> this<00:01:36.480> is<00:01:36.640> just<00:01:36.840> a<00:01:36.880> regular<00:01:37.200> coin," + }, + { + "start": 97.59, + "duration": 0.0, + "text": "And imagine this is just a regular coin," + }, + { + "start": 97.6, + "duration": 0.0, + "text": "And imagine this is just a regular coin, it's<00:01:37.720> a<00:01:37.760> fair<00:01:38.040> coin,<00:01:38.360> you<00:01:38.440> know,<00:01:38.560> nothing" + }, + { + "start": 98.91, + "duration": 0.0, + "text": "it's a fair coin, you know, nothing" + }, + { + "start": 98.92, + "duration": 0.0, + "text": "it's a fair coin, you know, nothing weird<00:01:39.120> is<00:01:39.200> going<00:01:39.520> on.<00:01:40.160> Why<00:01:40.320> do<00:01:40.440> we<00:01:40.560> say<00:01:40.760> the" + }, + { + "start": 100.91, + "duration": 0.0, + "text": "weird is going on. Why do we say the" + }, + { + "start": 100.92, + "duration": 0.0, + "text": "weird is going on. Why do we say the probability<00:01:41.440> is<00:01:41.600> 50%<00:01:42.440> uh<00:01:42.640> 50%?<00:01:43.720> The" + }, + { + "start": 103.79, + "duration": 0.0, + "text": "probability is 50% uh 50%? The" + }, + { + "start": 103.8, + "duration": 0.0, + "text": "probability is 50% uh 50%? The frequentists<00:01:44.320> would<00:01:44.480> say<00:01:44.720> that<00:01:44.960> because<00:01:45.760> if" + }, + { + "start": 106.11, + "duration": 0.0, + "text": "frequentists would say that because if" + }, + { + "start": 106.12, + "duration": 0.0, + "text": "frequentists would say that because if you<00:01:46.840> keep<00:01:47.400> flipping<00:01:47.720> the<00:01:47.800> coins,<00:01:48.800> the<00:01:49.080> ratio" + }, + { + "start": 109.59, + "duration": 0.0, + "text": "you keep flipping the coins, the ratio" + }, + { + "start": 109.6, + "duration": 0.0, + "text": "you keep flipping the coins, the ratio of<00:01:49.680> the<00:01:49.800> two<00:01:50.000> outcomes<00:01:50.840> will<00:01:51.120> eventually<00:01:52.120> uh" + }, + { + "start": 112.55, + "duration": 0.0, + "text": "of the two outcomes will eventually uh" + }, + { + "start": 112.56, + "duration": 0.0, + "text": "of the two outcomes will eventually uh approach<00:01:52.920> one,<00:01:53.280> meaning<00:01:53.960> uh" + }, + { + "start": 114.63, + "duration": 0.0, + "text": "approach one, meaning uh" + }, + { + "start": 114.64, + "duration": 0.0, + "text": "approach one, meaning uh that<00:01:54.840> the<00:01:54.920> probability<00:01:55.400> that<00:01:55.600> either<00:01:55.760> one<00:01:55.960> the" + }, + { + "start": 116.07, + "duration": 0.0, + "text": "that the probability that either one the" + }, + { + "start": 116.08, + "duration": 0.0, + "text": "that the probability that either one the the<00:01:56.200> fraction<00:01:56.640> of<00:01:56.720> either<00:01:56.920> of<00:01:57.040> one<00:01:57.200> happens" + }, + { + "start": 118.11, + "duration": 0.0, + "text": "the fraction of either of one happens" + }, + { + "start": 118.12, + "duration": 0.0, + "text": "the fraction of either of one happens approaches<00:01:58.520> one<00:01:58.680> half." + }, + { + "start": 119.95, + "duration": 0.0, + "text": "approaches one half." + }, + { + "start": 119.96, + "duration": 0.0, + "text": "approaches one half. Whereas<00:02:00.160> the<00:02:00.280> Bayesian<00:02:00.760> would<00:02:00.920> say<00:02:01.280> that<00:02:02.120> we" + }, + { + "start": 122.43, + "duration": 0.0, + "text": "Whereas the Bayesian would say that we" + }, + { + "start": 122.44, + "duration": 0.0, + "text": "Whereas the Bayesian would say that we say<00:02:02.680> the<00:02:02.760> probability<00:02:03.400> is<00:02:03.560> one<00:02:03.720> half<00:02:04.080> because" + }, + { + "start": 125.63, + "duration": 0.0, + "text": "say the probability is one half because" + }, + { + "start": 125.64, + "duration": 0.0, + "text": "say the probability is one half because we<00:02:05.760> don't<00:02:05.960> have<00:02:06.160> any<00:02:06.360> reason<00:02:06.680> for<00:02:06.840> prefer<00:02:07.200> any" + }, + { + "start": 127.39, + "duration": 0.0, + "text": "we don't have any reason for prefer any" + }, + { + "start": 127.4, + "duration": 0.0, + "text": "we don't have any reason for prefer any of<00:02:07.520> the<00:02:07.640> two<00:02:07.760> sides<00:02:08.320> given<00:02:08.600> our<00:02:09.000> ignorance<00:02:09.479> of" + }, + { + "start": 129.63, + "duration": 0.0, + "text": "of the two sides given our ignorance of" + }, + { + "start": 129.64, + "duration": 0.0, + "text": "of the two sides given our ignorance of the" + }, + { + "start": 130.43, + "duration": 0.0, + "text": "the" + }, + { + "start": 130.44, + "duration": 0.0, + "text": "the properties<00:02:10.800> of<00:02:10.960> the<00:02:11.800> the<00:02:11.880> coin." + }, + { + "start": 132.99, + "duration": 0.0, + "text": "properties of the the coin." + }, + { + "start": 133.0, + "duration": 0.0, + "text": "properties of the the coin. And<00:02:13.200> therefore<00:02:13.640> we<00:02:13.800> must<00:02:14.400> by<00:02:14.560> consistency" + }, + { + "start": 135.11, + "duration": 0.0, + "text": "And therefore we must by consistency" + }, + { + "start": 135.12, + "duration": 0.0, + "text": "And therefore we must by consistency assign<00:02:15.520> an<00:02:15.640> equal<00:02:15.880> probability<00:02:16.360> weight<00:02:16.680> to" + }, + { + "start": 136.79, + "duration": 0.0, + "text": "assign an equal probability weight to" + }, + { + "start": 136.8, + "duration": 0.0, + "text": "assign an equal probability weight to both<00:02:17.320> and<00:02:17.440> hence<00:02:18.360> 50%." + }, + { + "start": 141.32, + "duration": 0.0, + "text": "Now<00:02:21.760> there<00:02:22.040> are<00:02:22.120> some<00:02:22.920> important<00:02:23.320> problems" + }, + { + "start": 143.67, + "duration": 0.0, + "text": "Now there are some important problems" + }, + { + "start": 143.68, + "duration": 0.0, + "text": "Now there are some important problems with<00:02:23.800> the<00:02:23.920> frequentist<00:02:24.320> definition.<00:02:25.400> The" + }, + { + "start": 145.47, + "duration": 0.0, + "text": "with the frequentist definition. The" + }, + { + "start": 145.48, + "duration": 0.0, + "text": "with the frequentist definition. The most<00:02:25.640> important<00:02:26.000> one<00:02:26.160> is<00:02:26.320> it<00:02:26.440> cannot<00:02:26.720> assign" + }, + { + "start": 146.99, + "duration": 0.0, + "text": "most important one is it cannot assign" + }, + { + "start": 147.0, + "duration": 0.0, + "text": "most important one is it cannot assign probabilities<00:02:27.600> to<00:02:27.680> single<00:02:27.959> events.<00:02:28.320> So<00:02:28.720> in" + }, + { + "start": 148.83, + "duration": 0.0, + "text": "probabilities to single events. So in" + }, + { + "start": 148.84, + "duration": 0.0, + "text": "probabilities to single events. So in the<00:02:28.920> frequentist<00:02:29.480> framework<00:02:29.920> which<00:02:30.120> is<00:02:30.640> the" + }, + { + "start": 150.71, + "duration": 0.0, + "text": "the frequentist framework which is the" + }, + { + "start": 150.72, + "duration": 0.0, + "text": "the frequentist framework which is the orthodox<00:02:31.160> framework,<00:02:31.480> the<00:02:31.560> one<00:02:31.720> that<00:02:31.840> you" + }, + { + "start": 151.91, + "duration": 0.0, + "text": "orthodox framework, the one that you" + }, + { + "start": 151.92, + "duration": 0.0, + "text": "orthodox framework, the one that you were<00:02:32.000> taught<00:02:32.959> at<00:02:33.200> lectures<00:02:33.720> with<00:02:33.880> the<00:02:34.840> T<00:02:35.040> tests" + }, + { + "start": 155.51, + "duration": 0.0, + "text": "were taught at lectures with the T tests" + }, + { + "start": 155.52, + "duration": 0.0, + "text": "were taught at lectures with the T tests and<00:02:35.760> and<00:02:35.920> P<00:02:36.080> values<00:02:36.720> and<00:02:36.840> confidence" + }, + { + "start": 157.19, + "duration": 0.0, + "text": "and and P values and confidence" + }, + { + "start": 157.2, + "duration": 0.0, + "text": "and and P values and confidence intervals,<00:02:38.200> it<00:02:38.880> cannot<00:02:39.080> assign" + }, + { + "start": 159.27, + "duration": 0.0, + "text": "intervals, it cannot assign" + }, + { + "start": 159.28, + "duration": 0.0, + "text": "intervals, it cannot assign probabilities<00:02:39.680> to<00:02:39.760> single<00:02:40.000> events.<00:02:40.240> So<00:02:40.320> for" + }, + { + "start": 160.43, + "duration": 0.0, + "text": "probabilities to single events. So for" + }, + { + "start": 160.44, + "duration": 0.0, + "text": "probabilities to single events. So for example<00:02:40.760> the<00:02:40.800> statement<00:02:41.160> the<00:02:41.200> probability" + }, + { + "start": 161.67, + "duration": 0.0, + "text": "example the statement the probability" + }, + { + "start": 161.68, + "duration": 0.0, + "text": "example the statement the probability tomorrow<00:02:42.160> it<00:02:42.280> is<00:02:42.440> sunny" + }, + { + "start": 163.43, + "duration": 0.0, + "text": "tomorrow it is sunny" + }, + { + "start": 163.44, + "duration": 0.0, + "text": "tomorrow it is sunny doesn't<00:02:43.640> really<00:02:43.800> have<00:02:43.920> a<00:02:43.959> meaning<00:02:44.480> in<00:02:44.640> this" + }, + { + "start": 164.87, + "duration": 0.0, + "text": "doesn't really have a meaning in this" + }, + { + "start": 164.88, + "duration": 0.0, + "text": "doesn't really have a meaning in this interpretation.<00:02:45.400> The<00:02:45.480> probability<00:02:45.920> that" + }, + { + "start": 166.75, + "duration": 0.0, + "text": "interpretation. The probability that" + }, + { + "start": 166.76, + "duration": 0.0, + "text": "interpretation. The probability that a<00:02:46.800> certain<00:02:47.040> person<00:02:47.440> wins<00:02:47.680> an<00:02:47.760> election<00:02:48.160> or" + }, + { + "start": 168.23, + "duration": 0.0, + "text": "a certain person wins an election or" + }, + { + "start": 168.24, + "duration": 0.0, + "text": "a certain person wins an election or that<00:02:48.360> certain<00:02:48.800> person<00:02:49.120> wins<00:02:49.360> a<00:02:49.440> race<00:02:49.760> doesn't" + }, + { + "start": 169.99, + "duration": 0.0, + "text": "that certain person wins a race doesn't" + }, + { + "start": 170.0, + "duration": 0.0, + "text": "that certain person wins a race doesn't have<00:02:50.120> a<00:02:50.160> meaning<00:02:50.520> because<00:02:51.080> these<00:02:51.280> events<00:02:51.600> are" + }, + { + "start": 171.71, + "duration": 0.0, + "text": "have a meaning because these events are" + }, + { + "start": 171.72, + "duration": 0.0, + "text": "have a meaning because these events are unique." + }, + { + "start": 172.91, + "duration": 0.0, + "text": "unique." + }, + { + "start": 172.92, + "duration": 0.0, + "text": "unique. Also<00:02:53.760> the<00:02:54.080> the<00:02:54.600> notion" + }, + { + "start": 176.11, + "duration": 0.0, + "text": "Also the the notion" + }, + { + "start": 176.12, + "duration": 0.0, + "text": "Also the the notion the<00:02:56.200> validity<00:02:56.560> of<00:02:56.640> this<00:02:56.760> notion<00:02:57.040> of" + }, + { + "start": 177.07, + "duration": 0.0, + "text": "the validity of this notion of" + }, + { + "start": 177.08, + "duration": 0.0, + "text": "the validity of this notion of probability" + }, + { + "start": 178.31, + "duration": 0.0, + "text": "probability" + }, + { + "start": 178.32, + "duration": 0.0, + "text": "probability intuitively<00:02:58.959> relies<00:02:59.480> on<00:02:59.600> the<00:02:59.680> law<00:02:59.920> of<00:03:00.040> large" + }, + { + "start": 180.229, + "duration": 0.0, + "text": "intuitively relies on the law of large" + }, + { + "start": 180.239, + "duration": 0.0, + "text": "intuitively relies on the law of large numbers<00:03:00.959> which<00:03:01.320> in<00:03:01.440> turn<00:03:01.640> depends<00:03:02.239> of" + }, + { + "start": 182.79, + "duration": 0.0, + "text": "numbers which in turn depends of" + }, + { + "start": 182.8, + "duration": 0.0, + "text": "numbers which in turn depends of previous<00:03:03.080> definition<00:03:03.440> of<00:03:03.480> probability<00:03:03.959> which" + }, + { + "start": 184.229, + "duration": 0.0, + "text": "previous definition of probability which" + }, + { + "start": 184.239, + "duration": 0.0, + "text": "previous definition of probability which creates<00:03:04.600> some<00:03:05.040> circularity<00:03:05.600> issues." + }, + { + "start": 186.63, + "duration": 0.0, + "text": "creates some circularity issues." + }, + { + "start": 186.64, + "duration": 0.0, + "text": "creates some circularity issues. Also<00:03:07.680> this<00:03:07.880> is<00:03:08.120> probably<00:03:08.400> the<00:03:08.519> biggest" + }, + { + "start": 189.63, + "duration": 0.0, + "text": "Also this is probably the biggest" + }, + { + "start": 189.64, + "duration": 0.0, + "text": "Also this is probably the biggest the<00:03:09.680> biggest<00:03:10.000> hint<00:03:10.239> that<00:03:10.360> there's<00:03:10.560> something" + }, + { + "start": 190.87, + "duration": 0.0, + "text": "the biggest hint that there's something" + }, + { + "start": 190.88, + "duration": 0.0, + "text": "the biggest hint that there's something off." + }, + { + "start": 191.91, + "duration": 0.0, + "text": "off." + }, + { + "start": 191.92, + "duration": 0.0, + "text": "off. It<00:03:12.400> relies<00:03:12.840> on<00:03:13.040> some" + }, + { + "start": 193.79, + "duration": 0.0, + "text": "It relies on some" + }, + { + "start": 193.8, + "duration": 0.0, + "text": "It relies on some some<00:03:13.959> kind<00:03:14.080> of<00:03:14.200> notion<00:03:14.480> of<00:03:14.600> randomness<00:03:15.239> which" + }, + { + "start": 195.47, + "duration": 0.0, + "text": "some kind of notion of randomness which" + }, + { + "start": 195.48, + "duration": 0.0, + "text": "some kind of notion of randomness which is<00:03:15.760> very<00:03:16.120> hard<00:03:16.440> if<00:03:16.760> not<00:03:17.000> impossible<00:03:17.440> to" + }, + { + "start": 197.509, + "duration": 0.0, + "text": "is very hard if not impossible to" + }, + { + "start": 197.519, + "duration": 0.0, + "text": "is very hard if not impossible to define.<00:03:18.000> In<00:03:18.080> fact<00:03:18.280> many<00:03:18.480> textbooks<00:03:18.840> say<00:03:19.480> say" + }, + { + "start": 199.67, + "duration": 0.0, + "text": "define. In fact many textbooks say say" + }, + { + "start": 199.68, + "duration": 0.0, + "text": "define. In fact many textbooks say say it<00:03:19.800> out<00:03:20.000> loud<00:03:20.239> that<00:03:20.320> there<00:03:20.440> is<00:03:20.519> not<00:03:20.720> really" + }, + { + "start": 200.91, + "duration": 0.0, + "text": "it out loud that there is not really" + }, + { + "start": 200.92, + "duration": 0.0, + "text": "it out loud that there is not really good<00:03:21.080> definition<00:03:21.440> for<00:03:21.560> this.<00:03:22.080> Also<00:03:22.280> because" + }, + { + "start": 202.55, + "duration": 0.0, + "text": "good definition for this. Also because" + }, + { + "start": 202.56, + "duration": 0.0, + "text": "good definition for this. Also because simultaneously<00:03:23.280> it<00:03:23.400> requires<00:03:23.880> that<00:03:24.040> the" + }, + { + "start": 204.11, + "duration": 0.0, + "text": "simultaneously it requires that the" + }, + { + "start": 204.12, + "duration": 0.0, + "text": "simultaneously it requires that the different<00:03:24.800> events<00:03:25.160> and<00:03:25.239> different" + }, + { + "start": 205.43, + "duration": 0.0, + "text": "different events and different" + }, + { + "start": 205.44, + "duration": 0.0, + "text": "different events and different repetitions<00:03:26.280> have<00:03:26.519> to<00:03:26.600> be<00:03:26.720> also<00:03:26.920> be<00:03:27.200> equal<00:03:27.760> to" + }, + { + "start": 207.83, + "duration": 0.0, + "text": "repetitions have to be also be equal to" + }, + { + "start": 207.84, + "duration": 0.0, + "text": "repetitions have to be also be equal to the<00:03:27.959> others<00:03:28.320> in<00:03:28.480> some<00:03:28.720> way<00:03:28.880> that<00:03:29.000> is<00:03:29.320> well" + }, + { + "start": 209.59, + "duration": 0.0, + "text": "the others in some way that is well" + }, + { + "start": 209.6, + "duration": 0.0, + "text": "the others in some way that is well equally<00:03:30.120> hard<00:03:30.360> to<00:03:30.440> define." + }, + { + "start": 211.67, + "duration": 0.0, + "text": "equally hard to define." + }, + { + "start": 211.68, + "duration": 0.0, + "text": "equally hard to define. Um" + }, + { + "start": 213.509, + "duration": 0.0, + "text": "Um" + }, + { + "start": 213.519, + "duration": 0.0, + "text": "Um Also<00:03:33.760> it<00:03:33.880> is<00:03:34.640> you<00:03:34.800> could<00:03:34.920> say" + }, + { + "start": 215.15, + "duration": 0.0, + "text": "Also it is you could say" + }, + { + "start": 215.16, + "duration": 0.0, + "text": "Also it is you could say epistemologically<00:03:35.880> clunky<00:03:36.200> because<00:03:36.640> it<00:03:36.840> ever" + }, + { + "start": 217.229, + "duration": 0.0, + "text": "epistemologically clunky because it ever" + }, + { + "start": 217.239, + "duration": 0.0, + "text": "epistemologically clunky because it ever only<00:03:37.440> deals<00:03:37.800> with<00:03:37.959> the<00:03:38.080> probability<00:03:38.680> of<00:03:38.840> the" + }, + { + "start": 218.949, + "duration": 0.0, + "text": "only deals with the probability of the" + }, + { + "start": 218.959, + "duration": 0.0, + "text": "only deals with the probability of the data<00:03:39.640> given<00:03:39.880> the<00:03:39.959> hypothesis.<00:03:41.160> This<00:03:41.400> is<00:03:41.680> what" + }, + { + "start": 221.87, + "duration": 0.0, + "text": "data given the hypothesis. This is what" + }, + { + "start": 221.88, + "duration": 0.0, + "text": "data given the hypothesis. This is what you<00:03:42.000> do<00:03:42.360> when<00:03:42.560> you<00:03:42.680> do<00:03:42.880> what's<00:03:43.080> called<00:03:43.440> maximum" + }, + { + "start": 223.79, + "duration": 0.0, + "text": "you do when you do what's called maximum" + }, + { + "start": 223.8, + "duration": 0.0, + "text": "you do when you do what's called maximum likelihood.<00:03:44.840> You<00:03:45.000> just<00:03:45.200> see<00:03:46.000> what<00:03:46.880> values<00:03:47.239> of" + }, + { + "start": 227.31, + "duration": 0.0, + "text": "likelihood. You just see what values of" + }, + { + "start": 227.32, + "duration": 0.0, + "text": "likelihood. You just see what values of the<00:03:47.400> parameters<00:03:47.880> of<00:03:47.959> your<00:03:48.080> model<00:03:48.320> just" + }, + { + "start": 228.59, + "duration": 0.0, + "text": "the parameters of your model just" + }, + { + "start": 228.6, + "duration": 0.0, + "text": "the parameters of your model just maximizes<00:03:49.760> the<00:03:49.840> probability<00:03:50.320> of<00:03:50.400> the<00:03:50.519> data" + }, + { + "start": 230.83, + "duration": 0.0, + "text": "maximizes the probability of the data" + }, + { + "start": 230.84, + "duration": 0.0, + "text": "maximizes the probability of the data given<00:03:51.080> your<00:03:51.280> model." + }, + { + "start": 232.59, + "duration": 0.0, + "text": "given your model." + }, + { + "start": 232.6, + "duration": 0.0, + "text": "given your model. However,<00:03:52.960> it<00:03:53.080> doesn't<00:03:53.840> uh<00:03:53.920> give<00:03:54.200> you<00:03:54.640> a<00:03:54.720> way<00:03:54.920> to" + }, + { + "start": 235.43, + "duration": 0.0, + "text": "However, it doesn't uh give you a way to" + }, + { + "start": 235.44, + "duration": 0.0, + "text": "However, it doesn't uh give you a way to assign<00:03:55.720> probabilities." + }, + { + "start": 237.99, + "duration": 0.0, + "text": "assign probabilities." + }, + { + "start": 238.0, + "duration": 0.0, + "text": "assign probabilities. Sorry,<00:03:58.400> it<00:03:58.480> doesn't<00:03:58.680> have<00:03:58.760> a<00:03:58.800> way<00:03:58.880> to<00:03:59.000> assign" + }, + { + "start": 239.47, + "duration": 0.0, + "text": "Sorry, it doesn't have a way to assign" + }, + { + "start": 239.48, + "duration": 0.0, + "text": "Sorry, it doesn't have a way to assign probabilities<00:04:00.120> to<00:04:00.360> a<00:04:00.400> hypothesis<00:04:01.440> because" + }, + { + "start": 242.55, + "duration": 0.0, + "text": "probabilities to a hypothesis because" + }, + { + "start": 242.56, + "duration": 0.0, + "text": "probabilities to a hypothesis because um<00:04:03.120> hypothesis,<00:04:03.640> once<00:04:03.800> again,<00:04:04.280> I<00:04:04.960> are<00:04:05.120> only<00:04:06.080> or" + }, + { + "start": 246.99, + "duration": 0.0, + "text": "um hypothesis, once again, I are only or" + }, + { + "start": 247.0, + "duration": 0.0, + "text": "um hypothesis, once again, I are only or the<00:04:07.040> truth<00:04:07.600> is<00:04:07.920> sort<00:04:08.080> of<00:04:08.200> a<00:04:08.280> fixed<00:04:08.640> thing.<00:04:08.880> It's" + }, + { + "start": 248.99, + "duration": 0.0, + "text": "the truth is sort of a fixed thing. It's" + }, + { + "start": 249.0, + "duration": 0.0, + "text": "the truth is sort of a fixed thing. It's a<00:04:09.040> constant<00:04:09.600> and<00:04:09.680> therefore<00:04:10.400> this" + }, + { + "start": 250.55, + "duration": 0.0, + "text": "a constant and therefore this" + }, + { + "start": 250.56, + "duration": 0.0, + "text": "a constant and therefore this methodology<00:04:10.959> does<00:04:11.080> not<00:04:11.240> allow<00:04:12.000> you<00:04:12.080> to<00:04:12.200> give" + }, + { + "start": 252.35, + "duration": 0.0, + "text": "methodology does not allow you to give" + }, + { + "start": 252.36, + "duration": 0.0, + "text": "methodology does not allow you to give it<00:04:12.440> a<00:04:12.480> probability." + }, + { + "start": 253.949, + "duration": 0.0, + "text": "it a probability." + }, + { + "start": 253.959, + "duration": 0.0, + "text": "it a probability. Also,<00:04:14.360> it<00:04:14.560> has<00:04:14.800> the<00:04:15.320> sort<00:04:15.520> of<00:04:16.239> methodological" + }, + { + "start": 257.47, + "duration": 0.0, + "text": "Also, it has the sort of methodological" + }, + { + "start": 257.48, + "duration": 0.0, + "text": "Also, it has the sort of methodological uh<00:04:17.840> handicap<00:04:18.320> that<00:04:18.560> it<00:04:19.000> ignores<00:04:19.600> any<00:04:20.359> prior" + }, + { + "start": 260.59, + "duration": 0.0, + "text": "uh handicap that it ignores any prior" + }, + { + "start": 260.6, + "duration": 0.0, + "text": "uh handicap that it ignores any prior information<00:04:21.079> or<00:04:21.160> prior<00:04:21.400> knowledge<00:04:21.760> from" + }, + { + "start": 261.949, + "duration": 0.0, + "text": "information or prior knowledge from" + }, + { + "start": 261.959, + "duration": 0.0, + "text": "information or prior knowledge from either<00:04:22.160> previous<00:04:22.440> estimation<00:04:23.080> or" + }, + { + "start": 264.11, + "duration": 0.0, + "text": "either previous estimation or" + }, + { + "start": 264.12, + "duration": 0.0, + "text": "either previous estimation or uh<00:04:24.640> prior<00:04:24.840> results<00:04:25.240> in<00:04:25.360> a<00:04:25.400> completely" + }, + { + "start": 266.35, + "duration": 0.0, + "text": "uh prior results in a completely" + }, + { + "start": 266.36, + "duration": 0.0, + "text": "uh prior results in a completely different<00:04:26.640> experiment.<00:04:27.800> And<00:04:28.000> it<00:04:28.120> often" + }, + { + "start": 268.47, + "duration": 0.0, + "text": "different experiment. And it often" + }, + { + "start": 268.48, + "duration": 0.0, + "text": "different experiment. And it often provides<00:04:28.800> only<00:04:28.960> point<00:04:29.280> estimates,<00:04:29.800> which" + }, + { + "start": 270.35, + "duration": 0.0, + "text": "provides only point estimates, which" + }, + { + "start": 270.36, + "duration": 0.0, + "text": "provides only point estimates, which makes<00:04:30.640> it<00:04:31.160> very<00:04:31.360> hard<00:04:31.520> if<00:04:31.600> not<00:04:31.720> impossible<00:04:32.160> to" + }, + { + "start": 272.27, + "duration": 0.0, + "text": "makes it very hard if not impossible to" + }, + { + "start": 272.28, + "duration": 0.0, + "text": "makes it very hard if not impossible to assess<00:04:32.680> the<00:04:32.800> uncertainty<00:04:33.520> of<00:04:34.120> the<00:04:34.240> result." + }, + { + "start": 276.27, + "duration": 0.0, + "text": "assess the uncertainty of the result." + }, + { + "start": 276.28, + "duration": 0.0, + "text": "assess the uncertainty of the result. Uh<00:04:36.440> in<00:04:36.560> an<00:04:36.680> in<00:04:36.800> an<00:04:36.880> attempt<00:04:37.120> to<00:04:37.200> circumvent" + }, + { + "start": 277.71, + "duration": 0.0, + "text": "Uh in an in an attempt to circumvent" + }, + { + "start": 277.72, + "duration": 0.0, + "text": "Uh in an in an attempt to circumvent these<00:04:37.800> issues,<00:04:38.320> the<00:04:38.400> methodology<00:04:38.919> has<00:04:39.560> uh" + }, + { + "start": 280.23, + "duration": 0.0, + "text": "these issues, the methodology has uh" + }, + { + "start": 280.24, + "duration": 0.0, + "text": "these issues, the methodology has uh made<00:04:40.520> it<00:04:40.600> so<00:04:40.680> that<00:04:40.800> scientists<00:04:41.320> now<00:04:41.600> commonly" + }, + { + "start": 282.07, + "duration": 0.0, + "text": "made it so that scientists now commonly" + }, + { + "start": 282.08, + "duration": 0.0, + "text": "made it so that scientists now commonly reason<00:04:42.480> about<00:04:43.400> uh" + }, + { + "start": 283.99, + "duration": 0.0, + "text": "reason about uh" + }, + { + "start": 284.0, + "duration": 0.0, + "text": "reason about uh what<00:04:44.200> the<00:04:44.280> world<00:04:44.600> would<00:04:44.720> be<00:04:44.960> in<00:04:45.160> different" + }, + { + "start": 285.67, + "duration": 0.0, + "text": "what the world would be in different" + }, + { + "start": 285.68, + "duration": 0.0, + "text": "what the world would be in different imaginary<00:04:46.200> universes.<00:04:46.800> So,<00:04:46.919> for<00:04:47.040> example,<00:04:47.440> if" + }, + { + "start": 287.55, + "duration": 0.0, + "text": "imaginary universes. So, for example, if" + }, + { + "start": 287.56, + "duration": 0.0, + "text": "imaginary universes. So, for example, if you<00:04:47.640> wanted<00:04:47.880> to<00:04:47.960> estimate<00:04:48.520> the<00:04:48.640> mass<00:04:48.960> of<00:04:49.080> a" + }, + { + "start": 289.11, + "duration": 0.0, + "text": "you wanted to estimate the mass of a" + }, + { + "start": 289.12, + "duration": 0.0, + "text": "you wanted to estimate the mass of a planet,<00:04:50.160> uh<00:04:50.680> this<00:04:50.840> methodology<00:04:51.320> in<00:04:51.440> per<00:04:51.640> se" + }, + { + "start": 291.87, + "duration": 0.0, + "text": "planet, uh this methodology in per se" + }, + { + "start": 291.88, + "duration": 0.0, + "text": "planet, uh this methodology in per se would<00:04:52.040> not<00:04:52.200> allow<00:04:52.520> you<00:04:52.680> to<00:04:52.880> assign<00:04:53.160> a" + }, + { + "start": 293.19, + "duration": 0.0, + "text": "would not allow you to assign a" + }, + { + "start": 293.2, + "duration": 0.0, + "text": "would not allow you to assign a probability<00:04:53.840> to<00:04:53.960> this<00:04:54.160> value<00:04:54.480> because<00:04:54.720> it's<00:04:54.840> a" + }, + { + "start": 294.87, + "duration": 0.0, + "text": "probability to this value because it's a" + }, + { + "start": 294.88, + "duration": 0.0, + "text": "probability to this value because it's a constant.<00:04:55.440> There<00:04:55.560> is<00:04:55.680> only<00:04:55.840> one<00:04:56.000> true<00:04:56.200> value." + }, + { + "start": 297.07, + "duration": 0.0, + "text": "constant. There is only one true value." + }, + { + "start": 297.08, + "duration": 0.0, + "text": "constant. There is only one true value. And<00:04:57.200> so,<00:04:57.440> it<00:04:57.560> forces<00:04:57.840> you<00:04:57.919> to<00:04:58.040> imagine,<00:04:58.520> oh," + }, + { + "start": 298.71, + "duration": 0.0, + "text": "And so, it forces you to imagine, oh," + }, + { + "start": 298.72, + "duration": 0.0, + "text": "And so, it forces you to imagine, oh, what<00:04:58.960> if<00:04:59.160> there<00:04:59.280> were<00:04:59.480> different<00:04:59.840> worlds<00:05:00.400> in" + }, + { + "start": 300.51, + "duration": 0.0, + "text": "what if there were different worlds in" + }, + { + "start": 300.52, + "duration": 0.0, + "text": "what if there were different worlds in which<00:05:01.160> different<00:05:01.440> version<00:05:01.760> of<00:05:01.840> this<00:05:02.000> planet" + }, + { + "start": 302.67, + "duration": 0.0, + "text": "which different version of this planet" + }, + { + "start": 302.68, + "duration": 0.0, + "text": "which different version of this planet uh<00:05:02.760> existed?<00:05:03.760> And<00:05:03.960> so,<00:05:04.200> you<00:05:04.320> reason<00:05:04.600> about" + }, + { + "start": 304.83, + "duration": 0.0, + "text": "uh existed? And so, you reason about" + }, + { + "start": 304.84, + "duration": 0.0, + "text": "uh existed? And so, you reason about this.<00:05:05.640> And<00:05:06.480> uh<00:05:06.640> I'm<00:05:06.840> I'm<00:05:06.919> sure<00:05:07.120> you<00:05:07.240> are<00:05:07.400> all" + }, + { + "start": 307.59, + "duration": 0.0, + "text": "this. And uh I'm I'm sure you are all" + }, + { + "start": 307.6, + "duration": 0.0, + "text": "this. And uh I'm I'm sure you are all familiar<00:05:08.120> with<00:05:08.680> uh<00:05:08.720> what<00:05:08.840> a<00:05:08.919> p-value<00:05:09.520> is.<00:05:09.960> Uh" + }, + { + "start": 310.75, + "duration": 0.0, + "text": "familiar with uh what a p-value is. Uh" + }, + { + "start": 310.76, + "duration": 0.0, + "text": "familiar with uh what a p-value is. Uh And<00:05:10.880> there's<00:05:11.120> a<00:05:11.200> this<00:05:11.360> famous<00:05:11.680> critique<00:05:12.040> by" + }, + { + "start": 312.19, + "duration": 0.0, + "text": "And there's a this famous critique by" + }, + { + "start": 312.2, + "duration": 0.0, + "text": "And there's a this famous critique by Sir<00:05:12.400> Havel<00:05:12.919> Sir<00:05:13.160> Harold<00:05:13.480> Jeffreys<00:05:14.560> which<00:05:14.720> said" + }, + { + "start": 314.909, + "duration": 0.0, + "text": "Sir Havel Sir Harold Jeffreys which said" + }, + { + "start": 314.919, + "duration": 0.0, + "text": "Sir Havel Sir Harold Jeffreys which said that<00:05:15.200> what<00:05:15.400> p<00:05:15.640> implies<00:05:16.160> is<00:05:16.280> that<00:05:16.480> the" + }, + { + "start": 316.55, + "duration": 0.0, + "text": "that what p implies is that the" + }, + { + "start": 316.56, + "duration": 0.0, + "text": "that what p implies is that the hypothesis<00:05:17.200> that<00:05:17.440> may<00:05:17.600> be<00:05:17.720> true<00:05:18.520> may<00:05:18.600> be" + }, + { + "start": 318.67, + "duration": 0.0, + "text": "hypothesis that may be true may be" + }, + { + "start": 318.68, + "duration": 0.0, + "text": "hypothesis that may be true may be rejected<00:05:19.160> because<00:05:19.400> it<00:05:19.480> does<00:05:19.640> not<00:05:20.000> predict<00:05:20.320> it" + }, + { + "start": 320.39, + "duration": 0.0, + "text": "rejected because it does not predict it" + }, + { + "start": 320.4, + "duration": 0.0, + "text": "rejected because it does not predict it observable<00:05:20.880> results<00:05:21.240> that<00:05:21.400> have<00:05:21.560> not" + }, + { + "start": 321.75, + "duration": 0.0, + "text": "observable results that have not" + }, + { + "start": 321.76, + "duration": 0.0, + "text": "observable results that have not occurred." + }, + { + "start": 322.75, + "duration": 0.0, + "text": "occurred." + }, + { + "start": 322.76, + "duration": 0.0, + "text": "occurred. So,<00:05:23.400> uh<00:05:23.840> just<00:05:24.000> as<00:05:24.080> a<00:05:24.120> reminder,<00:05:24.480> the<00:05:24.600> p-value" + }, + { + "start": 325.27, + "duration": 0.0, + "text": "So, uh just as a reminder, the p-value" + }, + { + "start": 325.28, + "duration": 0.0, + "text": "So, uh just as a reminder, the p-value is<00:05:25.560> the<00:05:25.680> probability<00:05:26.360> that<00:05:26.560> you<00:05:26.640> get<00:05:26.800> a" + }, + { + "start": 326.83, + "duration": 0.0, + "text": "is the probability that you get a" + }, + { + "start": 326.84, + "duration": 0.0, + "text": "is the probability that you get a statistic<00:05:27.520> that<00:05:27.800> is<00:05:28.560> equal<00:05:28.760> to<00:05:28.840> the<00:05:28.960> one<00:05:29.080> that" + }, + { + "start": 329.19, + "duration": 0.0, + "text": "statistic that is equal to the one that" + }, + { + "start": 329.2, + "duration": 0.0, + "text": "statistic that is equal to the one that you<00:05:29.280> got<00:05:29.800> from<00:05:29.960> the<00:05:30.040> data<00:05:30.400> or<00:05:30.680> more<00:05:30.960> extreme," + }, + { + "start": 331.51, + "duration": 0.0, + "text": "you got from the data or more extreme," + }, + { + "start": 331.52, + "duration": 0.0, + "text": "you got from the data or more extreme, so<00:05:31.680> further<00:05:31.960> up<00:05:32.080> the<00:05:32.200> tails.<00:05:33.080> And<00:05:33.280> while<00:05:33.600> this" + }, + { + "start": 333.79, + "duration": 0.0, + "text": "so further up the tails. And while this" + }, + { + "start": 333.8, + "duration": 0.0, + "text": "so further up the tails. And while this intuitively<00:05:34.400> seems<00:05:34.640> like<00:05:34.760> a<00:05:34.800> good<00:05:35.040> reason<00:05:35.400> to" + }, + { + "start": 335.59, + "duration": 0.0, + "text": "intuitively seems like a good reason to" + }, + { + "start": 335.6, + "duration": 0.0, + "text": "intuitively seems like a good reason to reject<00:05:35.880> the<00:05:35.960> null<00:05:36.160> hypothesis," + }, + { + "start": 337.43, + "duration": 0.0, + "text": "reject the null hypothesis," + }, + { + "start": 337.44, + "duration": 0.0, + "text": "reject the null hypothesis, uh<00:05:37.720> it<00:05:37.840> also<00:05:38.080> means<00:05:38.320> that<00:05:38.440> the<00:05:38.520> null" + }, + { + "start": 338.71, + "duration": 0.0, + "text": "uh it also means that the null" + }, + { + "start": 338.72, + "duration": 0.0, + "text": "uh it also means that the null hypothesis<00:05:39.320> has<00:05:39.760> failed<00:05:40.320> to<00:05:40.440> produce<00:05:41.000> data" + }, + { + "start": 341.51, + "duration": 0.0, + "text": "hypothesis has failed to produce data" + }, + { + "start": 341.52, + "duration": 0.0, + "text": "hypothesis has failed to produce data that<00:05:41.760> is<00:05:41.960> further<00:05:42.240> up<00:05:42.360> the<00:05:42.440> tail.<00:05:42.760> So,<00:05:43.640> uh<00:05:43.840> it's" + }, + { + "start": 343.99, + "duration": 0.0, + "text": "that is further up the tail. So, uh it's" + }, + { + "start": 344.0, + "duration": 0.0, + "text": "that is further up the tail. So, uh it's actually<00:05:44.480> not<00:05:44.880> entirely<00:05:45.240> clear<00:05:45.600> why<00:05:46.160> this" + }, + { + "start": 346.39, + "duration": 0.0, + "text": "actually not entirely clear why this" + }, + { + "start": 346.4, + "duration": 0.0, + "text": "actually not entirely clear why this should<00:05:46.600> be<00:05:46.720> a<00:05:46.760> good<00:05:47.000> criterion<00:05:47.480> to<00:05:47.600> reject<00:05:47.880> it." + }, + { + "start": 348.79, + "duration": 0.0, + "text": "should be a good criterion to reject it." + }, + { + "start": 348.8, + "duration": 0.0, + "text": "should be a good criterion to reject it. Uh<00:05:49.280> and<00:05:49.400> also<00:05:49.640> it's<00:05:49.800> equally" + }, + { + "start": 351.07, + "duration": 0.0, + "text": "Uh and also it's equally" + }, + { + "start": 351.08, + "duration": 0.0, + "text": "Uh and also it's equally it's<00:05:51.120> equally<00:05:51.360> problematic<00:05:52.000> to<00:05:52.160> say<00:05:52.560> why<00:05:53.400> we" + }, + { + "start": 353.55, + "duration": 0.0, + "text": "it's equally problematic to say why we" + }, + { + "start": 353.56, + "duration": 0.0, + "text": "it's equally problematic to say why we can<00:05:53.800> interpret<00:05:54.160> the<00:05:54.240> confidence<00:05:54.720> interval<00:05:55.200> at" + }, + { + "start": 355.31, + "duration": 0.0, + "text": "can interpret the confidence interval at" + }, + { + "start": 355.32, + "duration": 0.0, + "text": "can interpret the confidence interval at all<00:05:55.480> because<00:05:55.760> a<00:05:55.800> confidence<00:05:56.280> interval" + }, + { + "start": 356.67, + "duration": 0.0, + "text": "all because a confidence interval" + }, + { + "start": 356.68, + "duration": 0.0, + "text": "all because a confidence interval doesn't<00:05:56.960> tell<00:05:57.200> us" + }, + { + "start": 358.03, + "duration": 0.0, + "text": "doesn't tell us" + }, + { + "start": 358.04, + "duration": 0.0, + "text": "doesn't tell us the<00:05:58.120> probability<00:05:58.920> of<00:05:59.560> the<00:06:00.120> null<00:06:00.720> being<00:06:00.960> true" + }, + { + "start": 361.19, + "duration": 0.0, + "text": "the probability of the null being true" + }, + { + "start": 361.2, + "duration": 0.0, + "text": "the probability of the null being true or<00:06:01.320> the<00:06:01.400> alternative<00:06:01.760> hypothesis<00:06:02.200> being" + }, + { + "start": 362.39, + "duration": 0.0, + "text": "or the alternative hypothesis being" + }, + { + "start": 362.4, + "duration": 0.0, + "text": "or the alternative hypothesis being true.<00:06:02.720> It<00:06:02.840> only<00:06:03.040> tells<00:06:03.320> us<00:06:03.440> that<00:06:03.680> if<00:06:04.560> we<00:06:04.760> repeat" + }, + { + "start": 365.11, + "duration": 0.0, + "text": "true. It only tells us that if we repeat" + }, + { + "start": 365.12, + "duration": 0.0, + "text": "true. It only tells us that if we repeat this<00:06:05.280> experiment<00:06:05.720> infinitely<00:06:06.080> many<00:06:06.240> times," + }, + { + "start": 366.67, + "duration": 0.0, + "text": "this experiment infinitely many times," + }, + { + "start": 366.68, + "duration": 0.0, + "text": "this experiment infinitely many times, then<00:06:07.040> say<00:06:07.240> 95%<00:06:07.880> of<00:06:07.960> the<00:06:08.040> time<00:06:08.360> or<00:06:08.480> whatever" + }, + { + "start": 369.19, + "duration": 0.0, + "text": "then say 95% of the time or whatever" + }, + { + "start": 369.2, + "duration": 0.0, + "text": "then say 95% of the time or whatever interval<00:06:09.560> we<00:06:09.640> choose,<00:06:10.440> the<00:06:10.520> statistics<00:06:10.960> is" + }, + { + "start": 371.07, + "duration": 0.0, + "text": "interval we choose, the statistics is" + }, + { + "start": 371.08, + "duration": 0.0, + "text": "interval we choose, the statistics is going<00:06:11.240> to<00:06:11.320> be<00:06:11.400> the<00:06:11.520> statistic<00:06:11.880> that<00:06:12.000> we<00:06:12.080> look" + }, + { + "start": 372.23, + "duration": 0.0, + "text": "going to be the statistic that we look" + }, + { + "start": 372.24, + "duration": 0.0, + "text": "going to be the statistic that we look is<00:06:12.320> going<00:06:12.440> to<00:06:12.520> be<00:06:12.600> in<00:06:12.760> there." + }, + { + "start": 375.4, + "duration": 0.0, + "text": "So,<00:06:15.520> again,<00:06:15.720> we<00:06:15.800> have<00:06:15.880> this<00:06:16.040> inversion" + }, + { + "start": 376.43, + "duration": 0.0, + "text": "So, again, we have this inversion" + }, + { + "start": 376.44, + "duration": 0.0, + "text": "So, again, we have this inversion problem<00:06:16.760> from<00:06:17.120> we<00:06:17.240> have<00:06:17.400> probability<00:06:17.840> of<00:06:17.919> the" + }, + { + "start": 378.03, + "duration": 0.0, + "text": "problem from we have probability of the" + }, + { + "start": 378.04, + "duration": 0.0, + "text": "problem from we have probability of the data<00:06:18.440> given<00:06:18.680> a<00:06:18.720> hypothesis,<00:06:19.280> but<00:06:19.400> not<00:06:19.520> the" + }, + { + "start": 379.59, + "duration": 0.0, + "text": "data given a hypothesis, but not the" + }, + { + "start": 379.6, + "duration": 0.0, + "text": "data given a hypothesis, but not the probability<00:06:19.960> of<00:06:20.040> the<00:06:20.120> hypothesis<00:06:20.840> given<00:06:21.120> the" + }, + { + "start": 381.19, + "duration": 0.0, + "text": "probability of the hypothesis given the" + }, + { + "start": 381.2, + "duration": 0.0, + "text": "probability of the hypothesis given the data." + }, + { + "start": 383.03, + "duration": 0.0, + "text": "data." + }, + { + "start": 383.04, + "duration": 0.0, + "text": "data. Now,<00:06:23.240> we're<00:06:23.360> going<00:06:23.480> to<00:06:23.520> talk<00:06:23.720> about<00:06:24.160> the" + }, + { + "start": 384.23, + "duration": 0.0, + "text": "Now, we're going to talk about the" + }, + { + "start": 384.24, + "duration": 0.0, + "text": "Now, we're going to talk about the alternative,<00:06:25.280> um" + }, + { + "start": 385.99, + "duration": 0.0, + "text": "alternative, um" + }, + { + "start": 386.0, + "duration": 0.0, + "text": "alternative, um which<00:06:26.560> has<00:06:26.760> been<00:06:26.880> famously<00:06:27.280> advocated<00:06:28.200> in<00:06:28.480> the" + }, + { + "start": 388.55, + "duration": 0.0, + "text": "which has been famously advocated in the" + }, + { + "start": 388.56, + "duration": 0.0, + "text": "which has been famously advocated in the last<00:06:28.760> century<00:06:29.080> by<00:06:29.919> Edwin<00:06:30.160> Thompson<00:06:30.440> James," + }, + { + "start": 390.909, + "duration": 0.0, + "text": "last century by Edwin Thompson James," + }, + { + "start": 390.919, + "duration": 0.0, + "text": "last century by Edwin Thompson James, legendary<00:06:31.400> physicist<00:06:31.960> and<00:06:32.160> statistician." + }, + { + "start": 393.27, + "duration": 0.0, + "text": "legendary physicist and statistician." + }, + { + "start": 393.28, + "duration": 0.0, + "text": "legendary physicist and statistician. And<00:06:33.960> to<00:06:34.040> do<00:06:34.160> that,<00:06:34.480> we're<00:06:34.600> going<00:06:34.720> to<00:06:34.800> do<00:06:35.000> first" + }, + { + "start": 395.23, + "duration": 0.0, + "text": "And to do that, we're going to do first" + }, + { + "start": 395.24, + "duration": 0.0, + "text": "And to do that, we're going to do first of<00:06:35.320> all<00:06:35.440> a<00:06:35.480> thought<00:06:35.560> experiment,<00:06:36.040> and<00:06:36.240> I" + }, + { + "start": 396.67, + "duration": 0.0, + "text": "of all a thought experiment, and I" + }, + { + "start": 396.68, + "duration": 0.0, + "text": "of all a thought experiment, and I really<00:06:36.919> want<00:06:37.080> you<00:06:37.160> to<00:06:37.840> think<00:06:38.040> along<00:06:38.240> with" + }, + { + "start": 398.35, + "duration": 0.0, + "text": "really want you to think along with" + }, + { + "start": 398.36, + "duration": 0.0, + "text": "really want you to think along with this.<00:06:38.919> Suppose<00:06:39.280> that<00:06:39.440> you<00:06:39.600> are<00:06:39.760> a<00:06:39.800> policeman." + }, + { + "start": 400.79, + "duration": 0.0, + "text": "this. Suppose that you are a policeman." + }, + { + "start": 400.8, + "duration": 0.0, + "text": "this. Suppose that you are a policeman. Uh<00:06:40.919> you<00:06:41.240> patrol<00:06:41.600> the<00:06:41.680> streets<00:06:41.960> at<00:06:42.080> night,<00:06:42.560> and" + }, + { + "start": 402.87, + "duration": 0.0, + "text": "Uh you patrol the streets at night, and" + }, + { + "start": 402.88, + "duration": 0.0, + "text": "Uh you patrol the streets at night, and suddenly<00:06:43.240> you<00:06:43.360> hear<00:06:43.520> an<00:06:43.640> alarm.<00:06:44.520> Uh<00:06:44.600> there's<00:06:44.760> a" + }, + { + "start": 404.79, + "duration": 0.0, + "text": "suddenly you hear an alarm. Uh there's a" + }, + { + "start": 404.8, + "duration": 0.0, + "text": "suddenly you hear an alarm. Uh there's a jewelry<00:06:45.160> store,<00:06:45.600> and<00:06:45.840> a<00:06:46.080> a<00:06:46.120> man<00:06:46.440> with<00:06:46.560> a<00:06:46.600> mask" + }, + { + "start": 406.99, + "duration": 0.0, + "text": "jewelry store, and a a man with a mask" + }, + { + "start": 407.0, + "duration": 0.0, + "text": "jewelry store, and a a man with a mask comes<00:06:47.320> out<00:06:47.560> with<00:06:47.720> a<00:06:48.520> with<00:06:48.640> a<00:06:48.720> bag<00:06:48.960> on<00:06:49.040> his" + }, + { + "start": 409.15, + "duration": 0.0, + "text": "comes out with a with a bag on his" + }, + { + "start": 409.16, + "duration": 0.0, + "text": "comes out with a with a bag on his shoulder.<00:06:49.880> You<00:06:50.040> arrest<00:06:50.400> him,<00:06:50.520> and<00:06:50.600> it<00:06:50.640> turns" + }, + { + "start": 410.87, + "duration": 0.0, + "text": "shoulder. You arrest him, and it turns" + }, + { + "start": 410.88, + "duration": 0.0, + "text": "shoulder. You arrest him, and it turns out<00:06:51.000> the<00:06:51.080> bag<00:06:51.480> is<00:06:51.680> full<00:06:51.880> of<00:06:52.120> jewels.<00:06:53.040> Now,<00:06:53.200> I" + }, + { + "start": 413.27, + "duration": 0.0, + "text": "out the bag is full of jewels. Now, I" + }, + { + "start": 413.28, + "duration": 0.0, + "text": "out the bag is full of jewels. Now, I want<00:06:53.440> you<00:06:53.520> to<00:06:53.640> tell<00:06:53.840> me,<00:06:54.000> do<00:06:54.160> you<00:06:54.240> think<00:06:54.520> this" + }, + { + "start": 414.67, + "duration": 0.0, + "text": "want you to tell me, do you think this" + }, + { + "start": 414.68, + "duration": 0.0, + "text": "want you to tell me, do you think this is<00:06:54.800> a<00:06:54.840> crime?<00:06:55.320> Did<00:06:55.480> you<00:06:55.600> witness<00:06:55.840> a<00:06:55.880> crime?" + }, + { + "start": 417.55, + "duration": 0.0, + "text": "is a crime? Did you witness a crime?" + }, + { + "start": 417.56, + "duration": 0.0, + "text": "is a crime? Did you witness a crime? What<00:06:57.680> would<00:06:57.800> you<00:06:57.880> say?<00:06:58.320> Your<00:06:58.440> gut<00:06:58.680> instinct." + }, + { + "start": 426.6, + "duration": 0.0, + "text": ">> Of<00:07:06.800> course,<00:07:07.000> it's<00:07:07.120> a<00:07:07.160> crime." + }, + { + "start": 430.12, + "duration": 0.0, + "text": ">> Is<00:07:10.200> it<00:07:10.280> a<00:07:10.320> crime<00:07:10.600> or<00:07:10.680> is<00:07:10.800> this<00:07:11.000> man<00:07:11.240> honest?" + }, + { + "start": 452.12, + "duration": 0.0, + "text": "Is<00:07:32.520> anybody<00:07:32.800> talking<00:07:33.160> or<00:07:33.280> am<00:07:33.400> I<00:07:33.440> not<00:07:33.600> hearing" + }, + { + "start": 453.87, + "duration": 0.0, + "text": "Is anybody talking or am I not hearing" + }, + { + "start": 453.88, + "duration": 0.0, + "text": "Is anybody talking or am I not hearing you?" + }, + { + "start": 456.15, + "duration": 0.0, + "text": "you?" + }, + { + "start": 456.16, + "duration": 0.0, + "text": "you? Okay." + }, + { + "start": 478.6, + "duration": 0.0, + "text": ">> That's<00:07:58.760> all<00:07:58.800> right." + }, + { + "start": 479.75, + "duration": 0.0, + "text": ">> That's all right." + }, + { + "start": 479.76, + "duration": 0.0, + "text": ">> That's all right. >> Don't<00:07:59.920> worry." + }, + { + "start": 480.71, + "duration": 0.0, + "text": ">> Don't worry." + }, + { + "start": 480.72, + "duration": 0.0, + "text": ">> Don't worry. >> Okay,<00:08:01.040> I<00:08:01.080> think<00:08:01.280> we<00:08:01.360> can<00:08:01.480> all<00:08:01.600> agree<00:08:01.880> this<00:08:02.200> this" + }, + { + "start": 482.39, + "duration": 0.0, + "text": ">> Okay, I think we can all agree this this" + }, + { + "start": 482.4, + "duration": 0.0, + "text": ">> Okay, I think we can all agree this this looks<00:08:02.600> like<00:08:02.760> a<00:08:02.800> crime,<00:08:03.160> right?<00:08:03.760> And<00:08:03.840> now<00:08:04.240> my" + }, + { + "start": 484.43, + "duration": 0.0, + "text": "looks like a crime, right? And now my" + }, + { + "start": 484.44, + "duration": 0.0, + "text": "looks like a crime, right? And now my second<00:08:04.760> question,<00:08:05.200> and<00:08:05.280> the<00:08:05.360> more<00:08:05.520> important" + }, + { + "start": 485.87, + "duration": 0.0, + "text": "second question, and the more important" + }, + { + "start": 485.88, + "duration": 0.0, + "text": "second question, and the more important question<00:08:06.200> actually,<00:08:06.480> is" + }, + { + "start": 487.27, + "duration": 0.0, + "text": "question actually, is" + }, + { + "start": 487.28, + "duration": 0.0, + "text": "question actually, is did<00:08:07.640> you<00:08:07.960> come<00:08:08.200> to<00:08:08.280> this<00:08:08.400> conclusion<00:08:08.840> through" + }, + { + "start": 489.31, + "duration": 0.0, + "text": "did you come to this conclusion through" + }, + { + "start": 489.32, + "duration": 0.0, + "text": "did you come to this conclusion through deduction<00:08:10.160> or<00:08:10.320> through<00:08:10.600> induction?" + }, + { + "start": 533.12, + "duration": 0.0, + "text": "No,<00:08:53.560> yes.<00:08:54.480> Uh<00:08:55.040> it<00:08:55.160> is<00:08:55.360> induction,<00:08:56.000> and<00:08:56.160> you<00:08:56.280> can" + }, + { + "start": 537.27, + "duration": 0.0, + "text": "No, yes. Uh it is induction, and you can" + }, + { + "start": 537.28, + "duration": 0.0, + "text": "No, yes. Uh it is induction, and you can easily<00:08:57.640> see<00:08:57.839> that<00:08:58.080> because<00:08:58.360> you<00:08:58.440> can<00:08:58.640> imagine" + }, + { + "start": 539.79, + "duration": 0.0, + "text": "easily see that because you can imagine" + }, + { + "start": 539.8, + "duration": 0.0, + "text": "easily see that because you can imagine you<00:08:59.920> can<00:09:00.080> come<00:09:00.280> up<00:09:00.480> with<00:09:00.920> reasons<00:09:01.400> why<00:09:01.480> this" + }, + { + "start": 541.67, + "duration": 0.0, + "text": "you can come up with reasons why this" + }, + { + "start": 541.68, + "duration": 0.0, + "text": "you can come up with reasons why this might<00:09:01.880> have<00:09:02.040> happened<00:09:02.560> that<00:09:02.720> are<00:09:02.800> completely" + }, + { + "start": 543.19, + "duration": 0.0, + "text": "might have happened that are completely" + }, + { + "start": 543.2, + "duration": 0.0, + "text": "might have happened that are completely innocent.<00:09:03.839> I<00:09:03.880> mean,<00:09:04.600> if<00:09:04.720> you<00:09:04.800> want<00:09:04.960> to<00:09:05.520> say<00:09:06.400> uh" + }, + { + "start": 546.47, + "duration": 0.0, + "text": "innocent. I mean, if you want to say uh" + }, + { + "start": 546.48, + "duration": 0.0, + "text": "innocent. I mean, if you want to say uh a<00:09:06.520> particularly<00:09:06.920> ridiculous<00:09:07.360> one,<00:09:07.520> imagine" + }, + { + "start": 547.91, + "duration": 0.0, + "text": "a particularly ridiculous one, imagine" + }, + { + "start": 547.92, + "duration": 0.0, + "text": "a particularly ridiculous one, imagine that<00:09:08.040> the" + }, + { + "start": 548.91, + "duration": 0.0, + "text": "that the" + }, + { + "start": 548.92, + "duration": 0.0, + "text": "that the the<00:09:09.120> the<00:09:09.360> the<00:09:09.480> man<00:09:09.760> in<00:09:09.839> the<00:09:09.920> mask<00:09:10.200> is<00:09:10.280> actually" + }, + { + "start": 550.55, + "duration": 0.0, + "text": "the the the man in the mask is actually" + }, + { + "start": 550.56, + "duration": 0.0, + "text": "the the the man in the mask is actually the<00:09:10.640> store<00:09:11.000> owner,<00:09:11.760> and<00:09:12.120> there<00:09:12.240> was<00:09:12.400> a<00:09:12.440> fire" + }, + { + "start": 552.75, + "duration": 0.0, + "text": "the store owner, and there was a fire" + }, + { + "start": 552.76, + "duration": 0.0, + "text": "the store owner, and there was a fire alarm,<00:09:13.120> and<00:09:13.240> he<00:09:13.360> just<00:09:14.040> uh" + }, + { + "start": 554.79, + "duration": 0.0, + "text": "alarm, and he just uh" + }, + { + "start": 554.8, + "duration": 0.0, + "text": "alarm, and he just uh wanted<00:09:15.000> to<00:09:15.080> save<00:09:15.280> his<00:09:15.440> jewels,<00:09:15.880> and<00:09:16.680> he" + }, + { + "start": 556.75, + "duration": 0.0, + "text": "wanted to save his jewels, and he" + }, + { + "start": 556.76, + "duration": 0.0, + "text": "wanted to save his jewels, and he somehow<00:09:17.000> forgot<00:09:17.280> his<00:09:17.400> keys,<00:09:17.680> so<00:09:17.760> he<00:09:17.880> saw<00:09:18.040> no" + }, + { + "start": 558.23, + "duration": 0.0, + "text": "somehow forgot his keys, so he saw no" + }, + { + "start": 558.24, + "duration": 0.0, + "text": "somehow forgot his keys, so he saw no other<00:09:18.440> way<00:09:18.760> than<00:09:19.080> to<00:09:19.520> uh<00:09:19.560> break<00:09:19.800> in,<00:09:20.000> and<00:09:20.600> he" + }, + { + "start": 560.67, + "duration": 0.0, + "text": "other way than to uh break in, and he" + }, + { + "start": 560.68, + "duration": 0.0, + "text": "other way than to uh break in, and he also<00:09:20.839> had<00:09:20.960> the<00:09:21.040> mask<00:09:21.360> on<00:09:21.480> hand<00:09:21.720> to<00:09:21.800> keep" + }, + { + "start": 561.91, + "duration": 0.0, + "text": "also had the mask on hand to keep" + }, + { + "start": 561.92, + "duration": 0.0, + "text": "also had the mask on hand to keep himself<00:09:22.240> warm.<00:09:22.440> You<00:09:22.520> know,<00:09:22.800> you<00:09:22.920> could<00:09:23.080> have" + }, + { + "start": 563.15, + "duration": 0.0, + "text": "himself warm. You know, you could have" + }, + { + "start": 563.16, + "duration": 0.0, + "text": "himself warm. You know, you could have imagined<00:09:23.440> some<00:09:23.560> ridiculous<00:09:24.040> scenario.<00:09:24.440> This" + }, + { + "start": 564.63, + "duration": 0.0, + "text": "imagined some ridiculous scenario. This" + }, + { + "start": 564.64, + "duration": 0.0, + "text": "imagined some ridiculous scenario. This is<00:09:24.839> possible.<00:09:25.720> So,<00:09:25.800> just<00:09:25.960> by<00:09:26.040> the<00:09:26.120> very<00:09:26.320> fact" + }, + { + "start": 566.59, + "duration": 0.0, + "text": "is possible. So, just by the very fact" + }, + { + "start": 566.6, + "duration": 0.0, + "text": "is possible. So, just by the very fact that<00:09:26.680> an<00:09:26.760> alternative<00:09:27.200> is<00:09:27.440> possible<00:09:27.880> tells" + }, + { + "start": 568.11, + "duration": 0.0, + "text": "that an alternative is possible tells" + }, + { + "start": 568.12, + "duration": 0.0, + "text": "that an alternative is possible tells you<00:09:28.160> that<00:09:28.280> this<00:09:28.440> cannot<00:09:28.760> be<00:09:28.839> deduction," + }, + { + "start": 569.79, + "duration": 0.0, + "text": "you that this cannot be deduction," + }, + { + "start": 569.8, + "duration": 0.0, + "text": "you that this cannot be deduction, because<00:09:30.120> otherwise<00:09:31.040> uh<00:09:31.120> the<00:09:31.200> result<00:09:31.480> would" + }, + { + "start": 571.59, + "duration": 0.0, + "text": "because otherwise uh the result would" + }, + { + "start": 571.6, + "duration": 0.0, + "text": "because otherwise uh the result would have<00:09:31.680> been<00:09:31.839> uniquely<00:09:32.200> determined." + }, + { + "start": 573.67, + "duration": 0.0, + "text": "have been uniquely determined." + }, + { + "start": 573.68, + "duration": 0.0, + "text": "have been uniquely determined. And<00:09:34.160> uh" + }, + { + "start": 574.79, + "duration": 0.0, + "text": "And uh" + }, + { + "start": 574.8, + "duration": 0.0, + "text": "And uh unfortunately,<00:09:35.160> induction<00:09:35.600> gets<00:09:35.800> a<00:09:35.839> bit<00:09:35.960> of<00:09:36.040> a" + }, + { + "start": 576.31, + "duration": 0.0, + "text": "unfortunately, induction gets a bit of a" + }, + { + "start": 576.32, + "duration": 0.0, + "text": "unfortunately, induction gets a bit of a bad<00:09:36.600> reputation,<00:09:37.400> mainly<00:09:37.800> because<00:09:38.080> of<00:09:38.160> some" + }, + { + "start": 578.43, + "duration": 0.0, + "text": "bad reputation, mainly because of some" + }, + { + "start": 578.44, + "duration": 0.0, + "text": "bad reputation, mainly because of some philosophers<00:09:39.040> that<00:09:39.200> were<00:09:39.320> not<00:09:39.840> scientists," + }, + { + "start": 580.39, + "duration": 0.0, + "text": "philosophers that were not scientists," + }, + { + "start": 580.4, + "duration": 0.0, + "text": "philosophers that were not scientists, but<00:09:40.560> that's<00:09:40.720> another<00:09:40.960> point.<00:09:41.840> Um," + }, + { + "start": 582.87, + "duration": 0.0, + "text": "but that's another point. Um," + }, + { + "start": 582.88, + "duration": 0.0, + "text": "but that's another point. Um, but<00:09:43.080> we're<00:09:43.200> going<00:09:43.320> to<00:09:43.360> show<00:09:43.600> that<00:09:44.200> what" + }, + { + "start": 584.43, + "duration": 0.0, + "text": "but we're going to show that what" + }, + { + "start": 584.44, + "duration": 0.0, + "text": "but we're going to show that what science<00:09:44.800> does<00:09:45.280> for<00:09:45.400> the<00:09:45.480> most<00:09:45.760> part<00:09:45.960> is" + }, + { + "start": 586.07, + "duration": 0.0, + "text": "science does for the most part is" + }, + { + "start": 586.08, + "duration": 0.0, + "text": "science does for the most part is actually<00:09:46.440> just<00:09:46.640> induction<00:09:47.160> and<00:09:47.480> deduction<00:09:47.960> is" + }, + { + "start": 588.15, + "duration": 0.0, + "text": "actually just induction and deduction is" + }, + { + "start": 588.16, + "duration": 0.0, + "text": "actually just induction and deduction is mostly<00:09:48.600> the" + }, + { + "start": 589.51, + "duration": 0.0, + "text": "mostly the" + }, + { + "start": 589.52, + "duration": 0.0, + "text": "mostly the realm<00:09:50.000> of<00:09:50.160> what<00:09:50.400> mathematics<00:09:50.920> does." + }, + { + "start": 593.15, + "duration": 0.0, + "text": "realm of what mathematics does." + }, + { + "start": 593.16, + "duration": 0.0, + "text": "realm of what mathematics does. And" + }, + { + "start": 593.87, + "duration": 0.0, + "text": "And" + }, + { + "start": 593.88, + "duration": 0.0, + "text": "And to<00:09:54.000> understand<00:09:54.360> the<00:09:54.440> difference<00:09:54.720> further," + }, + { + "start": 595.23, + "duration": 0.0, + "text": "to understand the difference further," + }, + { + "start": 595.24, + "duration": 0.0, + "text": "to understand the difference further, I'm<00:09:55.360> sure<00:09:55.560> you're<00:09:55.680> all<00:09:55.800> aware<00:09:56.120> of<00:09:56.240> the<00:09:56.320> kind<00:09:56.640> of" + }, + { + "start": 597.23, + "duration": 0.0, + "text": "I'm sure you're all aware of the kind of" + }, + { + "start": 597.24, + "duration": 0.0, + "text": "I'm sure you're all aware of the kind of reasoning<00:09:57.760> that<00:09:58.080> Sherlock<00:09:58.400> Holmes<00:09:58.720> does." + }, + { + "start": 599.39, + "duration": 0.0, + "text": "reasoning that Sherlock Holmes does." + }, + { + "start": 599.4, + "duration": 0.0, + "text": "reasoning that Sherlock Holmes does. Now,<00:09:59.800> if<00:09:59.920> I<00:09:59.960> remember<00:10:00.240> correctly,<00:10:00.600> he<00:10:00.720> usually" + }, + { + "start": 601.07, + "duration": 0.0, + "text": "Now, if I remember correctly, he usually" + }, + { + "start": 601.08, + "duration": 0.0, + "text": "Now, if I remember correctly, he usually says<00:10:01.400> this<00:10:01.560> is<00:10:01.680> deduction,<00:10:02.160> but<00:10:02.360> also<00:10:02.720> all<00:10:02.880> of" + }, + { + "start": 602.95, + "duration": 0.0, + "text": "says this is deduction, but also all of" + }, + { + "start": 602.96, + "duration": 0.0, + "text": "says this is deduction, but also all of these<00:10:03.120> kind<00:10:03.360> of<00:10:03.480> like<00:10:03.680> little<00:10:03.880> tricks<00:10:04.200> and" + }, + { + "start": 604.27, + "duration": 0.0, + "text": "these kind of like little tricks and" + }, + { + "start": 604.28, + "duration": 0.0, + "text": "these kind of like little tricks and connections<00:10:04.680> that<00:10:04.800> he<00:10:05.000> makes<00:10:05.560> actually<00:10:05.840> all" + }, + { + "start": 605.99, + "duration": 0.0, + "text": "connections that he makes actually all" + }, + { + "start": 606.0, + "duration": 0.0, + "text": "connections that he makes actually all inductive." + }, + { + "start": 607.35, + "duration": 0.0, + "text": "inductive." + }, + { + "start": 607.36, + "duration": 0.0, + "text": "inductive. And<00:10:07.520> so<00:10:07.880> you<00:10:08.000> can<00:10:08.240> you<00:10:08.360> can<00:10:08.480> tell<00:10:08.680> that" + }, + { + "start": 609.43, + "duration": 0.0, + "text": "And so you can you can tell that" + }, + { + "start": 609.44, + "duration": 0.0, + "text": "And so you can you can tell that even<00:10:09.680> though<00:10:09.920> induction<00:10:10.520> is<00:10:10.720> has<00:10:10.880> this" + }, + { + "start": 610.99, + "duration": 0.0, + "text": "even though induction is has this" + }, + { + "start": 611.0, + "duration": 0.0, + "text": "even though induction is has this reputation<00:10:11.400> of<00:10:11.520> being<00:10:11.880> very<00:10:12.120> weak<00:10:12.360> reasoning," + }, + { + "start": 612.79, + "duration": 0.0, + "text": "reputation of being very weak reasoning," + }, + { + "start": 612.8, + "duration": 0.0, + "text": "reputation of being very weak reasoning, we<00:10:12.920> can<00:10:13.080> be<00:10:13.240> very,<00:10:13.640> very<00:10:13.880> confident<00:10:14.400> about<00:10:14.800> the" + }, + { + "start": 614.87, + "duration": 0.0, + "text": "we can be very, very confident about the" + }, + { + "start": 614.88, + "duration": 0.0, + "text": "we can be very, very confident about the conclusions<00:10:15.320> that<00:10:15.440> we<00:10:15.560> take<00:10:15.720> from<00:10:15.920> it." + }, + { + "start": 617.31, + "duration": 0.0, + "text": "conclusions that we take from it." + }, + { + "start": 617.32, + "duration": 0.0, + "text": "conclusions that we take from it. Now,<00:10:17.960> here<00:10:18.400> are<00:10:18.600> the<00:10:19.200> most<00:10:19.480> fundamental<00:10:19.840> rules" + }, + { + "start": 620.03, + "duration": 0.0, + "text": "Now, here are the most fundamental rules" + }, + { + "start": 620.04, + "duration": 0.0, + "text": "Now, here are the most fundamental rules of<00:10:20.120> probability." + }, + { + "start": 621.75, + "duration": 0.0, + "text": "of probability." + }, + { + "start": 621.76, + "duration": 0.0, + "text": "of probability. Uh," + }, + { + "start": 622.35, + "duration": 0.0, + "text": "Uh," + }, + { + "start": 622.36, + "duration": 0.0, + "text": "Uh, in<00:10:22.440> general<00:10:22.800> in<00:10:22.920> the<00:10:23.000> Bayesian<00:10:23.320> methodology," + }, + { + "start": 623.79, + "duration": 0.0, + "text": "in general in the Bayesian methodology," + }, + { + "start": 623.8, + "duration": 0.0, + "text": "in general in the Bayesian methodology, probability<00:10:24.480> is<00:10:24.640> seen<00:10:24.880> as<00:10:25.040> a<00:10:25.160> quantification" + }, + { + "start": 625.87, + "duration": 0.0, + "text": "probability is seen as a quantification" + }, + { + "start": 625.88, + "duration": 0.0, + "text": "probability is seen as a quantification of<00:10:26.000> the<00:10:26.120> plausibility<00:10:26.840> of<00:10:27.000> a<00:10:27.080> proposition<00:10:27.600> of" + }, + { + "start": 627.71, + "duration": 0.0, + "text": "of the plausibility of a proposition of" + }, + { + "start": 627.72, + "duration": 0.0, + "text": "of the plausibility of a proposition of a<00:10:27.760> statement<00:10:28.680> where<00:10:28.840> a<00:10:28.880> value<00:10:29.200> of<00:10:29.360> zero" + }, + { + "start": 629.87, + "duration": 0.0, + "text": "a statement where a value of zero" + }, + { + "start": 629.88, + "duration": 0.0, + "text": "a statement where a value of zero corresponds<00:10:30.360> to<00:10:30.440> impossibility<00:10:31.200> and<00:10:31.320> a<00:10:31.360> value" + }, + { + "start": 631.63, + "duration": 0.0, + "text": "corresponds to impossibility and a value" + }, + { + "start": 631.64, + "duration": 0.0, + "text": "corresponds to impossibility and a value of<00:10:31.760> one<00:10:32.160> corresponds<00:10:32.560> to<00:10:32.640> certainty." + }, + { + "start": 633.71, + "duration": 0.0, + "text": "of one corresponds to certainty." + }, + { + "start": 633.72, + "duration": 0.0, + "text": "of one corresponds to certainty. Now,<00:10:33.960> if<00:10:34.160> your<00:10:34.280> mind<00:10:34.520> immediately<00:10:34.960> jumps<00:10:35.360> to" + }, + { + "start": 635.51, + "duration": 0.0, + "text": "Now, if your mind immediately jumps to" + }, + { + "start": 635.52, + "duration": 0.0, + "text": "Now, if your mind immediately jumps to the" + }, + { + "start": 636.67, + "duration": 0.0, + "text": "the" + }, + { + "start": 636.68, + "duration": 0.0, + "text": "the uh,<00:10:36.880> case<00:10:37.320> of<00:10:37.520> sampling<00:10:37.960> a<00:10:38.000> rational<00:10:38.400> number" + }, + { + "start": 638.71, + "duration": 0.0, + "text": "uh, case of sampling a rational number" + }, + { + "start": 638.72, + "duration": 0.0, + "text": "uh, case of sampling a rational number from<00:10:39.000> a<00:10:39.360> set<00:10:39.600> of<00:10:39.720> irrationals,<00:10:40.400> don't<00:10:40.560> worry," + }, + { + "start": 640.75, + "duration": 0.0, + "text": "from a set of irrationals, don't worry," + }, + { + "start": 640.76, + "duration": 0.0, + "text": "from a set of irrationals, don't worry, we'll<00:10:40.880> talk<00:10:41.080> about<00:10:41.280> that<00:10:41.400> later<00:10:41.680> because<00:10:41.920> this" + }, + { + "start": 642.03, + "duration": 0.0, + "text": "we'll talk about that later because this" + }, + { + "start": 642.04, + "duration": 0.0, + "text": "we'll talk about that later because this seems<00:10:42.200> to<00:10:42.280> contradict<00:10:42.720> this." + }, + { + "start": 643.75, + "duration": 0.0, + "text": "seems to contradict this." + }, + { + "start": 643.76, + "duration": 0.0, + "text": "seems to contradict this. Uh,<00:10:44.120> but<00:10:44.320> let's<00:10:44.440> start<00:10:44.640> with<00:10:44.720> the<00:10:44.800> basics." + }, + { + "start": 645.23, + "duration": 0.0, + "text": "Uh, but let's start with the basics." + }, + { + "start": 645.24, + "duration": 0.0, + "text": "Uh, but let's start with the basics. Now,<00:10:45.400> we<00:10:45.480> have<00:10:45.600> the<00:10:45.680> product<00:10:46.000> rule" + }, + { + "start": 647.11, + "duration": 0.0, + "text": "Now, we have the product rule" + }, + { + "start": 647.12, + "duration": 0.0, + "text": "Now, we have the product rule uh,<00:10:47.360> that<00:10:47.480> allows<00:10:47.839> us<00:10:48.560> uh,<00:10:48.880> to<00:10:49.000> find<00:10:49.560> the" + }, + { + "start": 649.63, + "duration": 0.0, + "text": "uh, that allows us uh, to find the" + }, + { + "start": 649.64, + "duration": 0.0, + "text": "uh, that allows us uh, to find the probability<00:10:50.320> of<00:10:51.040> uh,<00:10:51.360> compound<00:10:51.760> statement<00:10:52.120> of" + }, + { + "start": 652.31, + "duration": 0.0, + "text": "probability of uh, compound statement of" + }, + { + "start": 652.32, + "duration": 0.0, + "text": "probability of uh, compound statement of A<00:10:52.400> and<00:10:52.520> B<00:10:53.280> given<00:10:53.600> some<00:10:53.720> context<00:10:54.160> C.<00:10:54.880> And<00:10:55.000> then" + }, + { + "start": 655.15, + "duration": 0.0, + "text": "A and B given some context C. And then" + }, + { + "start": 655.16, + "duration": 0.0, + "text": "A and B given some context C. And then we<00:10:55.240> have<00:10:55.440> the<00:10:55.920> sum<00:10:56.240> rule<00:10:56.480> which<00:10:56.640> allows<00:10:57.000> us<00:10:57.160> to" + }, + { + "start": 658.15, + "duration": 0.0, + "text": "we have the sum rule which allows us to" + }, + { + "start": 658.16, + "duration": 0.0, + "text": "we have the sum rule which allows us to uh,<00:10:58.640> combine<00:10:58.920> the<00:10:59.000> probabilities<00:10:59.560> of<00:10:59.760> two" + }, + { + "start": 659.87, + "duration": 0.0, + "text": "uh, combine the probabilities of two" + }, + { + "start": 659.88, + "duration": 0.0, + "text": "uh, combine the probabilities of two complements." + }, + { + "start": 662.27, + "duration": 0.0, + "text": "complements." + }, + { + "start": 662.28, + "duration": 0.0, + "text": "complements. And<00:11:02.360> you<00:11:02.440> notice<00:11:02.760> that" + }, + { + "start": 663.87, + "duration": 0.0, + "text": "And you notice that" + }, + { + "start": 663.88, + "duration": 0.0, + "text": "And you notice that uh,<00:11:03.960> we<00:11:04.080> always<00:11:04.440> have<00:11:04.800> this<00:11:05.600> uh,<00:11:06.080> given<00:11:06.400> C" + }, + { + "start": 667.47, + "duration": 0.0, + "text": "uh, we always have this uh, given C" + }, + { + "start": 667.48, + "duration": 0.0, + "text": "uh, we always have this uh, given C at<00:11:07.560> the<00:11:07.680> end<00:11:08.360> on<00:11:08.440> the<00:11:08.560> right<00:11:08.839> side<00:11:09.600> of<00:11:09.720> the" + }, + { + "start": 669.79, + "duration": 0.0, + "text": "at the end on the right side of the" + }, + { + "start": 669.8, + "duration": 0.0, + "text": "at the end on the right side of the expression<00:11:10.320> and<00:11:10.480> this<00:11:11.160> uh,<00:11:11.280> means<00:11:11.600> that" + }, + { + "start": 672.67, + "duration": 0.0, + "text": "expression and this uh, means that" + }, + { + "start": 672.68, + "duration": 0.0, + "text": "expression and this uh, means that in<00:11:12.880> the" + }, + { + "start": 673.99, + "duration": 0.0, + "text": "in the" + }, + { + "start": 674.0, + "duration": 0.0, + "text": "in the in<00:11:14.040> the<00:11:14.120> Bayesian<00:11:14.360> methodology,<00:11:14.960> we<00:11:15.080> don't" + }, + { + "start": 675.63, + "duration": 0.0, + "text": "in the Bayesian methodology, we don't" + }, + { + "start": 675.64, + "duration": 0.0, + "text": "in the Bayesian methodology, we don't think<00:11:15.960> of<00:11:16.120> probabilities<00:11:16.600> as<00:11:16.720> somehow" + }, + { + "start": 676.99, + "duration": 0.0, + "text": "think of probabilities as somehow" + }, + { + "start": 677.0, + "duration": 0.0, + "text": "think of probabilities as somehow existing<00:11:17.400> in<00:11:17.520> a<00:11:17.560> vacuum<00:11:18.160> or<00:11:18.280> existing<00:11:18.640> as<00:11:18.720> a" + }, + { + "start": 678.75, + "duration": 0.0, + "text": "existing in a vacuum or existing as a" + }, + { + "start": 678.76, + "duration": 0.0, + "text": "existing in a vacuum or existing as a property<00:11:19.360> of<00:11:19.480> the<00:11:19.560> experiment<00:11:20.080> or<00:11:20.160> the<00:11:20.280> object" + }, + { + "start": 680.67, + "duration": 0.0, + "text": "property of the experiment or the object" + }, + { + "start": 680.68, + "duration": 0.0, + "text": "property of the experiment or the object that<00:11:20.760> you're<00:11:20.839> studying,<00:11:21.520> but<00:11:21.640> they<00:11:21.720> always" + }, + { + "start": 681.99, + "duration": 0.0, + "text": "that you're studying, but they always" + }, + { + "start": 682.0, + "duration": 0.0, + "text": "that you're studying, but they always depend<00:11:22.600> on<00:11:23.240> what<00:11:23.520> knowledge<00:11:24.200> you<00:11:24.680> have<00:11:25.000> at" + }, + { + "start": 685.07, + "duration": 0.0, + "text": "depend on what knowledge you have at" + }, + { + "start": 685.08, + "duration": 0.0, + "text": "depend on what knowledge you have at this<00:11:25.200> point<00:11:25.520> and<00:11:25.640> this<00:11:25.760> is<00:11:26.480> uh," + }, + { + "start": 687.07, + "duration": 0.0, + "text": "this point and this is uh," + }, + { + "start": 687.08, + "duration": 0.0, + "text": "this point and this is uh, why<00:11:27.240> we<00:11:27.360> always<00:11:27.640> write<00:11:27.839> this<00:11:27.960> at<00:11:28.080> the<00:11:28.200> end." + }, + { + "start": 688.43, + "duration": 0.0, + "text": "why we always write this at the end." + }, + { + "start": 688.44, + "duration": 0.0, + "text": "why we always write this at the end. Usually<00:11:28.839> then<00:11:29.080> it<00:11:29.200> is<00:11:29.400> omitted<00:11:30.000> when<00:11:30.120> it's" + }, + { + "start": 690.27, + "duration": 0.0, + "text": "Usually then it is omitted when it's" + }, + { + "start": 690.28, + "duration": 0.0, + "text": "Usually then it is omitted when it's clear<00:11:30.520> that<00:11:30.640> it's<00:11:30.800> always<00:11:31.040> going<00:11:31.160> to<00:11:31.240> be<00:11:31.320> the" + }, + { + "start": 691.47, + "duration": 0.0, + "text": "clear that it's always going to be the" + }, + { + "start": 691.48, + "duration": 0.0, + "text": "clear that it's always going to be the same<00:11:31.720> letter<00:11:31.960> in<00:11:32.040> your<00:11:32.160> calculations,<00:11:32.760> but<00:11:32.960> it" + }, + { + "start": 693.07, + "duration": 0.0, + "text": "same letter in your calculations, but it" + }, + { + "start": 693.08, + "duration": 0.0, + "text": "same letter in your calculations, but it is<00:11:33.200> always<00:11:33.520> there<00:11:33.680> implicitly." + }, + { + "start": 695.15, + "duration": 0.0, + "text": "is always there implicitly." + }, + { + "start": 695.16, + "duration": 0.0, + "text": "is always there implicitly. Um<00:11:35.600> now<00:11:35.720> we're<00:11:35.840> going<00:11:35.960> to<00:11:36.040> just<00:11:36.240> quickly<00:11:36.440> talk" + }, + { + "start": 696.63, + "duration": 0.0, + "text": "Um now we're going to just quickly talk" + }, + { + "start": 696.64, + "duration": 0.0, + "text": "Um now we're going to just quickly talk about<00:11:36.840> some<00:11:36.960> classical<00:11:37.400> logic.<00:11:37.960> Uh<00:11:38.240> I'm<00:11:38.360> sure" + }, + { + "start": 698.59, + "duration": 0.0, + "text": "about some classical logic. Uh I'm sure" + }, + { + "start": 698.6, + "duration": 0.0, + "text": "about some classical logic. Uh I'm sure you're<00:11:38.800> all<00:11:38.960> familiar<00:11:39.280> with<00:11:39.400> this,<00:11:39.520> so<00:11:39.640> we're" + }, + { + "start": 699.71, + "duration": 0.0, + "text": "you're all familiar with this, so we're" + }, + { + "start": 699.72, + "duration": 0.0, + "text": "you're all familiar with this, so we're going<00:11:39.840> to<00:11:39.920> be<00:11:40.320> quick<00:11:40.560> here." + }, + { + "start": 701.79, + "duration": 0.0, + "text": "going to be quick here." + }, + { + "start": 701.8, + "duration": 0.0, + "text": "going to be quick here. Um" + }, + { + "start": 703.07, + "duration": 0.0, + "text": "Um" + }, + { + "start": 703.08, + "duration": 0.0, + "text": "Um we're<00:11:43.200> going<00:11:43.320> to<00:11:43.400> see<00:11:43.480> the<00:11:43.600> probability<00:11:44.160> is<00:11:44.600> an" + }, + { + "start": 704.71, + "duration": 0.0, + "text": "we're going to see the probability is an" + }, + { + "start": 704.72, + "duration": 0.0, + "text": "we're going to see the probability is an extension<00:11:45.080> of<00:11:45.200> logic<00:11:45.720> where<00:11:46.200> classical<00:11:46.680> logic" + }, + { + "start": 707.67, + "duration": 0.0, + "text": "extension of logic where classical logic" + }, + { + "start": 707.68, + "duration": 0.0, + "text": "extension of logic where classical logic uh<00:11:47.840> is<00:11:48.000> just<00:11:48.360> a<00:11:48.400> special<00:11:48.680> case." + }, + { + "start": 709.75, + "duration": 0.0, + "text": "uh is just a special case." + }, + { + "start": 709.76, + "duration": 0.0, + "text": "uh is just a special case. And<00:11:50.200> the<00:11:50.400> most<00:11:50.960> straightforward<00:11:51.640> way<00:11:51.840> to" + }, + { + "start": 711.99, + "duration": 0.0, + "text": "And the most straightforward way to" + }, + { + "start": 712.0, + "duration": 0.0, + "text": "And the most straightforward way to approach<00:11:52.360> classical<00:11:52.720> logic<00:11:53.000> is<00:11:53.120> with<00:11:53.360> truth" + }, + { + "start": 713.59, + "duration": 0.0, + "text": "approach classical logic is with truth" + }, + { + "start": 713.6, + "duration": 0.0, + "text": "approach classical logic is with truth tables." + }, + { + "start": 714.67, + "duration": 0.0, + "text": "tables." + }, + { + "start": 714.68, + "duration": 0.0, + "text": "tables. Um" + }, + { + "start": 716.59, + "duration": 0.0, + "text": "Um" + }, + { + "start": 716.6, + "duration": 0.0, + "text": "Um and<00:11:56.800> with<00:11:56.920> the<00:11:57.000> operations<00:11:57.960> and<00:11:58.520> or<00:11:58.920> and<00:11:59.080> not." + }, + { + "start": 719.95, + "duration": 0.0, + "text": "and with the operations and or and not." + }, + { + "start": 719.96, + "duration": 0.0, + "text": "and with the operations and or and not. These<00:12:00.200> are<00:12:00.320> an<00:12:00.440> adequate<00:12:00.800> set,<00:12:01.120> meaning<00:12:01.560> that" + }, + { + "start": 721.95, + "duration": 0.0, + "text": "These are an adequate set, meaning that" + }, + { + "start": 721.96, + "duration": 0.0, + "text": "These are an adequate set, meaning that they<00:12:03.080> uh<00:12:03.240> allow<00:12:03.520> us<00:12:03.680> to<00:12:03.760> express<00:12:04.480> any<00:12:04.720> truth" + }, + { + "start": 724.99, + "duration": 0.0, + "text": "they uh allow us to express any truth" + }, + { + "start": 725.0, + "duration": 0.0, + "text": "they uh allow us to express any truth value<00:12:05.280> depending<00:12:05.640> on<00:12:05.760> any<00:12:06.000> input." + }, + { + "start": 727.11, + "duration": 0.0, + "text": "value depending on any input." + }, + { + "start": 727.12, + "duration": 0.0, + "text": "value depending on any input. Uh<00:12:07.240> there<00:12:07.400> are<00:12:07.480> actually<00:12:07.800> smaller<00:12:08.160> adequate" + }, + { + "start": 728.47, + "duration": 0.0, + "text": "Uh there are actually smaller adequate" + }, + { + "start": 728.48, + "duration": 0.0, + "text": "Uh there are actually smaller adequate sets,<00:12:08.760> the<00:12:08.840> most<00:12:09.120> famous<00:12:09.440> one<00:12:09.640> being<00:12:10.240> NAND," + }, + { + "start": 730.99, + "duration": 0.0, + "text": "sets, the most famous one being NAND," + }, + { + "start": 731.0, + "duration": 0.0, + "text": "sets, the most famous one being NAND, which<00:12:11.200> is<00:12:11.280> used<00:12:11.560> extensively<00:12:12.000> in<00:12:12.120> computer" + }, + { + "start": 732.43, + "duration": 0.0, + "text": "which is used extensively in computer" + }, + { + "start": 732.44, + "duration": 0.0, + "text": "which is used extensively in computer chip<00:12:12.600> design,<00:12:12.960> and<00:12:13.080> there's<00:12:13.200> also<00:12:13.400> XOR." + }, + { + "start": 734.63, + "duration": 0.0, + "text": "chip design, and there's also XOR." + }, + { + "start": 734.64, + "duration": 0.0, + "text": "chip design, and there's also XOR. Um<00:12:15.640> but<00:12:15.920> and<00:12:16.080> and<00:12:16.200> or<00:12:16.480> are<00:12:16.560> just<00:12:16.720> very" + }, + { + "start": 736.91, + "duration": 0.0, + "text": "Um but and and or are just very" + }, + { + "start": 736.92, + "duration": 0.0, + "text": "Um but and and or are just very convenient,<00:12:17.960> and<00:12:18.280> so<00:12:19.160> uh<00:12:19.400> we<00:12:19.560> will<00:12:19.960> use<00:12:20.080> them." + }, + { + "start": 741.43, + "duration": 0.0, + "text": "convenient, and so uh we will use them." + }, + { + "start": 741.44, + "duration": 0.0, + "text": "convenient, and so uh we will use them. Now<00:12:21.760> I<00:12:21.800> just<00:12:22.040> also<00:12:22.280> want<00:12:22.440> to<00:12:22.480> draw<00:12:22.640> your" + }, + { + "start": 742.71, + "duration": 0.0, + "text": "Now I just also want to draw your" + }, + { + "start": 742.72, + "duration": 0.0, + "text": "Now I just also want to draw your attention<00:12:23.040> to<00:12:23.120> these<00:12:23.600> to<00:12:23.680> this<00:12:23.840> notation.<00:12:24.760> And" + }, + { + "start": 744.91, + "duration": 0.0, + "text": "attention to these to this notation. And" + }, + { + "start": 744.92, + "duration": 0.0, + "text": "attention to these to this notation. And and<00:12:25.040> or<00:12:25.840> uh<00:12:25.880> the<00:12:26.080> and<00:12:26.480> is<00:12:26.640> somehow<00:12:27.040> described" + }, + { + "start": 747.43, + "duration": 0.0, + "text": "and or uh the and is somehow described" + }, + { + "start": 747.44, + "duration": 0.0, + "text": "and or uh the and is somehow described as<00:12:27.560> this<00:12:27.720> like<00:12:28.600> uh<00:12:28.720> hat." + }, + { + "start": 749.83, + "duration": 0.0, + "text": "as this like uh hat." + }, + { + "start": 749.84, + "duration": 0.0, + "text": "as this like uh hat. And<00:12:30.160> the<00:12:30.240> or<00:12:30.600> is<00:12:30.760> kind<00:12:30.960> of<00:12:31.040> this<00:12:31.440> valley.<00:12:32.400> While" + }, + { + "start": 752.67, + "duration": 0.0, + "text": "And the or is kind of this valley. While" + }, + { + "start": 752.68, + "duration": 0.0, + "text": "And the or is kind of this valley. While the<00:12:32.760> not<00:12:33.000> is<00:12:33.120> just<00:12:33.400> a<00:12:33.880> line<00:12:34.400> over<00:12:34.640> the<00:12:35.200> actual" + }, + { + "start": 755.43, + "duration": 0.0, + "text": "the not is just a line over the actual" + }, + { + "start": 755.44, + "duration": 0.0, + "text": "the not is just a line over the actual proposition." + }, + { + "start": 759.24, + "duration": 0.0, + "text": "Here<00:12:39.440> is<00:12:39.560> a<00:12:39.680> some<00:12:40.280> uh<00:12:40.360> famous<00:12:40.680> truth<00:12:40.880> tables" + }, + { + "start": 761.31, + "duration": 0.0, + "text": "Here is a some uh famous truth tables" + }, + { + "start": 761.32, + "duration": 0.0, + "text": "Here is a some uh famous truth tables and<00:12:41.440> what<00:12:41.600> they<00:12:41.680> might<00:12:41.880> look<00:12:42.040> like.<00:12:42.280> These<00:12:42.440> are" + }, + { + "start": 762.51, + "duration": 0.0, + "text": "and what they might look like. These are" + }, + { + "start": 762.52, + "duration": 0.0, + "text": "and what they might look like. These are actually<00:12:42.720> kind<00:12:42.880> of<00:12:42.960> important<00:12:43.960> uh<00:12:44.520> for<00:12:44.640> the" + }, + { + "start": 764.75, + "duration": 0.0, + "text": "actually kind of important uh for the" + }, + { + "start": 764.76, + "duration": 0.0, + "text": "actually kind of important uh for the rest<00:12:44.920> of<00:12:45.000> the<00:12:45.040> presentation,<00:12:45.680> but<00:12:45.960> I'm<00:12:46.080> not" + }, + { + "start": 766.51, + "duration": 0.0, + "text": "rest of the presentation, but I'm not" + }, + { + "start": 766.52, + "duration": 0.0, + "text": "rest of the presentation, but I'm not going<00:12:46.680> to<00:12:46.760> go<00:12:46.839> into<00:12:47.040> them.<00:12:47.360> If<00:12:47.960> you<00:12:48.120> ever<00:12:48.360> wish" + }, + { + "start": 768.79, + "duration": 0.0, + "text": "going to go into them. If you ever wish" + }, + { + "start": 768.8, + "duration": 0.0, + "text": "going to go into them. If you ever wish to<00:12:48.920> have<00:12:49.040> some<00:12:49.160> explanation<00:12:49.640> for<00:12:49.760> why" + }, + { + "start": 769.87, + "duration": 0.0, + "text": "to have some explanation for why" + }, + { + "start": 769.88, + "duration": 0.0, + "text": "to have some explanation for why something<00:12:50.160> happens,<00:12:50.880> I<00:12:50.960> can<00:12:51.080> come<00:12:51.240> back<00:12:51.440> here" + }, + { + "start": 771.67, + "duration": 0.0, + "text": "something happens, I can come back here" + }, + { + "start": 771.68, + "duration": 0.0, + "text": "something happens, I can come back here and<00:12:51.960> show<00:12:52.160> you." + }, + { + "start": 773.63, + "duration": 0.0, + "text": "and show you." + }, + { + "start": 773.64, + "duration": 0.0, + "text": "and show you. So<00:12:53.800> how<00:12:53.960> can<00:12:54.120> we<00:12:54.200> check<00:12:54.520> if<00:12:54.800> some<00:12:55.000> kind<00:12:55.240> of" + }, + { + "start": 775.47, + "duration": 0.0, + "text": "So how can we check if some kind of" + }, + { + "start": 775.48, + "duration": 0.0, + "text": "So how can we check if some kind of complex<00:12:56.080> compound<00:12:56.520> statement<00:12:57.360> uh<00:12:57.520> can<00:12:57.760> be" + }, + { + "start": 777.829, + "duration": 0.0, + "text": "complex compound statement uh can be" + }, + { + "start": 777.839, + "duration": 0.0, + "text": "complex compound statement uh can be true?" + }, + { + "start": 778.75, + "duration": 0.0, + "text": "true?" + }, + { + "start": 778.76, + "duration": 0.0, + "text": "true? We<00:12:59.000> can<00:12:59.200> do<00:12:59.440> what<00:12:59.600> is<00:12:59.720> called<00:13:00.200> reduction<00:13:00.680> to" + }, + { + "start": 780.87, + "duration": 0.0, + "text": "We can do what is called reduction to" + }, + { + "start": 780.88, + "duration": 0.0, + "text": "We can do what is called reduction to the<00:13:01.000> disjunctive<00:13:01.880> normal<00:13:02.120> form,<00:13:02.480> meaning<00:13:03.360> we" + }, + { + "start": 784.11, + "duration": 0.0, + "text": "the disjunctive normal form, meaning we" + }, + { + "start": 784.12, + "duration": 0.0, + "text": "the disjunctive normal form, meaning we uh" + }, + { + "start": 784.87, + "duration": 0.0, + "text": "uh" + }, + { + "start": 784.88, + "duration": 0.0, + "text": "uh split<00:13:05.600> the<00:13:05.680> statement<00:13:06.040> up<00:13:06.240> into<00:13:06.600> all<00:13:07.000> of<00:13:07.200> its" + }, + { + "start": 787.829, + "duration": 0.0, + "text": "split the statement up into all of its" + }, + { + "start": 787.839, + "duration": 0.0, + "text": "split the statement up into all of its uh<00:13:08.000> minus<00:13:08.800> uh<00:13:08.880> it's<00:13:09.080> all<00:13:09.200> of<00:13:09.280> its<00:13:09.440> smallest" + }, + { + "start": 789.87, + "duration": 0.0, + "text": "uh minus uh it's all of its smallest" + }, + { + "start": 789.88, + "duration": 0.0, + "text": "uh minus uh it's all of its smallest form,<00:13:10.120> so<00:13:10.200> to<00:13:10.280> speak,<00:13:11.160> by<00:13:11.640> applying<00:13:12.040> the<00:13:12.160> join" + }, + { + "start": 792.47, + "duration": 0.0, + "text": "form, so to speak, by applying the join" + }, + { + "start": 792.48, + "duration": 0.0, + "text": "form, so to speak, by applying the join operation,<00:13:13.160> meaning<00:13:13.400> the<00:13:13.520> and<00:13:13.760> operation." + }, + { + "start": 794.71, + "duration": 0.0, + "text": "operation, meaning the and operation." + }, + { + "start": 794.72, + "duration": 0.0, + "text": "operation, meaning the and operation. And<00:13:15.200> then<00:13:15.520> we<00:13:15.640> can<00:13:15.880> we<00:13:16.000> could<00:13:16.160> combine<00:13:16.520> those" + }, + { + "start": 796.75, + "duration": 0.0, + "text": "And then we can we could combine those" + }, + { + "start": 796.76, + "duration": 0.0, + "text": "And then we can we could combine those again<00:13:17.040> with<00:13:17.240> a<00:13:17.520> with<00:13:17.760> increasing<00:13:18.200> amount<00:13:18.400> of" + }, + { + "start": 798.51, + "duration": 0.0, + "text": "again with a with increasing amount of" + }, + { + "start": 798.52, + "duration": 0.0, + "text": "again with a with increasing amount of or<00:13:18.760> operations<00:13:19.280> to<00:13:19.360> come<00:13:19.560> back<00:13:19.760> to<00:13:19.839> the" + }, + { + "start": 799.91, + "duration": 0.0, + "text": "or operations to come back to the" + }, + { + "start": 799.92, + "duration": 0.0, + "text": "or operations to come back to the original<00:13:20.240> statement.<00:13:21.120> And<00:13:21.320> if<00:13:21.560> any<00:13:21.760> of<00:13:21.920> those" + }, + { + "start": 802.23, + "duration": 0.0, + "text": "original statement. And if any of those" + }, + { + "start": 802.24, + "duration": 0.0, + "text": "original statement. And if any of those atoms<00:13:23.400> that<00:13:23.520> cannot<00:13:23.800> be<00:13:23.960> further<00:13:24.520> combined" + }, + { + "start": 804.829, + "duration": 0.0, + "text": "atoms that cannot be further combined" + }, + { + "start": 804.839, + "duration": 0.0, + "text": "atoms that cannot be further combined with<00:13:25.440> with<00:13:25.640> meet<00:13:25.839> statements,<00:13:26.760> um<00:13:27.200> then<00:13:27.320> the" + }, + { + "start": 807.39, + "duration": 0.0, + "text": "with with meet statements, um then the" + }, + { + "start": 807.4, + "duration": 0.0, + "text": "with with meet statements, um then the whole<00:13:27.760> if<00:13:27.960> any<00:13:28.120> of<00:13:28.280> those<00:13:28.560> can<00:13:28.800> be<00:13:28.920> true,<00:13:29.160> then" + }, + { + "start": 809.31, + "duration": 0.0, + "text": "whole if any of those can be true, then" + }, + { + "start": 809.32, + "duration": 0.0, + "text": "whole if any of those can be true, then the<00:13:29.400> whole<00:13:29.560> statement<00:13:29.880> can<00:13:30.160> also<00:13:30.400> be<00:13:30.520> true." + }, + { + "start": 811.11, + "duration": 0.0, + "text": "the whole statement can also be true." + }, + { + "start": 811.12, + "duration": 0.0, + "text": "the whole statement can also be true. Now<00:13:31.240> this<00:13:31.400> doesn't<00:13:31.720> tell<00:13:31.880> us<00:13:32.000> whether<00:13:32.200> it<00:13:32.280> is" + }, + { + "start": 812.35, + "duration": 0.0, + "text": "Now this doesn't tell us whether it is" + }, + { + "start": 812.36, + "duration": 0.0, + "text": "Now this doesn't tell us whether it is true<00:13:32.480> or<00:13:32.560> not,<00:13:32.680> but<00:13:32.840> this<00:13:32.960> is<00:13:33.040> how<00:13:33.160> we<00:13:33.320> find<00:13:33.640> out" + }, + { + "start": 813.75, + "duration": 0.0, + "text": "true or not, but this is how we find out" + }, + { + "start": 813.76, + "duration": 0.0, + "text": "true or not, but this is how we find out if<00:13:33.880> it<00:13:34.080> could<00:13:34.320> possibly<00:13:34.720> be<00:13:34.800> true.<00:13:35.040> Because<00:13:35.400> if" + }, + { + "start": 815.87, + "duration": 0.0, + "text": "if it could possibly be true. Because if" + }, + { + "start": 815.88, + "duration": 0.0, + "text": "if it could possibly be true. Because if all<00:13:36.120> the<00:13:36.240> atoms" + }, + { + "start": 817.67, + "duration": 0.0, + "text": "all the atoms" + }, + { + "start": 817.68, + "duration": 0.0, + "text": "all the atoms have<00:13:37.840> contradictions<00:13:38.320> in<00:13:38.400> them,<00:13:38.600> then<00:13:38.760> the" + }, + { + "start": 818.87, + "duration": 0.0, + "text": "have contradictions in them, then the" + }, + { + "start": 818.88, + "duration": 0.0, + "text": "have contradictions in them, then the compound<00:13:39.200> statement<00:13:39.480> obviously<00:13:39.720> cannot<00:13:39.960> be" + }, + { + "start": 820.03, + "duration": 0.0, + "text": "compound statement obviously cannot be" + }, + { + "start": 820.04, + "duration": 0.0, + "text": "compound statement obviously cannot be true<00:13:40.240> either." + }, + { + "start": 821.87, + "duration": 0.0, + "text": "true either." + }, + { + "start": 821.88, + "duration": 0.0, + "text": "true either. For<00:13:41.960> example,<00:13:42.280> the<00:13:42.400> compound<00:13:42.840> statement<00:13:43.480> A<00:13:43.640> or" + }, + { + "start": 823.75, + "duration": 0.0, + "text": "For example, the compound statement A or" + }, + { + "start": 823.76, + "duration": 0.0, + "text": "For example, the compound statement A or B<00:13:44.000> can<00:13:44.160> be<00:13:44.240> divided<00:13:44.560> into<00:13:44.880> A" + }, + { + "start": 825.87, + "duration": 0.0, + "text": "B can be divided into A" + }, + { + "start": 825.88, + "duration": 0.0, + "text": "B can be divided into A individually<00:13:46.600> and<00:13:46.720> B<00:13:46.840> individually,<00:13:47.560> but" + }, + { + "start": 827.67, + "duration": 0.0, + "text": "individually and B individually, but" + }, + { + "start": 827.68, + "duration": 0.0, + "text": "individually and B individually, but then<00:13:47.800> also<00:13:47.960> further<00:13:48.280> into<00:13:48.920> A<00:13:49.040> and<00:13:49.200> not<00:13:49.440> B,<00:13:49.920> A" + }, + { + "start": 830.03, + "duration": 0.0, + "text": "then also further into A and not B, A" + }, + { + "start": 830.04, + "duration": 0.0, + "text": "then also further into A and not B, A and<00:13:50.200> B,<00:13:50.600> not<00:13:50.920> A<00:13:51.000> and<00:13:51.120> B,<00:13:51.400> and<00:13:51.560> not<00:13:51.840> A<00:13:51.880> and<00:13:52.000> not<00:13:52.160> B." + }, + { + "start": 832.39, + "duration": 0.0, + "text": "and B, not A and B, and not A and not B." + }, + { + "start": 832.4, + "duration": 0.0, + "text": "and B, not A and B, and not A and not B. And<00:13:52.560> these<00:13:52.800> are<00:13:52.920> then<00:13:53.080> the<00:13:53.200> atoms<00:13:53.560> because<00:13:53.840> you" + }, + { + "start": 833.91, + "duration": 0.0, + "text": "And these are then the atoms because you" + }, + { + "start": 833.92, + "duration": 0.0, + "text": "And these are then the atoms because you just<00:13:54.160> this<00:13:54.320> doesn't<00:13:54.560> make<00:13:54.680> sense<00:13:54.880> to<00:13:55.000> combine" + }, + { + "start": 835.31, + "duration": 0.0, + "text": "just this doesn't make sense to combine" + }, + { + "start": 835.32, + "duration": 0.0, + "text": "just this doesn't make sense to combine them<00:13:55.440> further" + }, + { + "start": 836.63, + "duration": 0.0, + "text": "them further" + }, + { + "start": 836.64, + "duration": 0.0, + "text": "them further with" + }, + { + "start": 837.55, + "duration": 0.0, + "text": "with" + }, + { + "start": 837.56, + "duration": 0.0, + "text": "with with<00:13:58.040> meet<00:13:58.200> operations." + }, + { + "start": 840.35, + "duration": 0.0, + "text": "with meet operations." + }, + { + "start": 840.36, + "duration": 0.0, + "text": "with meet operations. Now<00:14:00.880> from" + }, + { + "start": 842.15, + "duration": 0.0, + "text": "Now from" + }, + { + "start": 842.16, + "duration": 0.0, + "text": "Now from from<00:14:02.320> this<00:14:02.480> operation,<00:14:03.680> we<00:14:03.800> can<00:14:03.960> actually" + }, + { + "start": 844.35, + "duration": 0.0, + "text": "from this operation, we can actually" + }, + { + "start": 844.36, + "duration": 0.0, + "text": "from this operation, we can actually obtain<00:14:04.840> some<00:14:05.040> kind<00:14:05.280> of<00:14:05.720> order" + }, + { + "start": 847.27, + "duration": 0.0, + "text": "obtain some kind of order" + }, + { + "start": 847.28, + "duration": 0.0, + "text": "obtain some kind of order by<00:14:07.400> implication.<00:14:08.480> So" + }, + { + "start": 850.15, + "duration": 0.0, + "text": "by implication. So" + }, + { + "start": 850.16, + "duration": 0.0, + "text": "by implication. So we<00:14:10.280> know<00:14:10.560> that<00:14:10.960> these<00:14:11.280> are<00:14:11.360> the<00:14:11.440> most" + }, + { + "start": 851.67, + "duration": 0.0, + "text": "we know that these are the most" + }, + { + "start": 851.68, + "duration": 0.0, + "text": "we know that these are the most fundamental<00:14:12.160> building<00:14:12.400> blocks<00:14:13.360> here<00:14:13.560> at<00:14:13.640> the" + }, + { + "start": 853.71, + "duration": 0.0, + "text": "fundamental building blocks here at the" + }, + { + "start": 853.72, + "duration": 0.0, + "text": "fundamental building blocks here at the bottom,<00:14:14.520> and<00:14:14.760> these<00:14:14.920> statements<00:14:15.280> above<00:14:16.160> are" + }, + { + "start": 856.31, + "duration": 0.0, + "text": "bottom, and these statements above are" + }, + { + "start": 856.32, + "duration": 0.0, + "text": "bottom, and these statements above are just<00:14:16.560> the<00:14:16.640> combinations<00:14:17.360> with<00:14:17.520> the<00:14:18.000> with<00:14:18.080> the" + }, + { + "start": 858.19, + "duration": 0.0, + "text": "just the combinations with the with the" + }, + { + "start": 858.2, + "duration": 0.0, + "text": "just the combinations with the with the join<00:14:18.440> operations.<00:14:18.960> And<00:14:19.120> this<00:14:19.280> means<00:14:19.520> that" + }, + { + "start": 860.31, + "duration": 0.0, + "text": "join operations. And this means that" + }, + { + "start": 860.32, + "duration": 0.0, + "text": "join operations. And this means that this<00:14:21.600> element<00:14:22.000> here<00:14:22.320> implies<00:14:22.680> this<00:14:22.800> one" + }, + { + "start": 862.91, + "duration": 0.0, + "text": "this element here implies this one" + }, + { + "start": 862.92, + "duration": 0.0, + "text": "this element here implies this one above." + }, + { + "start": 866.12, + "duration": 0.0, + "text": "Um" + }, + { + "start": 867.67, + "duration": 0.0, + "text": "Um" + }, + { + "start": 867.68, + "duration": 0.0, + "text": "Um and<00:14:27.800> here<00:14:28.080> in<00:14:28.200> this<00:14:28.400> representation<00:14:29.000> actually" + }, + { + "start": 869.43, + "duration": 0.0, + "text": "and here in this representation actually" + }, + { + "start": 869.44, + "duration": 0.0, + "text": "and here in this representation actually not<00:14:29.880> A<00:14:30.040> and<00:14:30.200> not<00:14:30.400> B<00:14:30.560> is<00:14:30.720> ignored<00:14:31.200> because<00:14:31.680> our" + }, + { + "start": 871.829, + "duration": 0.0, + "text": "not A and not B is ignored because our" + }, + { + "start": 871.839, + "duration": 0.0, + "text": "not A and not B is ignored because our top<00:14:32.160> element<00:14:32.520> here<00:14:32.800> is<00:14:33.120> A<00:14:33.240> or<00:14:33.360> B." + }, + { + "start": 874.31, + "duration": 0.0, + "text": "top element here is A or B." + }, + { + "start": 874.32, + "duration": 0.0, + "text": "top element here is A or B. And<00:14:34.800> if<00:14:34.960> we<00:14:35.080> assume<00:14:35.360> that<00:14:35.480> this<00:14:35.640> is<00:14:35.760> true,<00:14:36.040> then" + }, + { + "start": 876.19, + "duration": 0.0, + "text": "And if we assume that this is true, then" + }, + { + "start": 876.2, + "duration": 0.0, + "text": "And if we assume that this is true, then at<00:14:36.320> least<00:14:36.560> one<00:14:37.240> of<00:14:37.440> A<00:14:38.240> and<00:14:38.360> B<00:14:38.520> must<00:14:38.839> be<00:14:38.920> true." + }, + { + "start": 879.23, + "duration": 0.0, + "text": "at least one of A and B must be true." + }, + { + "start": 879.24, + "duration": 0.0, + "text": "at least one of A and B must be true. And<00:14:39.360> so<00:14:39.480> the<00:14:39.600> statement<00:14:39.920> is<00:14:40.040> necessarily" + }, + { + "start": 880.51, + "duration": 0.0, + "text": "And so the statement is necessarily" + }, + { + "start": 880.52, + "duration": 0.0, + "text": "And so the statement is necessarily false.<00:14:40.800> So<00:14:41.480> A<00:14:42.000> and<00:14:42.160> not<00:14:42.400> B<00:14:42.560> is<00:14:42.680> sort<00:14:42.800> of<00:14:43.000> the" + }, + { + "start": 883.07, + "duration": 0.0, + "text": "false. So A and not B is sort of the" + }, + { + "start": 883.08, + "duration": 0.0, + "text": "false. So A and not B is sort of the complement<00:14:43.680> to<00:14:43.800> to<00:14:43.880> the<00:14:44.000> top<00:14:44.240> statement." + }, + { + "start": 884.55, + "duration": 0.0, + "text": "complement to to the top statement." + }, + { + "start": 884.56, + "duration": 0.0, + "text": "complement to to the top statement. Therefore,<00:14:44.880> it<00:14:44.960> doesn't<00:14:45.200> feature<00:14:45.560> in<00:14:45.640> this." + }, + { + "start": 886.67, + "duration": 0.0, + "text": "Therefore, it doesn't feature in this." + }, + { + "start": 886.68, + "duration": 0.0, + "text": "Therefore, it doesn't feature in this. And<00:14:47.400> um" + }, + { + "start": 888.87, + "duration": 0.0, + "text": "And um" + }, + { + "start": 888.88, + "duration": 0.0, + "text": "And um exactly.<00:14:49.680> We<00:14:49.800> assume<00:14:50.280> that<00:14:50.560> this<00:14:50.960> top" + }, + { + "start": 891.19, + "duration": 0.0, + "text": "exactly. We assume that this top" + }, + { + "start": 891.2, + "duration": 0.0, + "text": "exactly. We assume that this top statement<00:14:51.480> is<00:14:51.600> true." + }, + { + "start": 892.71, + "duration": 0.0, + "text": "statement is true." + }, + { + "start": 892.72, + "duration": 0.0, + "text": "statement is true. And<00:14:52.880> from<00:14:53.040> this" + }, + { + "start": 893.95, + "duration": 0.0, + "text": "And from this" + }, + { + "start": 893.96, + "duration": 0.0, + "text": "And from this we<00:14:54.120> write<00:14:54.280> it<00:14:54.360> with<00:14:54.520> this<00:14:54.720> arrow," + }, + { + "start": 895.63, + "duration": 0.0, + "text": "we write it with this arrow," + }, + { + "start": 895.64, + "duration": 0.0, + "text": "we write it with this arrow, skinny<00:14:56.280> T<00:14:56.480> symbol." + }, + { + "start": 899.04, + "duration": 0.0, + "text": "And<00:14:59.200> from<00:14:59.320> this<00:14:59.480> we<00:14:59.600> can<00:14:59.800> essentially<00:15:00.200> obtain" + }, + { + "start": 900.51, + "duration": 0.0, + "text": "And from this we can essentially obtain" + }, + { + "start": 900.52, + "duration": 0.0, + "text": "And from this we can essentially obtain a<00:15:00.600> lattice<00:15:01.160> that<00:15:01.760> has<00:15:02.000> elements<00:15:02.560> that<00:15:02.800> are" + }, + { + "start": 903.07, + "duration": 0.0, + "text": "a lattice that has elements that are" + }, + { + "start": 903.08, + "duration": 0.0, + "text": "a lattice that has elements that are ordered<00:15:03.440> by<00:15:03.960> implication." + }, + { + "start": 905.31, + "duration": 0.0, + "text": "ordered by implication." + }, + { + "start": 905.32, + "duration": 0.0, + "text": "ordered by implication. They<00:15:05.920> also<00:15:06.120> have<00:15:06.360> this<00:15:07.160> optional<00:15:07.960> bottom" + }, + { + "start": 908.23, + "duration": 0.0, + "text": "They also have this optional bottom" + }, + { + "start": 908.24, + "duration": 0.0, + "text": "They also have this optional bottom element<00:15:08.600> that<00:15:08.720> is<00:15:08.880> just<00:15:09.200> the<00:15:09.720> absurdity,<00:15:10.280> the" + }, + { + "start": 910.39, + "duration": 0.0, + "text": "element that is just the absurdity, the" + }, + { + "start": 910.4, + "duration": 0.0, + "text": "element that is just the absurdity, the contradiction,<00:15:10.960> this<00:15:11.160> makes<00:15:11.400> sense.<00:15:12.200> Um<00:15:13.240> and" + }, + { + "start": 914.07, + "duration": 0.0, + "text": "contradiction, this makes sense. Um and" + }, + { + "start": 914.08, + "duration": 0.0, + "text": "contradiction, this makes sense. Um and to<00:15:14.240> understand<00:15:14.800> what's<00:15:14.960> going<00:15:15.200> on<00:15:15.400> here,<00:15:15.600> it's" + }, + { + "start": 915.75, + "duration": 0.0, + "text": "to understand what's going on here, it's" + }, + { + "start": 915.76, + "duration": 0.0, + "text": "to understand what's going on here, it's very<00:15:16.000> easy<00:15:16.240> to<00:15:16.360> kind<00:15:16.560> of<00:15:16.640> see<00:15:16.800> an<00:15:17.160> isomorphism" + }, + { + "start": 917.87, + "duration": 0.0, + "text": "very easy to kind of see an isomorphism" + }, + { + "start": 917.88, + "duration": 0.0, + "text": "very easy to kind of see an isomorphism to" + }, + { + "start": 918.91, + "duration": 0.0, + "text": "to" + }, + { + "start": 918.92, + "duration": 0.0, + "text": "to a<00:15:18.960> lattice<00:15:19.640> of<00:15:20.000> sets<00:15:20.280> that<00:15:20.480> are<00:15:20.560> ordered<00:15:20.880> by" + }, + { + "start": 921.03, + "duration": 0.0, + "text": "a lattice of sets that are ordered by" + }, + { + "start": 921.04, + "duration": 0.0, + "text": "a lattice of sets that are ordered by inclusion,<00:15:21.560> by<00:15:21.680> set<00:15:21.920> inclusion," + }, + { + "start": 922.95, + "duration": 0.0, + "text": "inclusion, by set inclusion," + }, + { + "start": 922.96, + "duration": 0.0, + "text": "inclusion, by set inclusion, where<00:15:23.720> um<00:15:23.840> um" + }, + { + "start": 925.27, + "duration": 0.0, + "text": "where um um" + }, + { + "start": 925.28, + "duration": 0.0, + "text": "where um um you<00:15:25.440> just<00:15:25.600> see<00:15:26.520> uh<00:15:26.600> you<00:15:26.680> just<00:15:26.920> basically<00:15:27.200> make" + }, + { + "start": 927.35, + "duration": 0.0, + "text": "you just see uh you just basically make" + }, + { + "start": 927.36, + "duration": 0.0, + "text": "you just see uh you just basically make an<00:15:27.440> equivalent<00:15:27.960> of<00:15:28.320> the<00:15:28.560> the<00:15:28.720> labels<00:15:29.120> of<00:15:29.240> which" + }, + { + "start": 929.39, + "duration": 0.0, + "text": "an equivalent of the the labels of which" + }, + { + "start": 929.4, + "duration": 0.0, + "text": "an equivalent of the the labels of which statements<00:15:29.760> can<00:15:29.880> be<00:15:29.960> true,<00:15:30.560> and<00:15:30.680> you<00:15:30.760> can<00:15:30.880> see" + }, + { + "start": 931.03, + "duration": 0.0, + "text": "statements can be true, and you can see" + }, + { + "start": 931.04, + "duration": 0.0, + "text": "statements can be true, and you can see that<00:15:31.360> E<00:15:31.680> F<00:15:32.000> contains<00:15:32.480> E<00:15:32.680> and<00:15:32.840> F,<00:15:33.000> of<00:15:33.120> course," + }, + { + "start": 933.47, + "duration": 0.0, + "text": "that E F contains E and F, of course," + }, + { + "start": 933.48, + "duration": 0.0, + "text": "that E F contains E and F, of course, and<00:15:33.600> this<00:15:33.720> is<00:15:34.120> this<00:15:34.400> is<00:15:34.680> more<00:15:34.840> or<00:15:34.920> less<00:15:35.040> the" + }, + { + "start": 935.11, + "duration": 0.0, + "text": "and this is this is more or less the" + }, + { + "start": 935.12, + "duration": 0.0, + "text": "and this is this is more or less the fact<00:15:35.320> that<00:15:35.480> E<00:15:35.680> implies<00:15:36.200> E<00:15:36.320> or<00:15:36.480> F." + }, + { + "start": 938.96, + "duration": 0.0, + "text": "And<00:15:39.120> it<00:15:39.200> is<00:15:39.280> with<00:15:39.440> these<00:15:39.640> lattice<00:15:39.880> structures" + }, + { + "start": 940.27, + "duration": 0.0, + "text": "And it is with these lattice structures" + }, + { + "start": 940.28, + "duration": 0.0, + "text": "And it is with these lattice structures that<00:15:40.480> we're<00:15:40.600> going<00:15:40.800> to" + }, + { + "start": 941.71, + "duration": 0.0, + "text": "that we're going to" + }, + { + "start": 941.72, + "duration": 0.0, + "text": "that we're going to uh" + }, + { + "start": 942.47, + "duration": 0.0, + "text": "uh" + }, + { + "start": 942.48, + "duration": 0.0, + "text": "uh find<00:15:42.840> the<00:15:43.360> the<00:15:43.440> fundamental<00:15:43.840> rules<00:15:44.000> of" + }, + { + "start": 944.11, + "duration": 0.0, + "text": "find the the fundamental rules of" + }, + { + "start": 944.12, + "duration": 0.0, + "text": "find the the fundamental rules of probability.<00:15:44.960> If<00:15:45.160> any<00:15:45.360> of<00:15:45.720> you<00:15:46.000> are<00:15:46.120> familiar" + }, + { + "start": 946.55, + "duration": 0.0, + "text": "probability. If any of you are familiar" + }, + { + "start": 946.56, + "duration": 0.0, + "text": "probability. If any of you are familiar with<00:15:46.760> the<00:15:46.880> work<00:15:47.240> of<00:15:47.840> uh<00:15:48.080> of<00:15:48.240> Cox," + }, + { + "start": 949.47, + "duration": 0.0, + "text": "with the work of uh of Cox," + }, + { + "start": 949.48, + "duration": 0.0, + "text": "with the work of uh of Cox, he<00:15:49.800> derived<00:15:50.400> the<00:15:50.760> the<00:15:50.880> rules<00:15:51.080> of<00:15:51.160> probability" + }, + { + "start": 952.03, + "duration": 0.0, + "text": "he derived the the rules of probability" + }, + { + "start": 952.04, + "duration": 0.0, + "text": "he derived the the rules of probability by<00:15:52.240> assuming<00:15:52.960> also" + }, + { + "start": 954.23, + "duration": 0.0, + "text": "by assuming also" + }, + { + "start": 954.24, + "duration": 0.0, + "text": "by assuming also uh<00:15:54.720> continuity<00:15:55.560> of<00:15:55.920> the<00:15:56.040> probability<00:15:56.600> and" + }, + { + "start": 956.75, + "duration": 0.0, + "text": "uh continuity of the probability and" + }, + { + "start": 956.76, + "duration": 0.0, + "text": "uh continuity of the probability and differentiability,<00:15:57.680> and<00:15:57.880> the<00:15:58.000> beauty<00:15:58.360> of" + }, + { + "start": 958.47, + "duration": 0.0, + "text": "differentiability, and the beauty of" + }, + { + "start": 958.48, + "duration": 0.0, + "text": "differentiability, and the beauty of this<00:15:58.680> approach<00:15:59.120> is<00:15:59.320> that<00:15:59.640> it<00:15:59.760> does<00:15:59.960> not<00:16:00.200> assume" + }, + { + "start": 960.51, + "duration": 0.0, + "text": "this approach is that it does not assume" + }, + { + "start": 960.52, + "duration": 0.0, + "text": "this approach is that it does not assume these<00:16:00.680> things." + }, + { + "start": 961.63, + "duration": 0.0, + "text": "these things." + }, + { + "start": 961.64, + "duration": 0.0, + "text": "these things. Uh<00:16:01.800> it's<00:16:02.120> much<00:16:02.320> more<00:16:02.440> simpler." + }, + { + "start": 963.55, + "duration": 0.0, + "text": "Uh it's much more simpler." + }, + { + "start": 963.56, + "duration": 0.0, + "text": "Uh it's much more simpler. That's<00:16:03.800> much<00:16:03.920> simpler,<00:16:04.160> sorry." + }, + { + "start": 965.39, + "duration": 0.0, + "text": "That's much simpler, sorry." + }, + { + "start": 965.4, + "duration": 0.0, + "text": "That's much simpler, sorry. So,<00:16:05.760> what<00:16:05.920> is<00:16:06.000> a<00:16:06.040> lattice?<00:16:06.920> Um<00:16:07.240> I'm<00:16:07.360> assuming" + }, + { + "start": 967.99, + "duration": 0.0, + "text": "So, what is a lattice? Um I'm assuming" + }, + { + "start": 968.0, + "duration": 0.0, + "text": "So, what is a lattice? Um I'm assuming most<00:16:08.280> of<00:16:08.400> you<00:16:08.480> have<00:16:08.600> probably<00:16:09.320> seen<00:16:09.600> this" + }, + { + "start": 969.75, + "duration": 0.0, + "text": "most of you have probably seen this" + }, + { + "start": 969.76, + "duration": 0.0, + "text": "most of you have probably seen this before,<00:16:10.280> so<00:16:10.640> I'm<00:16:10.800> going<00:16:10.960> to<00:16:11.040> be<00:16:11.640> rather<00:16:11.800> quick" + }, + { + "start": 971.99, + "duration": 0.0, + "text": "before, so I'm going to be rather quick" + }, + { + "start": 972.0, + "duration": 0.0, + "text": "before, so I'm going to be rather quick here<00:16:12.120> as<00:16:12.240> well." + }, + { + "start": 973.15, + "duration": 0.0, + "text": "here as well." + }, + { + "start": 973.16, + "duration": 0.0, + "text": "here as well. Um<00:16:14.160> a<00:16:14.200> lattice<00:16:14.560> is<00:16:14.760> a<00:16:14.800> kind<00:16:15.080> of<00:16:15.240> partially" + }, + { + "start": 975.59, + "duration": 0.0, + "text": "Um a lattice is a kind of partially" + }, + { + "start": 975.6, + "duration": 0.0, + "text": "Um a lattice is a kind of partially ordered<00:16:15.840> set,<00:16:16.080> or<00:16:16.160> they<00:16:16.240> call<00:16:16.400> it<00:16:16.440> a<00:16:16.480> poset." + }, + { + "start": 977.15, + "duration": 0.0, + "text": "ordered set, or they call it a poset." + }, + { + "start": 977.16, + "duration": 0.0, + "text": "ordered set, or they call it a poset. What<00:16:17.280> is<00:16:17.360> a<00:16:17.400> poset?<00:16:17.840> A<00:16:17.920> poset<00:16:18.400> is<00:16:18.640> a<00:16:18.840> a<00:16:18.880> set<00:16:19.720> with" + }, + { + "start": 979.83, + "duration": 0.0, + "text": "What is a poset? A poset is a a set with" + }, + { + "start": 979.84, + "duration": 0.0, + "text": "What is a poset? A poset is a a set with a<00:16:19.880> binary<00:16:20.200> ordering<00:16:20.520> relation<00:16:21.400> with<00:16:21.680> the" + }, + { + "start": 982.35, + "duration": 0.0, + "text": "a binary ordering relation with the" + }, + { + "start": 982.36, + "duration": 0.0, + "text": "a binary ordering relation with the property<00:16:22.760> of<00:16:22.920> reflexivity,<00:16:23.680> antisymmetry," + }, + { + "start": 984.43, + "duration": 0.0, + "text": "property of reflexivity, antisymmetry," + }, + { + "start": 984.44, + "duration": 0.0, + "text": "property of reflexivity, antisymmetry, and<00:16:24.560> transitivity." + }, + { + "start": 986.829, + "duration": 0.0, + "text": "and transitivity." + }, + { + "start": 986.839, + "duration": 0.0, + "text": "and transitivity. And<00:16:27.600> we<00:16:27.720> can<00:16:27.839> also<00:16:28.080> say<00:16:28.400> that<00:16:28.839> if<00:16:29.080> A<00:16:29.200> is<00:16:29.320> not<00:16:29.560> B" + }, + { + "start": 989.99, + "duration": 0.0, + "text": "And we can also say that if A is not B" + }, + { + "start": 990.0, + "duration": 0.0, + "text": "And we can also say that if A is not B and<00:16:30.640> B<00:16:31.200> is<00:16:31.960> uh<00:16:32.080> above<00:16:32.520> A,<00:16:32.760> then<00:16:33.400> B<00:16:33.560> is<00:16:33.680> strictly" + }, + { + "start": 993.99, + "duration": 0.0, + "text": "and B is uh above A, then B is strictly" + }, + { + "start": 994.0, + "duration": 0.0, + "text": "and B is uh above A, then B is strictly above<00:16:34.320> A." + }, + { + "start": 994.946, + "duration": 0.0, + "text": "above A." + }, + { + "start": 994.956, + "duration": 0.0, + "text": "above A. >> [snorts]" + }, + { + "start": 994.99, + "duration": 0.0, + "text": ">> [snorts]" + }, + { + "start": 995.0, + "duration": 0.0, + "text": ">> [snorts] >> Um" + }, + { + "start": 997.15, + "duration": 0.0, + "text": ">> Um" + }, + { + "start": 997.16, + "duration": 0.0, + "text": ">> Um And<00:16:37.360> we<00:16:37.440> can<00:16:37.640> also<00:16:38.000> use<00:16:38.760> uh<00:16:39.000> this<00:16:39.160> notation" + }, + { + "start": 1000.19, + "duration": 0.0, + "text": "And we can also use uh this notation" + }, + { + "start": 1000.2, + "duration": 0.0, + "text": "And we can also use uh this notation to<00:16:40.400> show<00:16:40.800> that<00:16:41.240> B<00:16:41.560> is<00:16:41.800> sort<00:16:41.960> of<00:16:42.280> the<00:16:42.440> immediate" + }, + { + "start": 1002.829, + "duration": 0.0, + "text": "to show that B is sort of the immediate" + }, + { + "start": 1002.839, + "duration": 0.0, + "text": "to show that B is sort of the immediate superior<00:16:43.600> to<00:16:43.760> A." + }, + { + "start": 1008.88, + "duration": 0.0, + "text": "To<00:16:49.000> understand<00:16:49.240> what<00:16:49.360> a<00:16:49.400> lattice<00:16:49.680> is,<00:16:49.839> you" + }, + { + "start": 1009.91, + "duration": 0.0, + "text": "To understand what a lattice is, you" + }, + { + "start": 1009.92, + "duration": 0.0, + "text": "To understand what a lattice is, you need<00:16:50.160> to<00:16:50.200> understand<00:16:50.600> two<00:16:50.720> more<00:16:50.839> concepts." + }, + { + "start": 1011.95, + "duration": 0.0, + "text": "need to understand two more concepts." + }, + { + "start": 1011.96, + "duration": 0.0, + "text": "need to understand two more concepts. Um<00:16:52.360> imagine<00:16:53.240> uh<00:16:53.600> a<00:16:53.680> subset<00:16:54.839> X<00:16:55.320> of<00:16:55.440> a<00:16:55.720> of<00:16:55.880> a<00:16:55.920> poset" + }, + { + "start": 1016.31, + "duration": 0.0, + "text": "Um imagine uh a subset X of a of a poset" + }, + { + "start": 1016.32, + "duration": 0.0, + "text": "Um imagine uh a subset X of a of a poset P." + }, + { + "start": 1017.15, + "duration": 0.0, + "text": "P." + }, + { + "start": 1017.16, + "duration": 0.0, + "text": "P. We<00:16:57.360> can<00:16:57.880> talk<00:16:58.080> about<00:16:58.320> an<00:16:58.400> element<00:16:59.040> A<00:16:59.360> in<00:16:59.520> P<00:16:59.760> that" + }, + { + "start": 1019.95, + "duration": 0.0, + "text": "We can talk about an element A in P that" + }, + { + "start": 1019.96, + "duration": 0.0, + "text": "We can talk about an element A in P that contains<00:17:00.760> every<00:17:01.000> element<00:17:02.080> uh<00:17:02.320> of<00:17:02.520> X,<00:17:03.280> meaning" + }, + { + "start": 1023.55, + "duration": 0.0, + "text": "contains every element uh of X, meaning" + }, + { + "start": 1023.56, + "duration": 0.0, + "text": "contains every element uh of X, meaning it<00:17:03.680> is<00:17:03.800> superior<00:17:04.240> to<00:17:04.400> all<00:17:04.600> of<00:17:04.720> them<00:17:04.959> in<00:17:05.079> the" + }, + { + "start": 1026.23, + "duration": 0.0, + "text": "it is superior to all of them in the" + }, + { + "start": 1026.24, + "duration": 0.0, + "text": "it is superior to all of them in the in<00:17:06.400> the<00:17:06.480> hierarchy.<00:17:07.400> Then<00:17:07.680> A<00:17:07.839> is<00:17:08.000> called<00:17:08.400> an" + }, + { + "start": 1028.55, + "duration": 0.0, + "text": "in the hierarchy. Then A is called an" + }, + { + "start": 1028.56, + "duration": 0.0, + "text": "in the hierarchy. Then A is called an upper<00:17:08.800> bound<00:17:09.199> of<00:17:09.319> the<00:17:09.400> subset<00:17:09.839> X.<00:17:10.560> Then<00:17:10.760> the" + }, + { + "start": 1030.87, + "duration": 0.0, + "text": "upper bound of the subset X. Then the" + }, + { + "start": 1030.88, + "duration": 0.0, + "text": "upper bound of the subset X. Then the least<00:17:11.280> upper<00:17:11.520> bound<00:17:12.280> is<00:17:12.959> sort<00:17:13.160> of<00:17:13.280> the<00:17:13.640> the" + }, + { + "start": 1033.75, + "duration": 0.0, + "text": "least upper bound is sort of the the" + }, + { + "start": 1033.76, + "duration": 0.0, + "text": "least upper bound is sort of the the notion<00:17:14.199> of<00:17:14.319> the<00:17:14.800> thing<00:17:15.079> that<00:17:15.280> we<00:17:15.400> would<00:17:15.640> most" + }, + { + "start": 1035.87, + "duration": 0.0, + "text": "notion of the thing that we would most" + }, + { + "start": 1035.88, + "duration": 0.0, + "text": "notion of the thing that we would most intuitively<00:17:16.920> associate<00:17:17.400> with<00:17:17.520> an<00:17:17.600> upper" + }, + { + "start": 1037.829, + "duration": 0.0, + "text": "intuitively associate with an upper" + }, + { + "start": 1037.839, + "duration": 0.0, + "text": "intuitively associate with an upper bound,<00:17:18.600> and<00:17:18.760> it<00:17:18.880> is<00:17:19.120> the<00:17:19.640> element<00:17:20.000> in<00:17:20.120> P<00:17:20.600> which" + }, + { + "start": 1040.75, + "duration": 0.0, + "text": "bound, and it is the element in P which" + }, + { + "start": 1040.76, + "duration": 0.0, + "text": "bound, and it is the element in P which is<00:17:20.839> an<00:17:20.959> upper<00:17:21.120> bound<00:17:21.319> of<00:17:21.480> X<00:17:22.079> and<00:17:22.400> is<00:17:22.520> contained" + }, + { + "start": 1042.99, + "duration": 0.0, + "text": "is an upper bound of X and is contained" + }, + { + "start": 1043.0, + "duration": 0.0, + "text": "is an upper bound of X and is contained in<00:17:23.160> every<00:17:23.439> other<00:17:23.720> upper<00:17:23.920> bound<00:17:24.199> of<00:17:24.280> the" + }, + { + "start": 1044.35, + "duration": 0.0, + "text": "in every other upper bound of the" + }, + { + "start": 1044.36, + "duration": 0.0, + "text": "in every other upper bound of the subset.<00:17:24.800> So,<00:17:25.040> it<00:17:25.199> is,<00:17:25.800> as<00:17:25.920> the<00:17:26.040> name<00:17:26.199> suggests," + }, + { + "start": 1046.59, + "duration": 0.0, + "text": "subset. So, it is, as the name suggests," + }, + { + "start": 1046.6, + "duration": 0.0, + "text": "subset. So, it is, as the name suggests, the" + }, + { + "start": 1047.55, + "duration": 0.0, + "text": "the" + }, + { + "start": 1047.56, + "duration": 0.0, + "text": "the the<00:17:27.680> lowest<00:17:28.400> of<00:17:28.560> all<00:17:28.680> the<00:17:28.840> upper<00:17:29.000> bounds." + }, + { + "start": 1050.11, + "duration": 0.0, + "text": "the lowest of all the upper bounds." + }, + { + "start": 1050.12, + "duration": 0.0, + "text": "the lowest of all the upper bounds. And<00:17:31.160> uh<00:17:31.240> dually,<00:17:31.520> we<00:17:31.600> can<00:17:31.760> also<00:17:32.040> define<00:17:32.440> the" + }, + { + "start": 1052.51, + "duration": 0.0, + "text": "And uh dually, we can also define the" + }, + { + "start": 1052.52, + "duration": 0.0, + "text": "And uh dually, we can also define the greatest<00:17:32.880> lower<00:17:33.120> bound,<00:17:33.480> which<00:17:33.680> is<00:17:33.760> simply" + }, + { + "start": 1054.07, + "duration": 0.0, + "text": "greatest lower bound, which is simply" + }, + { + "start": 1054.08, + "duration": 0.0, + "text": "greatest lower bound, which is simply the<00:17:34.200> lower<00:17:34.480> bound<00:17:34.680> that<00:17:34.800> contains<00:17:35.520> all<00:17:35.760> other" + }, + { + "start": 1056.23, + "duration": 0.0, + "text": "the lower bound that contains all other" + }, + { + "start": 1056.24, + "duration": 0.0, + "text": "the lower bound that contains all other lower<00:17:36.480> bounds.<00:17:36.840> We<00:17:36.920> must<00:17:37.120> invert<00:17:37.480> the" + }, + { + "start": 1057.67, + "duration": 0.0, + "text": "lower bounds. We must invert the" + }, + { + "start": 1057.68, + "duration": 0.0, + "text": "lower bounds. We must invert the containment<00:17:38.160> operation.<00:17:39.240> And<00:17:39.320> a<00:17:39.360> lattice<00:17:39.680> is" + }, + { + "start": 1059.75, + "duration": 0.0, + "text": "containment operation. And a lattice is" + }, + { + "start": 1059.76, + "duration": 0.0, + "text": "containment operation. And a lattice is simply<00:17:40.040> a<00:17:40.120> poset<00:17:40.560> for<00:17:40.760> which<00:17:40.960> the<00:17:41.040> least<00:17:41.640> and" + }, + { + "start": 1062.15, + "duration": 0.0, + "text": "simply a poset for which the least and" + }, + { + "start": 1062.16, + "duration": 0.0, + "text": "simply a poset for which the least and upper<00:17:42.400> bound<00:17:42.720> and<00:17:42.800> the<00:17:42.880> greatest<00:17:43.160> lower<00:17:43.360> bound" + }, + { + "start": 1063.59, + "duration": 0.0, + "text": "upper bound and the greatest lower bound" + }, + { + "start": 1063.6, + "duration": 0.0, + "text": "upper bound and the greatest lower bound exist<00:17:44.000> for<00:17:44.160> all<00:17:44.360> pairs<00:17:44.600> of<00:17:44.720> elements<00:17:45.600> in<00:17:45.720> the" + }, + { + "start": 1065.79, + "duration": 0.0, + "text": "exist for all pairs of elements in the" + }, + { + "start": 1065.8, + "duration": 0.0, + "text": "exist for all pairs of elements in the set." + }, + { + "start": 1069.72, + "duration": 0.0, + "text": "The<00:17:50.320> lower<00:17:50.720> upper<00:17:50.960> bound<00:17:51.160> between<00:17:51.560> A<00:17:51.640> and<00:17:51.760> B<00:17:52.000> is" + }, + { + "start": 1072.11, + "duration": 0.0, + "text": "The lower upper bound between A and B is" + }, + { + "start": 1072.12, + "duration": 0.0, + "text": "The lower upper bound between A and B is commonly<00:17:52.400> denoted<00:17:52.840> as,<00:17:53.760> again,<00:17:54.080> with<00:17:54.280> this" + }, + { + "start": 1074.43, + "duration": 0.0, + "text": "commonly denoted as, again, with this" + }, + { + "start": 1074.44, + "duration": 0.0, + "text": "commonly denoted as, again, with this kind<00:17:54.600> of<00:17:54.720> valley<00:17:54.960> notation,<00:17:55.400> which<00:17:55.520> is<00:17:55.600> called" + }, + { + "start": 1075.79, + "duration": 0.0, + "text": "kind of valley notation, which is called" + }, + { + "start": 1075.8, + "duration": 0.0, + "text": "kind of valley notation, which is called the<00:17:55.880> join<00:17:56.200> operation,<00:17:57.160> whereas<00:17:57.480> the<00:17:57.520> greatest" + }, + { + "start": 1077.79, + "duration": 0.0, + "text": "the join operation, whereas the greatest" + }, + { + "start": 1077.8, + "duration": 0.0, + "text": "the join operation, whereas the greatest lower<00:17:58.080> bound<00:17:58.560> is<00:17:59.040> this<00:17:59.280> hat,<00:18:00.200> meet,<00:18:00.560> and<00:18:00.960> these" + }, + { + "start": 1081.23, + "duration": 0.0, + "text": "lower bound is this hat, meet, and these" + }, + { + "start": 1081.24, + "duration": 0.0, + "text": "lower bound is this hat, meet, and these symbols<00:18:01.720> mirror<00:18:02.040> those<00:18:02.360> used<00:18:02.560> in<00:18:02.680> Boolean" + }, + { + "start": 1082.99, + "duration": 0.0, + "text": "symbols mirror those used in Boolean" + }, + { + "start": 1083.0, + "duration": 0.0, + "text": "symbols mirror those used in Boolean algebra," + }, + { + "start": 1084.31, + "duration": 0.0, + "text": "algebra," + }, + { + "start": 1084.32, + "duration": 0.0, + "text": "algebra, uh<00:18:04.560> because" + }, + { + "start": 1085.91, + "duration": 0.0, + "text": "uh because" + }, + { + "start": 1085.92, + "duration": 0.0, + "text": "uh because uh<00:18:06.240> when<00:18:06.480> you<00:18:06.720> treat<00:18:07.240> propositions<00:18:07.840> as" + }, + { + "start": 1087.95, + "duration": 0.0, + "text": "uh when you treat propositions as" + }, + { + "start": 1087.96, + "duration": 0.0, + "text": "uh when you treat propositions as ordered<00:18:08.160> by<00:18:08.240> implication,<00:18:08.840> the<00:18:09.000> logical<00:18:09.440> or" + }, + { + "start": 1089.83, + "duration": 0.0, + "text": "ordered by implication, the logical or" + }, + { + "start": 1089.84, + "duration": 0.0, + "text": "ordered by implication, the logical or and<00:18:09.960> logical<00:18:10.360> and<00:18:10.560> operation<00:18:11.360> act<00:18:11.720> exactly<00:18:12.400> as" + }, + { + "start": 1092.63, + "duration": 0.0, + "text": "and logical and operation act exactly as" + }, + { + "start": 1092.64, + "duration": 0.0, + "text": "and logical and operation act exactly as the<00:18:12.760> join<00:18:13.120> and<00:18:13.280> the<00:18:13.360> meet<00:18:13.560> operation." + }, + { + "start": 1094.75, + "duration": 0.0, + "text": "the join and the meet operation." + }, + { + "start": 1094.76, + "duration": 0.0, + "text": "the join and the meet operation. So,<00:18:14.880> there's<00:18:15.160> this<00:18:15.600> uh<00:18:15.720> nice<00:18:15.920> correspondence" + }, + { + "start": 1096.83, + "duration": 0.0, + "text": "So, there's this uh nice correspondence" + }, + { + "start": 1096.84, + "duration": 0.0, + "text": "So, there's this uh nice correspondence that<00:18:17.040> also<00:18:17.240> makes<00:18:17.440> the<00:18:17.480> notation<00:18:18.080> much<00:18:18.680> uh" + }, + { + "start": 1098.83, + "duration": 0.0, + "text": "that also makes the notation much uh" + }, + { + "start": 1098.84, + "duration": 0.0, + "text": "that also makes the notation much uh nicer<00:18:19.160> for<00:18:19.280> us<00:18:19.400> to<00:18:19.520> use." + }, + { + "start": 1100.83, + "duration": 0.0, + "text": "nicer for us to use." + }, + { + "start": 1100.84, + "duration": 0.0, + "text": "nicer for us to use. Uh<00:18:21.160> then<00:18:21.360> also<00:18:21.640> we<00:18:21.760> speak<00:18:22.040> of<00:18:22.160> a<00:18:22.200> distributive" + }, + { + "start": 1102.71, + "duration": 0.0, + "text": "Uh then also we speak of a distributive" + }, + { + "start": 1102.72, + "duration": 0.0, + "text": "Uh then also we speak of a distributive lattice<00:18:23.080> if<00:18:23.280> we<00:18:23.400> have<00:18:23.560> some<00:18:23.680> kind<00:18:23.840> of" + }, + { + "start": 1103.91, + "duration": 0.0, + "text": "lattice if we have some kind of" + }, + { + "start": 1103.92, + "duration": 0.0, + "text": "lattice if we have some kind of distributivity<00:18:25.160> property" + }, + { + "start": 1106.43, + "duration": 0.0, + "text": "distributivity property" + }, + { + "start": 1106.44, + "duration": 0.0, + "text": "distributivity property uh<00:18:26.760> of<00:18:26.960> the" + }, + { + "start": 1108.03, + "duration": 0.0, + "text": "uh of the" + }, + { + "start": 1108.04, + "duration": 0.0, + "text": "uh of the and<00:18:28.240> over<00:18:28.400> the<00:18:28.520> or." + }, + { + "start": 1111.64, + "duration": 0.0, + "text": "And<00:18:32.160> there's<00:18:32.480> also<00:18:32.800> an<00:18:33.200> even<00:18:33.440> more" + }, + { + "start": 1113.59, + "duration": 0.0, + "text": "And there's also an even more" + }, + { + "start": 1113.6, + "duration": 0.0, + "text": "And there's also an even more restrictive<00:18:34.000> class<00:18:34.280> of<00:18:34.400> lattices<00:18:34.920> which<00:18:35.080> are" + }, + { + "start": 1115.15, + "duration": 0.0, + "text": "restrictive class of lattices which are" + }, + { + "start": 1115.16, + "duration": 0.0, + "text": "restrictive class of lattices which are called<00:18:35.440> Boolean<00:18:35.800> lattices<00:18:36.280> for<00:18:36.520> which<00:18:37.000> each" + }, + { + "start": 1117.23, + "duration": 0.0, + "text": "called Boolean lattices for which each" + }, + { + "start": 1117.24, + "duration": 0.0, + "text": "called Boolean lattices for which each element<00:18:37.600> has<00:18:37.720> a<00:18:37.800> complement.<00:18:38.880> And<00:18:39.080> a" + }, + { + "start": 1119.11, + "duration": 0.0, + "text": "element has a complement. And a" + }, + { + "start": 1119.12, + "duration": 0.0, + "text": "element has a complement. And a complement<00:18:39.600> is<00:18:39.720> simply<00:18:40.160> an<00:18:40.320> element<00:18:41.400> um" + }, + { + "start": 1122.75, + "duration": 0.0, + "text": "complement is simply an element um" + }, + { + "start": 1122.76, + "duration": 0.0, + "text": "complement is simply an element um for<00:18:42.920> which<00:18:43.600> the<00:18:43.800> join<00:18:44.760> is<00:18:44.920> the<00:18:45.000> top<00:18:45.240> element" + }, + { + "start": 1125.75, + "duration": 0.0, + "text": "for which the join is the top element" + }, + { + "start": 1125.76, + "duration": 0.0, + "text": "for which the join is the top element and<00:18:46.000> the<00:18:46.280> the<00:18:46.400> meet<00:18:46.720> is<00:18:46.920> the<00:18:47.280> most<00:18:47.480> bottom" + }, + { + "start": 1127.75, + "duration": 0.0, + "text": "and the the meet is the most bottom" + }, + { + "start": 1127.76, + "duration": 0.0, + "text": "and the the meet is the most bottom element.<00:18:48.280> That's<00:18:48.440> simply<00:18:48.680> what<00:18:48.800> that<00:18:48.920> means." + }, + { + "start": 1129.27, + "duration": 0.0, + "text": "element. That's simply what that means." + }, + { + "start": 1129.28, + "duration": 0.0, + "text": "element. That's simply what that means. However,<00:18:50.200> in<00:18:50.320> this<00:18:50.520> derivation,<00:18:51.120> we're<00:18:51.240> not" + }, + { + "start": 1131.55, + "duration": 0.0, + "text": "However, in this derivation, we're not" + }, + { + "start": 1131.56, + "duration": 0.0, + "text": "However, in this derivation, we're not going<00:18:51.680> to<00:18:51.760> need<00:18:52.240> Boolean<00:18:52.480> lattices." + }, + { + "start": 1133.03, + "duration": 0.0, + "text": "going to need Boolean lattices." + }, + { + "start": 1133.04, + "duration": 0.0, + "text": "going to need Boolean lattices. Distributive<00:18:53.680> lattices<00:18:54.160> are<00:18:54.360> completely" + }, + { + "start": 1134.71, + "duration": 0.0, + "text": "Distributive lattices are completely" + }, + { + "start": 1134.72, + "duration": 0.0, + "text": "Distributive lattices are completely sufficient." + }, + { + "start": 1136.11, + "duration": 0.0, + "text": "sufficient." + }, + { + "start": 1136.12, + "duration": 0.0, + "text": "sufficient. Uh<00:18:56.240> which<00:18:56.440> is<00:18:57.080> has<00:18:57.280> some<00:18:57.480> practical" + }, + { + "start": 1137.79, + "duration": 0.0, + "text": "Uh which is has some practical" + }, + { + "start": 1137.8, + "duration": 0.0, + "text": "Uh which is has some practical implication,<00:18:58.280> but<00:18:58.880> this<00:18:59.080> is<00:18:59.800> not<00:18:59.960> important" + }, + { + "start": 1140.27, + "duration": 0.0, + "text": "implication, but this is not important" + }, + { + "start": 1140.28, + "duration": 0.0, + "text": "implication, but this is not important right<00:19:00.400> now.<00:19:01.000> So,<00:19:01.520> now<00:19:01.800> we're<00:19:01.920> going<00:19:02.080> to<00:19:02.120> define" + }, + { + "start": 1142.87, + "duration": 0.0, + "text": "right now. So, now we're going to define" + }, + { + "start": 1142.88, + "duration": 0.0, + "text": "right now. So, now we're going to define the<00:19:03.200> objective<00:19:03.840> of<00:19:03.960> this<00:19:04.080> derivation.<00:19:05.080> We" + }, + { + "start": 1145.43, + "duration": 0.0, + "text": "the objective of this derivation. We" + }, + { + "start": 1145.44, + "duration": 0.0, + "text": "the objective of this derivation. We want<00:19:05.760> to<00:19:06.240> basically<00:19:06.640> generalize<00:19:07.320> the<00:19:07.720> zeta" + }, + { + "start": 1147.99, + "duration": 0.0, + "text": "want to basically generalize the zeta" + }, + { + "start": 1148.0, + "duration": 0.0, + "text": "want to basically generalize the zeta function.<00:19:08.360> The<00:19:08.440> zeta<00:19:08.680> function" + }, + { + "start": 1149.83, + "duration": 0.0, + "text": "function. The zeta function" + }, + { + "start": 1149.84, + "duration": 0.0, + "text": "function. The zeta function uh<00:19:10.000> Uh,<00:19:10.200> is<00:19:10.880> in<00:19:11.080> its<00:19:11.360> classical<00:19:11.800> form<00:19:12.200> just<00:19:12.720> an" + }, + { + "start": 1152.91, + "duration": 0.0, + "text": "uh Uh, is in its classical form just an" + }, + { + "start": 1152.92, + "duration": 0.0, + "text": "uh Uh, is in its classical form just an indicator<00:19:13.680> that<00:19:13.800> tells<00:19:14.080> us<00:19:14.200> if<00:19:14.320> an<00:19:14.440> element<00:19:14.840> is" + }, + { + "start": 1155.35, + "duration": 0.0, + "text": "indicator that tells us if an element is" + }, + { + "start": 1155.36, + "duration": 0.0, + "text": "indicator that tells us if an element is below<00:19:16.360> uh<00:19:16.440> or<00:19:16.600> equal<00:19:16.800> to<00:19:16.920> another<00:19:17.240> and<00:19:17.400> zero" + }, + { + "start": 1157.75, + "duration": 0.0, + "text": "below uh or equal to another and zero" + }, + { + "start": 1157.76, + "duration": 0.0, + "text": "below uh or equal to another and zero otherwise." + }, + { + "start": 1158.91, + "duration": 0.0, + "text": "otherwise." + }, + { + "start": 1158.92, + "duration": 0.0, + "text": "otherwise. Uh,<00:19:19.080> in<00:19:19.200> our<00:19:19.320> context,<00:19:19.760> we're<00:19:19.880> looking<00:19:20.160> to" + }, + { + "start": 1160.23, + "duration": 0.0, + "text": "Uh, in our context, we're looking to" + }, + { + "start": 1160.24, + "duration": 0.0, + "text": "Uh, in our context, we're looking to kind<00:19:20.480> of<00:19:20.600> generalize<00:19:21.120> the<00:19:21.240> inverse,<00:19:21.880> so<00:19:22.200> the" + }, + { + "start": 1162.27, + "duration": 0.0, + "text": "kind of generalize the inverse, so the" + }, + { + "start": 1162.28, + "duration": 0.0, + "text": "kind of generalize the inverse, so the one<00:19:22.520> that<00:19:22.640> tells<00:19:22.960> us<00:19:23.680> um" + }, + { + "start": 1164.83, + "duration": 0.0, + "text": "one that tells us um" + }, + { + "start": 1164.84, + "duration": 0.0, + "text": "one that tells us um if<00:19:24.920> a<00:19:25.000> proposition<00:19:25.440> is<00:19:25.640> above.<00:19:26.560> However,<00:19:27.000> we" + }, + { + "start": 1167.23, + "duration": 0.0, + "text": "if a proposition is above. However, we" + }, + { + "start": 1167.24, + "duration": 0.0, + "text": "if a proposition is above. However, we want<00:19:27.560> it<00:19:27.720> to<00:19:27.800> be<00:19:27.920> not<00:19:28.120> only<00:19:28.280> just<00:19:28.440> a<00:19:28.520> binary" + }, + { + "start": 1168.87, + "duration": 0.0, + "text": "want it to be not only just a binary" + }, + { + "start": 1168.88, + "duration": 0.0, + "text": "want it to be not only just a binary indicator,<00:19:29.600> but<00:19:29.920> to<00:19:30.080> also<00:19:30.320> have<00:19:30.440> some<00:19:30.560> kind<00:19:30.840> of" + }, + { + "start": 1171.35, + "duration": 0.0, + "text": "indicator, but to also have some kind of" + }, + { + "start": 1171.36, + "duration": 0.0, + "text": "indicator, but to also have some kind of um" + }, + { + "start": 1172.59, + "duration": 0.0, + "text": "um" + }, + { + "start": 1172.6, + "duration": 0.0, + "text": "um some<00:19:32.760> continuity,<00:19:33.440> meaning<00:19:33.720> we<00:19:33.840> have<00:19:34.160> some" + }, + { + "start": 1174.79, + "duration": 0.0, + "text": "some continuity, meaning we have some" + }, + { + "start": 1174.8, + "duration": 0.0, + "text": "some continuity, meaning we have some degree<00:19:35.280> of<00:19:35.440> implication.<00:19:36.160> This<00:19:36.280> is<00:19:36.360> what" + }, + { + "start": 1176.47, + "duration": 0.0, + "text": "degree of implication. This is what" + }, + { + "start": 1176.48, + "duration": 0.0, + "text": "degree of implication. This is what we're<00:19:36.560> looking<00:19:36.800> for.<00:19:37.040> So,<00:19:37.240> we're<00:19:37.360> looking<00:19:37.600> to" + }, + { + "start": 1177.75, + "duration": 0.0, + "text": "we're looking for. So, we're looking to" + }, + { + "start": 1177.76, + "duration": 0.0, + "text": "we're looking for. So, we're looking to have<00:19:38.000> something<00:19:38.320> like<00:19:38.480> this<00:19:38.640> function<00:19:38.960> Z" + }, + { + "start": 1180.11, + "duration": 0.0, + "text": "have something like this function Z" + }, + { + "start": 1180.12, + "duration": 0.0, + "text": "have something like this function Z such<00:19:40.400> that<00:19:40.640> it<00:19:40.760> is<00:19:41.000> one<00:19:41.400> if<00:19:41.560> the<00:19:41.680> element<00:19:42.040> X<00:19:42.280> is" + }, + { + "start": 1182.43, + "duration": 0.0, + "text": "such that it is one if the element X is" + }, + { + "start": 1182.44, + "duration": 0.0, + "text": "such that it is one if the element X is above<00:19:42.840> T." + }, + { + "start": 1183.83, + "duration": 0.0, + "text": "above T." + }, + { + "start": 1183.84, + "duration": 0.0, + "text": "above T. It<00:19:44.000> is<00:19:44.160> zero<00:19:44.920> if<00:19:45.160> the<00:19:45.320> two<00:19:45.720> meet<00:19:46.000> at<00:19:46.080> the<00:19:46.160> bottom" + }, + { + "start": 1186.47, + "duration": 0.0, + "text": "It is zero if the two meet at the bottom" + }, + { + "start": 1186.48, + "duration": 0.0, + "text": "It is zero if the two meet at the bottom of<00:19:46.600> the<00:19:46.680> lattice," + }, + { + "start": 1187.71, + "duration": 0.0, + "text": "of the lattice," + }, + { + "start": 1187.72, + "duration": 0.0, + "text": "of the lattice, meaning" + }, + { + "start": 1188.99, + "duration": 0.0, + "text": "meaning" + }, + { + "start": 1189.0, + "duration": 0.0, + "text": "meaning uh<00:19:49.120> they<00:19:49.280> don't<00:19:49.520> imply<00:19:49.800> each<00:19:50.000> other<00:19:50.240> at<00:19:50.400> all." + }, + { + "start": 1191.03, + "duration": 0.0, + "text": "uh they don't imply each other at all." + }, + { + "start": 1191.04, + "duration": 0.0, + "text": "uh they don't imply each other at all. And<00:19:51.600> we<00:19:51.720> have<00:19:51.880> some<00:19:52.040> value<00:19:52.280> between<00:19:52.600> zero<00:19:52.880> and" + }, + { + "start": 1193.03, + "duration": 0.0, + "text": "And we have some value between zero and" + }, + { + "start": 1193.04, + "duration": 0.0, + "text": "And we have some value between zero and one<00:19:53.480> otherwise." + }, + { + "start": 1195.67, + "duration": 0.0, + "text": "one otherwise." + }, + { + "start": 1195.68, + "duration": 0.0, + "text": "one otherwise. And<00:19:56.160> this<00:19:56.640> generally<00:19:57.200> this<00:19:57.480> generalization" + }, + { + "start": 1198.07, + "duration": 0.0, + "text": "And this generally this generalization" + }, + { + "start": 1198.08, + "duration": 0.0, + "text": "And this generally this generalization of<00:19:58.160> the<00:19:58.400> inverse<00:19:58.720> zeta<00:19:58.920> function<00:19:59.280> is<00:19:59.480> then" + }, + { + "start": 1199.67, + "duration": 0.0, + "text": "of the inverse zeta function is then" + }, + { + "start": 1199.68, + "duration": 0.0, + "text": "of the inverse zeta function is then what<00:19:59.800> we're<00:19:59.880> going<00:20:00.000> to<00:20:00.080> call<00:20:00.320> probability.<00:20:01.480> It" + }, + { + "start": 1201.91, + "duration": 0.0, + "text": "what we're going to call probability. It" + }, + { + "start": 1201.92, + "duration": 0.0, + "text": "what we're going to call probability. It respects<00:20:02.200> the<00:20:02.240> ordering<00:20:02.480> of<00:20:02.560> the<00:20:02.640> zeta" + }, + { + "start": 1202.83, + "duration": 0.0, + "text": "respects the ordering of the zeta" + }, + { + "start": 1202.84, + "duration": 0.0, + "text": "respects the ordering of the zeta function,<00:20:03.160> but<00:20:03.320> allows<00:20:03.800> for<00:20:04.200> incomplete" + }, + { + "start": 1204.67, + "duration": 0.0, + "text": "function, but allows for incomplete" + }, + { + "start": 1204.68, + "duration": 0.0, + "text": "function, but allows for incomplete information." + }, + { + "start": 1207.23, + "duration": 0.0, + "text": "information." + }, + { + "start": 1207.24, + "duration": 0.0, + "text": "information. And<00:20:07.760> we're<00:20:07.920> going<00:20:08.080> to<00:20:08.600> derive<00:20:09.480> the<00:20:09.640> rules<00:20:09.800> of" + }, + { + "start": 1209.91, + "duration": 0.0, + "text": "And we're going to derive the rules of" + }, + { + "start": 1209.92, + "duration": 0.0, + "text": "And we're going to derive the rules of probability<00:20:10.480> by<00:20:10.640> looking<00:20:10.960> at<00:20:11.120> some" + }, + { + "start": 1211.47, + "duration": 0.0, + "text": "probability by looking at some" + }, + { + "start": 1211.48, + "duration": 0.0, + "text": "probability by looking at some symmetries<00:20:12.360> in<00:20:12.480> these<00:20:12.640> lattices.<00:20:13.520> The<00:20:13.600> most" + }, + { + "start": 1214.07, + "duration": 0.0, + "text": "symmetries in these lattices. The most" + }, + { + "start": 1214.08, + "duration": 0.0, + "text": "symmetries in these lattices. The most the<00:20:14.160> first<00:20:14.400> one<00:20:14.600> is<00:20:14.840> not<00:20:15.000> really<00:20:15.200> a<00:20:15.240> symmetry," + }, + { + "start": 1215.63, + "duration": 0.0, + "text": "the first one is not really a symmetry," + }, + { + "start": 1215.64, + "duration": 0.0, + "text": "the first one is not really a symmetry, it's<00:20:15.800> more<00:20:15.920> of<00:20:16.000> a<00:20:16.080> convention,<00:20:17.080> and<00:20:17.400> it's" + }, + { + "start": 1217.55, + "duration": 0.0, + "text": "it's more of a convention, and it's" + }, + { + "start": 1217.56, + "duration": 0.0, + "text": "it's more of a convention, and it's simply<00:20:17.880> that<00:20:18.280> elements<00:20:18.720> that<00:20:18.880> are<00:20:19.080> higher<00:20:19.520> up" + }, + { + "start": 1219.99, + "duration": 0.0, + "text": "simply that elements that are higher up" + }, + { + "start": 1220.0, + "duration": 0.0, + "text": "simply that elements that are higher up in<00:20:20.120> the<00:20:20.200> order<00:20:20.640> in<00:20:20.760> the<00:20:20.880> hierarchy<00:20:21.880> uh<00:20:22.240> are" + }, + { + "start": 1222.35, + "duration": 0.0, + "text": "in the order in the hierarchy uh are" + }, + { + "start": 1222.36, + "duration": 0.0, + "text": "in the order in the hierarchy uh are just<00:20:23.320> evaluated<00:20:24.000> by<00:20:24.480> higher<00:20:25.000> real<00:20:25.280> numbers." + }, + { + "start": 1225.99, + "duration": 0.0, + "text": "just evaluated by higher real numbers." + }, + { + "start": 1226.0, + "duration": 0.0, + "text": "just evaluated by higher real numbers. That's<00:20:26.400> all<00:20:26.520> it<00:20:26.600> means.<00:20:26.880> And<00:20:27.160> in<00:20:27.320> general,<00:20:27.760> for" + }, + { + "start": 1227.87, + "duration": 0.0, + "text": "That's all it means. And in general, for" + }, + { + "start": 1227.88, + "duration": 0.0, + "text": "That's all it means. And in general, for the<00:20:28.000> rest<00:20:28.280> of<00:20:28.640> the<00:20:28.760> presentation,<00:20:29.880> uh" + }, + { + "start": 1230.55, + "duration": 0.0, + "text": "the rest of the presentation, uh" + }, + { + "start": 1230.56, + "duration": 0.0, + "text": "the rest of the presentation, uh the<00:20:30.680> capital<00:20:31.040> letters<00:20:31.480> will<00:20:31.960> represent" + }, + { + "start": 1232.43, + "duration": 0.0, + "text": "the capital letters will represent" + }, + { + "start": 1232.44, + "duration": 0.0, + "text": "the capital letters will represent lattice<00:20:32.720> elements," + }, + { + "start": 1233.71, + "duration": 0.0, + "text": "lattice elements," + }, + { + "start": 1233.72, + "duration": 0.0, + "text": "lattice elements, and<00:20:34.600> small" + }, + { + "start": 1235.71, + "duration": 0.0, + "text": "and small" + }, + { + "start": 1235.72, + "duration": 0.0, + "text": "and small letters<00:20:36.120> will<00:20:36.240> represent<00:20:36.760> the<00:20:36.880> real<00:20:37.120> numbers," + }, + { + "start": 1237.47, + "duration": 0.0, + "text": "letters will represent the real numbers," + }, + { + "start": 1237.48, + "duration": 0.0, + "text": "letters will represent the real numbers, which<00:20:37.600> correspond<00:20:38.040> to<00:20:38.120> their<00:20:38.760> evaluations." + }, + { + "start": 1241.76, + "duration": 0.0, + "text": "Now,<00:20:41.960> the<00:20:42.080> first<00:20:42.320> symmetry<00:20:42.760> is<00:20:43.000> that<00:20:43.240> the" + }, + { + "start": 1243.31, + "duration": 0.0, + "text": "Now, the first symmetry is that the" + }, + { + "start": 1243.32, + "duration": 0.0, + "text": "Now, the first symmetry is that the combination<00:20:44.640> uh<00:20:45.200> preserves<00:20:45.760> order<00:20:46.000> from<00:20:46.160> the" + }, + { + "start": 1246.27, + "duration": 0.0, + "text": "combination uh preserves order from the" + }, + { + "start": 1246.28, + "duration": 0.0, + "text": "combination uh preserves order from the right<00:20:46.520> and<00:20:46.640> from<00:20:46.840> the<00:20:46.920> left.<00:20:47.280> So,<00:20:47.560> if<00:20:47.720> we<00:20:47.800> have" + }, + { + "start": 1248.35, + "duration": 0.0, + "text": "right and from the left. So, if we have" + }, + { + "start": 1248.36, + "duration": 0.0, + "text": "right and from the left. So, if we have two<00:20:48.520> elements,<00:20:49.000> one<00:20:49.120> is<00:20:49.200> strictly<00:20:49.880> above<00:20:50.120> the" + }, + { + "start": 1250.23, + "duration": 0.0, + "text": "two elements, one is strictly above the" + }, + { + "start": 1250.24, + "duration": 0.0, + "text": "two elements, one is strictly above the other,<00:20:51.040> then<00:20:52.000> uh" + }, + { + "start": 1252.55, + "duration": 0.0, + "text": "other, then uh" + }, + { + "start": 1252.56, + "duration": 0.0, + "text": "other, then uh the<00:20:52.680> join<00:20:52.960> operation" + }, + { + "start": 1254.27, + "duration": 0.0, + "text": "the join operation" + }, + { + "start": 1254.28, + "duration": 0.0, + "text": "the join operation uh<00:20:54.400> makes<00:20:54.680> it<00:20:54.800> so<00:20:55.080> that<00:20:55.600> kind<00:20:55.760> of<00:20:56.160> the<00:20:56.280> compound" + }, + { + "start": 1256.71, + "duration": 0.0, + "text": "uh makes it so that kind of the compound" + }, + { + "start": 1256.72, + "duration": 0.0, + "text": "uh makes it so that kind of the compound statements<00:20:57.840> uh<00:20:57.920> also<00:20:58.240> have<00:20:58.560> the<00:20:59.000> preserve<00:20:59.320> the" + }, + { + "start": 1259.43, + "duration": 0.0, + "text": "statements uh also have the preserve the" + }, + { + "start": 1259.44, + "duration": 0.0, + "text": "statements uh also have the preserve the order,<00:20:59.720> and<00:21:00.000> no<00:21:00.160> matter<00:21:00.480> from<00:21:00.640> which<00:21:00.840> side<00:21:01.240> you" + }, + { + "start": 1261.39, + "duration": 0.0, + "text": "order, and no matter from which side you" + }, + { + "start": 1261.4, + "duration": 0.0, + "text": "order, and no matter from which side you add<00:21:01.680> new<00:21:01.920> element.<00:21:02.920> And<00:21:03.560> by<00:21:03.720> extension,<00:21:04.280> this" + }, + { + "start": 1264.47, + "duration": 0.0, + "text": "add new element. And by extension, this" + }, + { + "start": 1264.48, + "duration": 0.0, + "text": "add new element. And by extension, this must<00:21:04.680> also<00:21:04.920> hold<00:21:05.240> for<00:21:05.360> the<00:21:05.480> operations<00:21:06.280> that" + }, + { + "start": 1266.39, + "duration": 0.0, + "text": "must also hold for the operations that" + }, + { + "start": 1266.4, + "duration": 0.0, + "text": "must also hold for the operations that quantifies<00:21:07.560> the<00:21:07.640> join<00:21:07.960> operation<00:21:08.680> of<00:21:08.920> these" + }, + { + "start": 1269.39, + "duration": 0.0, + "text": "quantifies the join operation of these" + }, + { + "start": 1269.4, + "duration": 0.0, + "text": "quantifies the join operation of these disjoint<00:21:09.800> elements.<00:21:10.240> So,<00:21:10.520> if<00:21:10.720> you<00:21:10.800> have<00:21:11.080> this" + }, + { + "start": 1272.63, + "duration": 0.0, + "text": "disjoint elements. So, if you have this" + }, + { + "start": 1272.64, + "duration": 0.0, + "text": "disjoint elements. So, if you have this uh<00:21:12.720> state<00:21:13.080> this<00:21:13.360> proposition<00:21:13.880> D,<00:21:14.600> which<00:21:14.760> is<00:21:14.920> A" + }, + { + "start": 1275.03, + "duration": 0.0, + "text": "uh state this proposition D, which is A" + }, + { + "start": 1275.04, + "duration": 0.0, + "text": "uh state this proposition D, which is A or<00:21:15.160> C,<00:21:16.000> then<00:21:16.120> the<00:21:16.160> valuation<00:21:16.680> of<00:21:16.800> D<00:21:16.960> must<00:21:17.200> be" + }, + { + "start": 1277.27, + "duration": 0.0, + "text": "or C, then the valuation of D must be" + }, + { + "start": 1277.28, + "duration": 0.0, + "text": "or C, then the valuation of D must be somehow<00:21:17.640> a<00:21:17.720> combination<00:21:18.240> of<00:21:18.360> the<00:21:18.440> valuation" + }, + { + "start": 1278.79, + "duration": 0.0, + "text": "somehow a combination of the valuation" + }, + { + "start": 1278.8, + "duration": 0.0, + "text": "somehow a combination of the valuation of<00:21:19.000> A<00:21:19.120> and<00:21:19.200> the<00:21:19.280> combination<00:21:19.600> of<00:21:19.680> C<00:21:19.880> for<00:21:20.000> this" + }, + { + "start": 1281.03, + "duration": 0.0, + "text": "of A and the combination of C for this" + }, + { + "start": 1281.04, + "duration": 0.0, + "text": "of A and the combination of C for this uh" + }, + { + "start": 1281.79, + "duration": 0.0, + "text": "uh" + }, + { + "start": 1281.8, + "duration": 0.0, + "text": "uh kind<00:21:21.920> of<00:21:22.040> plus<00:21:22.240> operator<00:21:22.600> that<00:21:22.760> we<00:21:22.920> will<00:21:23.080> see" + }, + { + "start": 1283.67, + "duration": 0.0, + "text": "kind of plus operator that we will see" + }, + { + "start": 1283.68, + "duration": 0.0, + "text": "kind of plus operator that we will see is<00:21:23.880> going<00:21:24.000> to<00:21:24.040> turn<00:21:24.240> out<00:21:24.360> to<00:21:24.440> be<00:21:24.560> the<00:21:24.640> sum.<00:21:25.360> And" + }, + { + "start": 1285.47, + "duration": 0.0, + "text": "is going to turn out to be the sum. And" + }, + { + "start": 1285.48, + "duration": 0.0, + "text": "is going to turn out to be the sum. And so,<00:21:25.560> here<00:21:25.760> we<00:21:25.840> have<00:21:25.960> the<00:21:26.040> same<00:21:26.280> symmetry" + }, + { + "start": 1286.59, + "duration": 0.0, + "text": "so, here we have the same symmetry" + }, + { + "start": 1286.6, + "duration": 0.0, + "text": "so, here we have the same symmetry reflected<00:21:27.560> uh<00:21:27.800> with<00:21:27.960> it." + }, + { + "start": 1289.75, + "duration": 0.0, + "text": "reflected uh with it." + }, + { + "start": 1289.76, + "duration": 0.0, + "text": "reflected uh with it. And<00:21:29.880> this<00:21:30.040> basically<00:21:30.360> means<00:21:30.760> that<00:21:31.040> the" + }, + { + "start": 1291.19, + "duration": 0.0, + "text": "And this basically means that the" + }, + { + "start": 1291.2, + "duration": 0.0, + "text": "And this basically means that the ordering<00:21:31.920> has<00:21:32.120> to<00:21:32.200> survive<00:21:32.560> a<00:21:32.640> combination" + }, + { + "start": 1293.35, + "duration": 0.0, + "text": "ordering has to survive a combination" + }, + { + "start": 1293.36, + "duration": 0.0, + "text": "ordering has to survive a combination with<00:21:33.680> any<00:21:33.880> arbitrary<00:21:34.360> context.<00:21:34.880> Otherwise," + }, + { + "start": 1295.63, + "duration": 0.0, + "text": "with any arbitrary context. Otherwise," + }, + { + "start": 1295.64, + "duration": 0.0, + "text": "with any arbitrary context. Otherwise, it<00:21:35.800> it's<00:21:36.240> basically<00:21:36.640> useless<00:21:37.200> for<00:21:37.600> any<00:21:37.760> kind" + }, + { + "start": 1297.91, + "duration": 0.0, + "text": "it it's basically useless for any kind" + }, + { + "start": 1297.92, + "duration": 0.0, + "text": "it it's basically useless for any kind of<00:21:38.000> reasoning." + }, + { + "start": 1299.23, + "duration": 0.0, + "text": "of reasoning." + }, + { + "start": 1299.24, + "duration": 0.0, + "text": "of reasoning. Uh<00:21:39.480> to<00:21:39.560> put<00:21:39.720> it<00:21:39.800> in<00:21:39.920> set<00:21:40.160> theoretical" + }, + { + "start": 1300.55, + "duration": 0.0, + "text": "Uh to put it in set theoretical" + }, + { + "start": 1300.56, + "duration": 0.0, + "text": "Uh to put it in set theoretical language,<00:21:41.000> if<00:21:41.360> X<00:21:41.560> is<00:21:41.920> strictly<00:21:42.440> in<00:21:42.560> contained" + }, + { + "start": 1302.95, + "duration": 0.0, + "text": "language, if X is strictly in contained" + }, + { + "start": 1302.96, + "duration": 0.0, + "text": "language, if X is strictly in contained in<00:21:43.000> Y,<00:21:43.480> then<00:21:44.320> if<00:21:44.680> you<00:21:44.960> uh" + }, + { + "start": 1305.63, + "duration": 0.0, + "text": "in Y, then if you uh" + }, + { + "start": 1305.64, + "duration": 0.0, + "text": "in Y, then if you uh if<00:21:45.840> you<00:21:46.160> add<00:21:46.400> another<00:21:46.720> set<00:21:47.200> to<00:21:47.320> both<00:21:47.640> sides," + }, + { + "start": 1308.47, + "duration": 0.0, + "text": "if you add another set to both sides," + }, + { + "start": 1308.48, + "duration": 0.0, + "text": "if you add another set to both sides, this<00:21:48.680> kind<00:21:48.880> of<00:21:49.240> ordering<00:21:49.600> relation,<00:21:50.080> this" + }, + { + "start": 1310.19, + "duration": 0.0, + "text": "this kind of ordering relation, this" + }, + { + "start": 1310.2, + "duration": 0.0, + "text": "this kind of ordering relation, this containment,<00:21:50.760> does<00:21:50.920> not<00:21:51.080> change.<00:21:52.000> And<00:21:52.120> to<00:21:52.240> put" + }, + { + "start": 1312.39, + "duration": 0.0, + "text": "containment, does not change. And to put" + }, + { + "start": 1312.4, + "duration": 0.0, + "text": "containment, does not change. And to put it<00:21:52.560> into<00:21:52.800> a<00:21:52.880> more<00:21:53.160> practical<00:21:53.600> example," + }, + { + "start": 1314.79, + "duration": 0.0, + "text": "it into a more practical example," + }, + { + "start": 1314.8, + "duration": 0.0, + "text": "it into a more practical example, um<00:21:55.280> we<00:21:55.480> all<00:21:55.880> we<00:21:56.000> know<00:21:56.200> that<00:21:56.480> all<00:21:56.640> dogs<00:21:56.920> are" + }, + { + "start": 1316.99, + "duration": 0.0, + "text": "um we all we know that all dogs are" + }, + { + "start": 1317.0, + "duration": 0.0, + "text": "um we all we know that all dogs are mammals,<00:21:57.600> but<00:21:57.760> not<00:21:57.960> all<00:21:58.080> mammals<00:21:58.440> are<00:21:58.520> dogs." + }, + { + "start": 1318.91, + "duration": 0.0, + "text": "mammals, but not all mammals are dogs." + }, + { + "start": 1318.92, + "duration": 0.0, + "text": "mammals, but not all mammals are dogs. Therefore,<00:21:59.200> being<00:21:59.440> a<00:21:59.480> dog<00:22:00.040> implies<00:22:00.480> being<00:22:00.720> a" + }, + { + "start": 1320.75, + "duration": 0.0, + "text": "Therefore, being a dog implies being a" + }, + { + "start": 1320.76, + "duration": 0.0, + "text": "Therefore, being a dog implies being a mammal,<00:22:01.520> which<00:22:01.720> we<00:22:01.800> could<00:22:02.000> write<00:22:02.320> as<00:22:02.720> uh" + }, + { + "start": 1323.75, + "duration": 0.0, + "text": "mammal, which we could write as uh" + }, + { + "start": 1323.76, + "duration": 0.0, + "text": "mammal, which we could write as uh like<00:22:03.840> this." + }, + { + "start": 1325.87, + "duration": 0.0, + "text": "like this." + }, + { + "start": 1325.88, + "duration": 0.0, + "text": "like this. Now,<00:22:06.040> if<00:22:06.160> combination<00:22:06.680> didn't<00:22:06.960> preserve" + }, + { + "start": 1327.35, + "duration": 0.0, + "text": "Now, if combination didn't preserve" + }, + { + "start": 1327.36, + "duration": 0.0, + "text": "Now, if combination didn't preserve order,<00:22:07.560> we<00:22:07.680> could<00:22:07.880> encounter<00:22:08.280> situations<00:22:09.440> uh" + }, + { + "start": 1329.71, + "duration": 0.0, + "text": "order, we could encounter situations uh" + }, + { + "start": 1329.72, + "duration": 0.0, + "text": "order, we could encounter situations uh such<00:22:10.000> that<00:22:10.680> dog<00:22:11.040> or<00:22:11.200> fish<00:22:11.800> is<00:22:11.960> somehow<00:22:12.800> implied" + }, + { + "start": 1333.35, + "duration": 0.0, + "text": "such that dog or fish is somehow implied" + }, + { + "start": 1333.36, + "duration": 0.0, + "text": "such that dog or fish is somehow implied by<00:22:13.760> mammal<00:22:14.120> or<00:22:14.200> fish,<00:22:14.480> which<00:22:14.680> is<00:22:14.760> of<00:22:14.880> course" + }, + { + "start": 1335.07, + "duration": 0.0, + "text": "by mammal or fish, which is of course" + }, + { + "start": 1335.08, + "duration": 0.0, + "text": "by mammal or fish, which is of course absurd.<00:22:16.120> Um" + }, + { + "start": 1337.23, + "duration": 0.0, + "text": "absurd. Um" + }, + { + "start": 1337.24, + "duration": 0.0, + "text": "absurd. Um because<00:22:17.600> uh" + }, + { + "start": 1339.11, + "duration": 0.0, + "text": "because uh" + }, + { + "start": 1339.12, + "duration": 0.0, + "text": "because uh these<00:22:19.560> additional<00:22:20.000> possibilities<00:22:20.760> for<00:22:20.880> what" + }, + { + "start": 1341.07, + "duration": 0.0, + "text": "these additional possibilities for what" + }, + { + "start": 1341.08, + "duration": 0.0, + "text": "these additional possibilities for what it<00:22:21.360> the<00:22:21.560> the<00:22:21.680> animal<00:22:21.960> really<00:22:22.160> is<00:22:22.400> shouldn't" + }, + { + "start": 1342.75, + "duration": 0.0, + "text": "it the the animal really is shouldn't" + }, + { + "start": 1342.76, + "duration": 0.0, + "text": "it the the animal really is shouldn't change<00:22:23.080> the<00:22:23.440> the<00:22:23.520> order<00:22:23.880> of<00:22:24.240> dog<00:22:24.440> and<00:22:24.560> mammal." + }, + { + "start": 1347.48, + "duration": 0.0, + "text": "And<00:22:27.640> then<00:22:27.800> there's<00:22:27.920> another<00:22:28.120> symmetry,<00:22:28.520> which" + }, + { + "start": 1348.75, + "duration": 0.0, + "text": "And then there's another symmetry, which" + }, + { + "start": 1348.76, + "duration": 0.0, + "text": "And then there's another symmetry, which is<00:22:28.840> simply<00:22:29.120> that<00:22:29.400> the<00:22:29.520> combination<00:22:30.240> is" + }, + { + "start": 1350.51, + "duration": 0.0, + "text": "is simply that the combination is" + }, + { + "start": 1350.52, + "duration": 0.0, + "text": "is simply that the combination is associative." + }, + { + "start": 1351.95, + "duration": 0.0, + "text": "associative." + }, + { + "start": 1351.96, + "duration": 0.0, + "text": "associative. Um" + }, + { + "start": 1353.43, + "duration": 0.0, + "text": "Um" + }, + { + "start": 1353.44, + "duration": 0.0, + "text": "Um And<00:22:33.520> this<00:22:33.680> follows<00:22:33.960> simply<00:22:34.160> from<00:22:34.320> the<00:22:34.400> fact" + }, + { + "start": 1354.71, + "duration": 0.0, + "text": "And this follows simply from the fact" + }, + { + "start": 1354.72, + "duration": 0.0, + "text": "And this follows simply from the fact that<00:22:35.000> the<00:22:35.240> the" + }, + { + "start": 1356.23, + "duration": 0.0, + "text": "that the the" + }, + { + "start": 1356.24, + "duration": 0.0, + "text": "that the the join<00:22:36.520> operator<00:22:37.040> is<00:22:37.280> defined<00:22:37.680> as<00:22:37.800> the<00:22:37.920> lower" + }, + { + "start": 1358.15, + "duration": 0.0, + "text": "join operator is defined as the lower" + }, + { + "start": 1358.16, + "duration": 0.0, + "text": "join operator is defined as the lower upper<00:22:38.400> bound,<00:22:39.120> which<00:22:39.360> just<00:22:39.600> takes<00:22:39.880> in<00:22:40.120> a<00:22:40.200> set" + }, + { + "start": 1360.39, + "duration": 0.0, + "text": "upper bound, which just takes in a set" + }, + { + "start": 1360.4, + "duration": 0.0, + "text": "upper bound, which just takes in a set of<00:22:40.520> elements,<00:22:41.080> and<00:22:41.280> so<00:22:41.480> it<00:22:41.640> doesn't<00:22:42.120> care<00:22:42.400> what" + }, + { + "start": 1362.55, + "duration": 0.0, + "text": "of elements, and so it doesn't care what" + }, + { + "start": 1362.56, + "duration": 0.0, + "text": "of elements, and so it doesn't care what grouping<00:22:42.840> at<00:22:42.960> all." + }, + { + "start": 1363.91, + "duration": 0.0, + "text": "grouping at all." + }, + { + "start": 1363.92, + "duration": 0.0, + "text": "grouping at all. Uh<00:22:44.000> it<00:22:44.080> doesn't<00:22:44.280> matter<00:22:44.480> which<00:22:44.720> order<00:22:45.480> uh<00:22:45.520> we" + }, + { + "start": 1365.63, + "duration": 0.0, + "text": "Uh it doesn't matter which order uh we" + }, + { + "start": 1365.64, + "duration": 0.0, + "text": "Uh it doesn't matter which order uh we do<00:22:45.800> it." + }, + { + "start": 1366.83, + "duration": 0.0, + "text": "do it." + }, + { + "start": 1366.84, + "duration": 0.0, + "text": "do it. And<00:22:47.000> so,<00:22:47.160> we're<00:22:47.280> going<00:22:47.400> to<00:22:47.520> use<00:22:47.840> the<00:22:47.960> most" + }, + { + "start": 1368.15, + "duration": 0.0, + "text": "And so, we're going to use the most" + }, + { + "start": 1368.16, + "duration": 0.0, + "text": "And so, we're going to use the most important<00:22:48.520> theorem<00:22:48.800> of<00:22:48.920> this<00:22:49.040> whole" + }, + { + "start": 1369.15, + "duration": 0.0, + "text": "important theorem of this whole" + }, + { + "start": 1369.16, + "duration": 0.0, + "text": "important theorem of this whole derivation,<00:22:49.720> which<00:22:49.880> is<00:22:50.000> the<00:22:50.320> associativity" + }, + { + "start": 1370.95, + "duration": 0.0, + "text": "derivation, which is the associativity" + }, + { + "start": 1370.96, + "duration": 0.0, + "text": "derivation, which is the associativity theorem." + }, + { + "start": 1372.07, + "duration": 0.0, + "text": "theorem." + }, + { + "start": 1372.08, + "duration": 0.0, + "text": "theorem. Uh" + }, + { + "start": 1372.83, + "duration": 0.0, + "text": "Uh" + }, + { + "start": 1372.84, + "duration": 0.0, + "text": "Uh we<00:22:52.960> have<00:22:53.440> two<00:22:53.520> disjoint<00:22:53.880> elements<00:22:54.560> of<00:22:54.680> a" + }, + { + "start": 1374.75, + "duration": 0.0, + "text": "we have two disjoint elements of a" + }, + { + "start": 1374.76, + "duration": 0.0, + "text": "we have two disjoint elements of a distributive<00:22:55.280> lattice<00:22:55.680> with<00:22:56.120> their" + }, + { + "start": 1376.55, + "duration": 0.0, + "text": "distributive lattice with their" + }, + { + "start": 1376.56, + "duration": 0.0, + "text": "distributive lattice with their respective<00:22:56.960> valuations." + }, + { + "start": 1378.23, + "duration": 0.0, + "text": "respective valuations." + }, + { + "start": 1378.24, + "duration": 0.0, + "text": "respective valuations. Now,<00:22:58.640> the<00:22:58.800> two<00:22:59.000> symmetries" + }, + { + "start": 1380.35, + "duration": 0.0, + "text": "Now, the two symmetries" + }, + { + "start": 1380.36, + "duration": 0.0, + "text": "Now, the two symmetries uh<00:23:00.480> it<00:23:00.560> translates<00:23:01.000> to<00:23:01.120> axioms<00:23:01.720> imply<00:23:02.360> that" + }, + { + "start": 1383.03, + "duration": 0.0, + "text": "uh it translates to axioms imply that" + }, + { + "start": 1383.04, + "duration": 0.0, + "text": "uh it translates to axioms imply that there<00:23:03.200> is<00:23:03.320> some<00:23:03.760> order<00:23:03.960> preserving<00:23:04.360> mapping" + }, + { + "start": 1384.71, + "duration": 0.0, + "text": "there is some order preserving mapping" + }, + { + "start": 1384.72, + "duration": 0.0, + "text": "there is some order preserving mapping for<00:23:04.920> which<00:23:05.760> this<00:23:06.320> kind<00:23:06.560> of<00:23:07.160> combination" + }, + { + "start": 1387.75, + "duration": 0.0, + "text": "for which this kind of combination" + }, + { + "start": 1387.76, + "duration": 0.0, + "text": "for which this kind of combination operator<00:23:08.800> uh,<00:23:09.320> is<00:23:10.040> addition." + }, + { + "start": 1391.27, + "duration": 0.0, + "text": "operator uh, is addition." + }, + { + "start": 1391.28, + "duration": 0.0, + "text": "operator uh, is addition. And<00:23:11.560> the<00:23:11.640> beauty<00:23:11.960> is<00:23:12.080> that<00:23:12.280> since<00:23:12.680> this<00:23:12.960> is<00:23:13.200> the" + }, + { + "start": 1393.27, + "duration": 0.0, + "text": "And the beauty is that since this is the" + }, + { + "start": 1393.28, + "duration": 0.0, + "text": "And the beauty is that since this is the first<00:23:13.960> operator<00:23:14.440> that<00:23:14.680> we<00:23:14.880> are<00:23:15.000> kind<00:23:15.200> of" + }, + { + "start": 1395.27, + "duration": 0.0, + "text": "first operator that we are kind of" + }, + { + "start": 1395.28, + "duration": 0.0, + "text": "first operator that we are kind of looking<00:23:15.560> at,<00:23:16.320> we<00:23:16.680> can<00:23:16.880> also<00:23:17.120> just<00:23:17.440> immediately" + }, + { + "start": 1398.19, + "duration": 0.0, + "text": "looking at, we can also just immediately" + }, + { + "start": 1398.2, + "duration": 0.0, + "text": "looking at, we can also just immediately throw<00:23:18.480> away<00:23:19.040> these<00:23:19.280> original<00:23:19.760> valuations" + }, + { + "start": 1400.35, + "duration": 0.0, + "text": "throw away these original valuations" + }, + { + "start": 1400.36, + "duration": 0.0, + "text": "throw away these original valuations that<00:23:20.520> we<00:23:20.640> had<00:23:20.880> for<00:23:21.040> all<00:23:21.200> of<00:23:21.280> these" + }, + { + "start": 1401.43, + "duration": 0.0, + "text": "that we had for all of these" + }, + { + "start": 1401.44, + "duration": 0.0, + "text": "that we had for all of these propositions<00:23:22.120> and<00:23:22.320> just<00:23:22.560> use<00:23:23.320> sort<00:23:23.600> of<00:23:24.040> theta" + }, + { + "start": 1404.75, + "duration": 0.0, + "text": "propositions and just use sort of theta" + }, + { + "start": 1404.76, + "duration": 0.0, + "text": "propositions and just use sort of theta theta<00:23:25.200> of<00:23:25.680> the<00:23:25.760> original<00:23:26.080> valuation<00:23:26.560> and<00:23:26.640> just" + }, + { + "start": 1406.91, + "duration": 0.0, + "text": "theta of the original valuation and just" + }, + { + "start": 1406.92, + "duration": 0.0, + "text": "theta of the original valuation and just use<00:23:27.120> addition<00:23:27.480> with<00:23:27.640> them." + }, + { + "start": 1408.67, + "duration": 0.0, + "text": "use addition with them." + }, + { + "start": 1408.68, + "duration": 0.0, + "text": "use addition with them. So,<00:23:28.880> we<00:23:29.000> can<00:23:29.640> use" + }, + { + "start": 1411.07, + "duration": 0.0, + "text": "So, we can use" + }, + { + "start": 1411.08, + "duration": 0.0, + "text": "So, we can use uh," + }, + { + "start": 1412.43, + "duration": 0.0, + "text": "uh," + }, + { + "start": 1412.44, + "duration": 0.0, + "text": "uh, sum<00:23:33.440> for<00:23:33.600> the<00:23:33.680> combination<00:23:34.240> of<00:23:34.400> disjoint" + }, + { + "start": 1414.79, + "duration": 0.0, + "text": "sum for the combination of disjoint" + }, + { + "start": 1414.8, + "duration": 0.0, + "text": "sum for the combination of disjoint elements<00:23:35.200> without<00:23:35.560> any<00:23:35.720> loss<00:23:35.920> of<00:23:36.000> generality." + }, + { + "start": 1416.95, + "duration": 0.0, + "text": "elements without any loss of generality." + }, + { + "start": 1416.96, + "duration": 0.0, + "text": "elements without any loss of generality. Now,<00:23:37.160> the<00:23:37.280> proof<00:23:37.520> of<00:23:37.640> this<00:23:37.960> is<00:23:38.640> too<00:23:38.760> long<00:23:39.040> for" + }, + { + "start": 1419.15, + "duration": 0.0, + "text": "Now, the proof of this is too long for" + }, + { + "start": 1419.16, + "duration": 0.0, + "text": "Now, the proof of this is too long for me<00:23:39.280> to<00:23:39.400> present<00:23:39.720> it<00:23:39.840> here.<00:23:40.000> It's<00:23:40.120> also<00:23:40.320> kind<00:23:40.440> of" + }, + { + "start": 1420.51, + "duration": 0.0, + "text": "me to present it here. It's also kind of" + }, + { + "start": 1420.52, + "duration": 0.0, + "text": "me to present it here. It's also kind of annoying,<00:23:41.480> uh,<00:23:41.640> but<00:23:42.080> uh,<00:23:42.400> you<00:23:42.560> can<00:23:42.760> find<00:23:42.960> it<00:23:43.080> in" + }, + { + "start": 1423.15, + "duration": 0.0, + "text": "annoying, uh, but uh, you can find it in" + }, + { + "start": 1423.16, + "duration": 0.0, + "text": "annoying, uh, but uh, you can find it in this<00:23:43.400> paper<00:23:44.040> by<00:23:44.320> Kevin<00:23:44.680> Knuth<00:23:45.000> and<00:23:45.120> John" + }, + { + "start": 1425.35, + "duration": 0.0, + "text": "this paper by Kevin Knuth and John" + }, + { + "start": 1425.36, + "duration": 0.0, + "text": "this paper by Kevin Knuth and John Skilling.<00:23:45.760> It<00:23:45.840> was<00:23:45.960> published<00:23:46.320> in<00:23:46.520> 2012." + }, + { + "start": 1427.83, + "duration": 0.0, + "text": "Skilling. It was published in 2012." + }, + { + "start": 1427.84, + "duration": 0.0, + "text": "Skilling. It was published in 2012. And<00:23:48.360> uh," + }, + { + "start": 1429.47, + "duration": 0.0, + "text": "And uh," + }, + { + "start": 1429.48, + "duration": 0.0, + "text": "And uh, there<00:23:49.840> there<00:23:50.040> the<00:23:50.160> paper<00:23:50.440> is<00:23:50.560> super" + }, + { + "start": 1430.83, + "duration": 0.0, + "text": "there there the paper is super" + }, + { + "start": 1430.84, + "duration": 0.0, + "text": "there there the paper is super interesting<00:23:51.280> and<00:23:51.400> it<00:23:51.520> rests<00:23:51.800> on<00:23:51.960> like<00:23:52.120> a" + }, + { + "start": 1432.15, + "duration": 0.0, + "text": "interesting and it rests on like a" + }, + { + "start": 1432.16, + "duration": 0.0, + "text": "interesting and it rests on like a decade<00:23:52.480> of<00:23:52.640> work<00:23:53.120> by<00:23:54.160> uh,<00:23:54.640> those<00:23:54.880> two." + }, + { + "start": 1436.43, + "duration": 0.0, + "text": "decade of work by uh, those two." + }, + { + "start": 1436.44, + "duration": 0.0, + "text": "decade of work by uh, those two. The<00:23:56.680> only<00:23:56.880> thing<00:23:57.120> that<00:23:57.880> I<00:23:58.000> would<00:23:58.160> recommend" + }, + { + "start": 1438.51, + "duration": 0.0, + "text": "The only thing that I would recommend" + }, + { + "start": 1438.52, + "duration": 0.0, + "text": "The only thing that I would recommend that<00:23:58.680> I<00:23:58.720> would<00:23:59.080> say<00:23:59.480> if<00:23:59.640> you<00:23:59.720> ever<00:23:59.960> read<00:24:00.160> this" + }, + { + "start": 1440.31, + "duration": 0.0, + "text": "that I would say if you ever read this" + }, + { + "start": 1440.32, + "duration": 0.0, + "text": "that I would say if you ever read this is<00:24:00.600> to<00:24:01.240> be<00:24:01.320> very<00:24:01.520> careful<00:24:01.840> because<00:24:02.200> they<00:24:02.440> are<00:24:02.600> a" + }, + { + "start": 1442.67, + "duration": 0.0, + "text": "is to be very careful because they are a" + }, + { + "start": 1442.68, + "duration": 0.0, + "text": "is to be very careful because they are a bit<00:24:02.880> rough<00:24:03.080> sometimes,<00:24:03.520> but<00:24:03.720> it's<00:24:03.960> absolutely" + }, + { + "start": 1444.51, + "duration": 0.0, + "text": "bit rough sometimes, but it's absolutely" + }, + { + "start": 1444.52, + "duration": 0.0, + "text": "bit rough sometimes, but it's absolutely a<00:24:04.600> gorgeous<00:24:05.400> paper." + }, + { + "start": 1449.56, + "duration": 0.0, + "text": "And<00:24:09.840> now<00:24:10.120> we<00:24:10.240> can<00:24:10.440> already<00:24:10.720> find<00:24:11.000> one<00:24:11.120> of<00:24:11.160> the" + }, + { + "start": 1451.27, + "duration": 0.0, + "text": "And now we can already find one of the" + }, + { + "start": 1451.28, + "duration": 0.0, + "text": "And now we can already find one of the familiar<00:24:12.320> rules<00:24:12.600> of<00:24:12.640> probability" + }, + { + "start": 1453.95, + "duration": 0.0, + "text": "familiar rules of probability" + }, + { + "start": 1453.96, + "duration": 0.0, + "text": "familiar rules of probability where<00:24:14.120> we<00:24:14.200> can<00:24:14.680> kind<00:24:14.800> of<00:24:14.880> get<00:24:15.000> a<00:24:15.040> hint<00:24:15.480> of<00:24:15.600> what" + }, + { + "start": 1455.71, + "duration": 0.0, + "text": "where we can kind of get a hint of what" + }, + { + "start": 1455.72, + "duration": 0.0, + "text": "where we can kind of get a hint of what it's<00:24:15.800> going<00:24:16.000> to<00:24:16.080> be.<00:24:16.880> Um,<00:24:17.520> we<00:24:17.640> can<00:24:17.880> send<00:24:18.120> out" + }, + { + "start": 1458.27, + "duration": 0.0, + "text": "it's going to be. Um, we can send out" + }, + { + "start": 1458.28, + "duration": 0.0, + "text": "it's going to be. Um, we can send out kind<00:24:18.440> of<00:24:18.520> see<00:24:18.640> the<00:24:18.720> combination<00:24:19.200> of<00:24:19.360> arbitrary" + }, + { + "start": 1459.75, + "duration": 0.0, + "text": "kind of see the combination of arbitrary" + }, + { + "start": 1459.76, + "duration": 0.0, + "text": "kind of see the combination of arbitrary arguments<00:24:20.160> because<00:24:20.480> before<00:24:20.920> we<00:24:21.080> only" + }, + { + "start": 1462.19, + "duration": 0.0, + "text": "arguments because before we only" + }, + { + "start": 1462.2, + "duration": 0.0, + "text": "arguments because before we only saw<00:24:22.400> that<00:24:22.520> we<00:24:22.600> could<00:24:22.760> use<00:24:22.960> addition<00:24:23.480> for" + }, + { + "start": 1463.75, + "duration": 0.0, + "text": "saw that we could use addition for" + }, + { + "start": 1463.76, + "duration": 0.0, + "text": "saw that we could use addition for disjoint<00:24:24.440> elements,<00:24:25.320> but<00:24:25.520> imagine<00:24:25.880> that<00:24:25.960> we" + }, + { + "start": 1466.03, + "duration": 0.0, + "text": "disjoint elements, but imagine that we" + }, + { + "start": 1466.04, + "duration": 0.0, + "text": "disjoint elements, but imagine that we have<00:24:26.280> X<00:24:26.440> and<00:24:26.600> Y<00:24:26.880> which<00:24:27.080> are<00:24:27.160> kind<00:24:27.400> of<00:24:27.680> have<00:24:27.920> some" + }, + { + "start": 1468.39, + "duration": 0.0, + "text": "have X and Y which are kind of have some" + }, + { + "start": 1468.4, + "duration": 0.0, + "text": "have X and Y which are kind of have some elements<00:24:28.960> in<00:24:29.080> common,<00:24:29.400> so<00:24:29.480> to<00:24:29.560> speak." + }, + { + "start": 1470.91, + "duration": 0.0, + "text": "elements in common, so to speak." + }, + { + "start": 1470.92, + "duration": 0.0, + "text": "elements in common, so to speak. And<00:24:31.400> then<00:24:31.760> if<00:24:31.880> we<00:24:31.960> want<00:24:32.120> to<00:24:32.200> have<00:24:32.360> their" + }, + { + "start": 1472.47, + "duration": 0.0, + "text": "And then if we want to have their" + }, + { + "start": 1472.48, + "duration": 0.0, + "text": "And then if we want to have their valuations,<00:24:33.760> uh,<00:24:34.080> we<00:24:34.280> would<00:24:34.720> could<00:24:34.880> of<00:24:34.960> course" + }, + { + "start": 1475.35, + "duration": 0.0, + "text": "valuations, uh, we would could of course" + }, + { + "start": 1475.36, + "duration": 0.0, + "text": "valuations, uh, we would could of course just<00:24:35.600> sum<00:24:36.480> their<00:24:36.720> their<00:24:36.920> components" + }, + { + "start": 1477.39, + "duration": 0.0, + "text": "just sum their their components" + }, + { + "start": 1477.4, + "duration": 0.0, + "text": "just sum their their components together,<00:24:37.920> but<00:24:38.200> if<00:24:38.280> you<00:24:38.360> want<00:24:38.520> to<00:24:38.560> express<00:24:38.960> it" + }, + { + "start": 1479.31, + "duration": 0.0, + "text": "together, but if you want to express it" + }, + { + "start": 1479.32, + "duration": 0.0, + "text": "together, but if you want to express it as<00:24:39.520> a<00:24:39.760> function<00:24:40.120> of<00:24:40.240> the<00:24:40.320> elements<00:24:41.400> of<00:24:41.640> X,<00:24:42.080> uh," + }, + { + "start": 1482.67, + "duration": 0.0, + "text": "as a function of the elements of X, uh," + }, + { + "start": 1482.68, + "duration": 0.0, + "text": "as a function of the elements of X, uh, sorry,<00:24:42.920> of<00:24:43.040> the" + }, + { + "start": 1484.19, + "duration": 0.0, + "text": "sorry, of the" + }, + { + "start": 1484.2, + "duration": 0.0, + "text": "sorry, of the the<00:24:44.400> valuation<00:24:44.760> of<00:24:44.880> X<00:24:45.080> and<00:24:45.200> the<00:24:45.240> valuation<00:24:45.640> of" + }, + { + "start": 1485.75, + "duration": 0.0, + "text": "the valuation of X and the valuation of" + }, + { + "start": 1485.76, + "duration": 0.0, + "text": "the valuation of X and the valuation of Y,<00:24:46.680> then<00:24:47.240> we<00:24:47.360> find<00:24:47.600> that<00:24:47.720> we<00:24:47.800> need<00:24:47.920> to<00:24:48.000> subtract" + }, + { + "start": 1488.39, + "duration": 0.0, + "text": "Y, then we find that we need to subtract" + }, + { + "start": 1488.4, + "duration": 0.0, + "text": "Y, then we find that we need to subtract the<00:24:48.440> valuation<00:24:48.960> of<00:24:49.080> the<00:24:49.160> element<00:24:49.520> that<00:24:49.640> they" + }, + { + "start": 1489.75, + "duration": 0.0, + "text": "the valuation of the element that they" + }, + { + "start": 1489.76, + "duration": 0.0, + "text": "the valuation of the element that they have<00:24:49.920> in<00:24:50.000> common.<00:24:50.720> And<00:24:50.880> this<00:24:51.120> leads<00:24:51.400> to<00:24:51.520> the" + }, + { + "start": 1491.63, + "duration": 0.0, + "text": "have in common. And this leads to the" + }, + { + "start": 1491.64, + "duration": 0.0, + "text": "have in common. And this leads to the general<00:24:52.160> inclusion-exclusion<00:24:53.640> uh,<00:24:53.720> sum<00:24:54.000> rule" + }, + { + "start": 1494.75, + "duration": 0.0, + "text": "general inclusion-exclusion uh, sum rule" + }, + { + "start": 1494.76, + "duration": 0.0, + "text": "general inclusion-exclusion uh, sum rule that<00:24:54.920> is<00:24:55.040> usually<00:24:55.480> taught<00:24:55.800> with<00:24:56.640> uh,<00:24:56.720> Venn" + }, + { + "start": 1496.99, + "duration": 0.0, + "text": "that is usually taught with uh, Venn" + }, + { + "start": 1497.0, + "duration": 0.0, + "text": "that is usually taught with uh, Venn diagrams<00:24:57.480> that<00:24:57.560> are<00:24:57.640> overlapping<00:24:58.360> stuff<00:24:58.560> like" + }, + { + "start": 1498.67, + "duration": 0.0, + "text": "diagrams that are overlapping stuff like" + }, + { + "start": 1498.68, + "duration": 0.0, + "text": "diagrams that are overlapping stuff like this.<00:24:58.920> I'm<00:24:59.040> sure<00:24:59.240> you've<00:24:59.400> all<00:24:59.520> seen<00:24:59.760> those." + }, + { + "start": 1500.71, + "duration": 0.0, + "text": "this. I'm sure you've all seen those." + }, + { + "start": 1500.72, + "duration": 0.0, + "text": "this. I'm sure you've all seen those. Um<00:25:01.280> and<00:25:01.440> interestingly,<00:25:02.160> we<00:25:02.320> also<00:25:02.520> can" + }, + { + "start": 1502.67, + "duration": 0.0, + "text": "Um and interestingly, we also can" + }, + { + "start": 1502.68, + "duration": 0.0, + "text": "Um and interestingly, we also can immediately" + }, + { + "start": 1504.35, + "duration": 0.0, + "text": "immediately" + }, + { + "start": 1504.36, + "duration": 0.0, + "text": "immediately find<00:25:04.720> out<00:25:04.960> that<00:25:05.840> this<00:25:06.240> um" + }, + { + "start": 1509.56, + "duration": 0.0, + "text": "uh<00:25:09.800> the<00:25:09.920> commit<00:25:10.280> the<00:25:10.760> join<00:25:11.040> and<00:25:11.160> meet" + }, + { + "start": 1511.35, + "duration": 0.0, + "text": "uh the commit the join and meet" + }, + { + "start": 1511.36, + "duration": 0.0, + "text": "uh the commit the join and meet operations<00:25:12.200> are<00:25:12.440> associative<00:25:13.080> because<00:25:13.640> the" + }, + { + "start": 1513.75, + "duration": 0.0, + "text": "operations are associative because the" + }, + { + "start": 1513.76, + "duration": 0.0, + "text": "operations are associative because the commit<00:25:14.400> these<00:25:14.640> the<00:25:14.760> real<00:25:15.000> numbers<00:25:15.480> are<00:25:15.680> commit" + }, + { + "start": 1516.23, + "duration": 0.0, + "text": "commit these the real numbers are commit" + }, + { + "start": 1516.24, + "duration": 0.0, + "text": "commit these the real numbers are commit No,<00:25:16.400> sorry.<00:25:16.800> Did<00:25:16.960> I<00:25:17.000> say<00:25:17.120> associative?<00:25:17.600> I" + }, + { + "start": 1517.63, + "duration": 0.0, + "text": "No, sorry. Did I say associative? I" + }, + { + "start": 1517.64, + "duration": 0.0, + "text": "No, sorry. Did I say associative? I meant<00:25:17.840> commutative,<00:25:18.400> sorry.<00:25:19.240> Because<00:25:19.440> the" + }, + { + "start": 1519.51, + "duration": 0.0, + "text": "meant commutative, sorry. Because the" + }, + { + "start": 1519.52, + "duration": 0.0, + "text": "meant commutative, sorry. Because the real<00:25:19.720> numbers<00:25:20.040> are<00:25:20.160> of<00:25:20.280> course<00:25:20.440> commutative." + }, + { + "start": 1520.95, + "duration": 0.0, + "text": "real numbers are of course commutative." + }, + { + "start": 1520.96, + "duration": 0.0, + "text": "real numbers are of course commutative. So,<00:25:21.120> since<00:25:21.320> the<00:25:21.440> valuations<00:25:22.360> have<00:25:22.520> this" + }, + { + "start": 1522.67, + "duration": 0.0, + "text": "So, since the valuations have this" + }, + { + "start": 1522.68, + "duration": 0.0, + "text": "So, since the valuations have this property," + }, + { + "start": 1523.95, + "duration": 0.0, + "text": "property," + }, + { + "start": 1523.96, + "duration": 0.0, + "text": "property, uh<00:25:24.320> also<00:25:24.560> the<00:25:25.280> algebra<00:25:25.520> of<00:25:25.560> the<00:25:25.640> individual" + }, + { + "start": 1526.03, + "duration": 0.0, + "text": "uh also the algebra of the individual" + }, + { + "start": 1526.04, + "duration": 0.0, + "text": "uh also the algebra of the individual elements<00:25:26.400> must<00:25:26.600> have<00:25:26.760> it." + }, + { + "start": 1529.88, + "duration": 0.0, + "text": "Then<00:25:29.960> we're<00:25:30.040> going<00:25:30.160> to<00:25:30.240> introduce<00:25:30.520> a<00:25:30.560> new" + }, + { + "start": 1530.75, + "duration": 0.0, + "text": "Then we're going to introduce a new" + }, + { + "start": 1530.76, + "duration": 0.0, + "text": "Then we're going to introduce a new symmetry," + }, + { + "start": 1531.79, + "duration": 0.0, + "text": "symmetry," + }, + { + "start": 1531.8, + "duration": 0.0, + "text": "symmetry, uh<00:25:32.000> which<00:25:32.200> is<00:25:32.520> Imagine<00:25:32.840> we<00:25:32.920> have<00:25:33.080> two" + }, + { + "start": 1533.27, + "duration": 0.0, + "text": "uh which is Imagine we have two" + }, + { + "start": 1533.28, + "duration": 0.0, + "text": "uh which is Imagine we have two independent<00:25:33.760> systems,<00:25:34.400> S1<00:25:34.720> S2,<00:25:35.200> that<00:25:35.440> we<00:25:35.560> want" + }, + { + "start": 1535.75, + "duration": 0.0, + "text": "independent systems, S1 S2, that we want" + }, + { + "start": 1535.76, + "duration": 0.0, + "text": "independent systems, S1 S2, that we want to<00:25:35.840> combine.<00:25:36.440> We<00:25:36.520> sort<00:25:36.680> of<00:25:36.760> want<00:25:36.960> to<00:25:37.440> consider" + }, + { + "start": 1537.75, + "duration": 0.0, + "text": "to combine. We sort of want to consider" + }, + { + "start": 1537.76, + "duration": 0.0, + "text": "to combine. We sort of want to consider them<00:25:37.920> together.<00:25:38.880> And<00:25:39.200> these<00:25:39.360> two<00:25:39.480> systems" + }, + { + "start": 1539.83, + "duration": 0.0, + "text": "them together. And these two systems" + }, + { + "start": 1539.84, + "duration": 0.0, + "text": "them together. And these two systems would<00:25:40.000> then<00:25:40.160> be<00:25:40.360> two<00:25:40.520> different<00:25:40.840> lattices.<00:25:41.840> Um" + }, + { + "start": 1542.43, + "duration": 0.0, + "text": "would then be two different lattices. Um" + }, + { + "start": 1542.44, + "duration": 0.0, + "text": "would then be two different lattices. Um And<00:25:42.480> this<00:25:42.640> forms<00:25:42.840> a<00:25:42.880> new<00:25:43.080> space<00:25:43.800> with<00:25:44.080> the<00:25:44.160> type" + }, + { + "start": 1544.95, + "duration": 0.0, + "text": "And this forms a new space with the type" + }, + { + "start": 1544.96, + "duration": 0.0, + "text": "And this forms a new space with the type uh" + }, + { + "start": 1545.59, + "duration": 0.0, + "text": "uh" + }, + { + "start": 1545.6, + "duration": 0.0, + "text": "uh defined<00:25:45.840> by<00:25:45.960> some<00:25:46.080> kind<00:25:46.240> of<00:25:46.320> cross<00:25:46.600> product<00:25:46.880> of" + }, + { + "start": 1546.99, + "duration": 0.0, + "text": "defined by some kind of cross product of" + }, + { + "start": 1547.0, + "duration": 0.0, + "text": "defined by some kind of cross product of their<00:25:47.120> elements." + }, + { + "start": 1548.19, + "duration": 0.0, + "text": "their elements." + }, + { + "start": 1548.2, + "duration": 0.0, + "text": "their elements. And<00:25:48.600> the<00:25:49.080> direct<00:25:49.400> product<00:25:49.800> operator<00:25:51.040> uh<00:25:51.400> is" + }, + { + "start": 1551.87, + "duration": 0.0, + "text": "And the direct product operator uh is" + }, + { + "start": 1551.88, + "duration": 0.0, + "text": "And the direct product operator uh is right<00:25:52.160> distributive" + }, + { + "start": 1553.71, + "duration": 0.0, + "text": "right distributive" + }, + { + "start": 1553.72, + "duration": 0.0, + "text": "right distributive over<00:25:53.880> the<00:25:54.560> join<00:25:54.840> operation." + }, + { + "start": 1557.64, + "duration": 0.0, + "text": "Therefore,<00:25:58.480> the<00:25:59.120> operator<00:25:59.440> that<00:25:59.560> we<00:25:59.640> choose" + }, + { + "start": 1559.91, + "duration": 0.0, + "text": "Therefore, the operator that we choose" + }, + { + "start": 1559.92, + "duration": 0.0, + "text": "Therefore, the operator that we choose for<00:26:00.080> this<00:26:00.600> this<00:26:00.800> kind<00:26:01.000> of<00:26:01.560> uh" + }, + { + "start": 1562.23, + "duration": 0.0, + "text": "for this this kind of uh" + }, + { + "start": 1562.24, + "duration": 0.0, + "text": "for this this kind of uh crossed<00:26:02.480> out<00:26:02.920> o" + }, + { + "start": 1563.71, + "duration": 0.0, + "text": "crossed out o" + }, + { + "start": 1563.72, + "duration": 0.0, + "text": "crossed out o must<00:26:04.000> also<00:26:04.240> satisfy<00:26:04.560> this<00:26:04.720> property.<00:26:05.160> And<00:26:05.400> why" + }, + { + "start": 1565.59, + "duration": 0.0, + "text": "must also satisfy this property. And why" + }, + { + "start": 1565.6, + "duration": 0.0, + "text": "must also satisfy this property. And why do<00:26:05.720> we<00:26:05.800> think<00:26:05.960> this<00:26:06.120> is<00:26:06.280> reasonable?" + }, + { + "start": 1567.71, + "duration": 0.0, + "text": "do we think this is reasonable?" + }, + { + "start": 1567.72, + "duration": 0.0, + "text": "do we think this is reasonable? Uh<00:26:08.000> because<00:26:08.800> um" + }, + { + "start": 1569.59, + "duration": 0.0, + "text": "Uh because um" + }, + { + "start": 1569.6, + "duration": 0.0, + "text": "Uh because um if<00:26:09.760> we<00:26:09.840> imagine<00:26:10.160> that<00:26:10.240> we<00:26:10.320> want<00:26:10.520> to<00:26:10.640> analyze" + }, + { + "start": 1571.03, + "duration": 0.0, + "text": "if we imagine that we want to analyze" + }, + { + "start": 1571.04, + "duration": 0.0, + "text": "if we imagine that we want to analyze coin<00:26:11.400> tosses<00:26:11.800> and<00:26:11.960> die<00:26:12.200> tosses<00:26:12.560> at<00:26:12.680> the<00:26:12.760> same" + }, + { + "start": 1573.03, + "duration": 0.0, + "text": "coin tosses and die tosses at the same" + }, + { + "start": 1573.04, + "duration": 0.0, + "text": "coin tosses and die tosses at the same time," + }, + { + "start": 1574.23, + "duration": 0.0, + "text": "time," + }, + { + "start": 1574.24, + "duration": 0.0, + "text": "time, um<00:26:15.240> let's<00:26:15.480> propositions<00:26:16.480> refer<00:26:16.680> to<00:26:16.800> the" + }, + { + "start": 1576.91, + "duration": 0.0, + "text": "um let's propositions refer to the" + }, + { + "start": 1576.92, + "duration": 0.0, + "text": "um let's propositions refer to the outcome<00:26:17.160> of<00:26:17.240> the<00:26:17.320> coin<00:26:17.560> toss<00:26:17.720> just<00:26:17.880> be<00:26:18.160> H<00:26:18.360> and" + }, + { + "start": 1578.47, + "duration": 0.0, + "text": "outcome of the coin toss just be H and" + }, + { + "start": 1578.48, + "duration": 0.0, + "text": "outcome of the coin toss just be H and T,<00:26:18.800> and<00:26:18.960> then<00:26:19.120> we<00:26:19.240> have<00:26:19.440> D<00:26:20.280> K<00:26:20.920> with<00:26:21.120> K<00:26:21.440> just" + }, + { + "start": 1581.63, + "duration": 0.0, + "text": "T, and then we have D K with K just" + }, + { + "start": 1581.64, + "duration": 0.0, + "text": "T, and then we have D K with K just being<00:26:22.120> one<00:26:22.240> of<00:26:22.320> six," + }, + { + "start": 1583.27, + "duration": 0.0, + "text": "being one of six," + }, + { + "start": 1583.28, + "duration": 0.0, + "text": "being one of six, not<00:26:23.480> outcomes<00:26:23.840> of<00:26:23.880> the<00:26:23.960> die." + }, + { + "start": 1584.99, + "duration": 0.0, + "text": "not outcomes of the die." + }, + { + "start": 1585.0, + "duration": 0.0, + "text": "not outcomes of the die. Um<00:26:25.560> and<00:26:25.680> we<00:26:25.800> treat<00:26:26.000> them<00:26:26.120> as<00:26:26.280> independent.<00:26:27.440> Um" + }, + { + "start": 1588.23, + "duration": 0.0, + "text": "Um and we treat them as independent. Um" + }, + { + "start": 1588.24, + "duration": 0.0, + "text": "Um and we treat them as independent. Um if<00:26:28.480> we<00:26:28.600> had<00:26:28.840> it<00:26:28.920> that<00:26:29.080> somehow" + }, + { + "start": 1590.27, + "duration": 0.0, + "text": "if we had it that somehow" + }, + { + "start": 1590.28, + "duration": 0.0, + "text": "if we had it that somehow this<00:26:30.920> uh" + }, + { + "start": 1591.47, + "duration": 0.0, + "text": "this uh" + }, + { + "start": 1591.48, + "duration": 0.0, + "text": "this uh the<00:26:31.600> quality<00:26:31.840> before<00:26:32.120> was<00:26:32.240> not<00:26:32.440> true<00:26:32.720> as<00:26:32.840> shown" + }, + { + "start": 1593.07, + "duration": 0.0, + "text": "the quality before was not true as shown" + }, + { + "start": 1593.08, + "duration": 0.0, + "text": "the quality before was not true as shown here,<00:26:33.800> then<00:26:33.960> it<00:26:34.040> would<00:26:34.240> mean<00:26:34.520> that<00:26:34.680> somehow" + }, + { + "start": 1595.15, + "duration": 0.0, + "text": "here, then it would mean that somehow" + }, + { + "start": 1595.16, + "duration": 0.0, + "text": "here, then it would mean that somehow considering<00:26:35.840> the<00:26:35.960> outcomes<00:26:36.440> of<00:26:36.480> the<00:26:36.600> coin" + }, + { + "start": 1596.95, + "duration": 0.0, + "text": "considering the outcomes of the coin" + }, + { + "start": 1596.96, + "duration": 0.0, + "text": "considering the outcomes of the coin toss<00:26:37.560> would<00:26:37.960> be<00:26:38.200> different<00:26:38.640> if<00:26:38.800> you<00:26:38.880> paired" + }, + { + "start": 1599.15, + "duration": 0.0, + "text": "toss would be different if you paired" + }, + { + "start": 1599.16, + "duration": 0.0, + "text": "toss would be different if you paired them<00:26:39.280> with<00:26:39.400> the<00:26:39.520> outcomes<00:26:39.800> of<00:26:39.880> the<00:26:40.000> die,<00:26:40.960> uh" + }, + { + "start": 1601.15, + "duration": 0.0, + "text": "them with the outcomes of the die, uh" + }, + { + "start": 1601.16, + "duration": 0.0, + "text": "them with the outcomes of the die, uh which<00:26:41.400> would<00:26:41.720> kind<00:26:41.920> of<00:26:42.000> mean<00:26:42.200> that<00:26:42.320> there<00:26:42.400> was" + }, + { + "start": 1602.59, + "duration": 0.0, + "text": "which would kind of mean that there was" + }, + { + "start": 1602.6, + "duration": 0.0, + "text": "which would kind of mean that there was some<00:26:42.800> kind<00:26:43.000> of<00:26:43.280> entanglement,<00:26:43.840> some<00:26:43.960> kind<00:26:44.120> of" + }, + { + "start": 1604.23, + "duration": 0.0, + "text": "some kind of entanglement, some kind of" + }, + { + "start": 1604.24, + "duration": 0.0, + "text": "some kind of entanglement, some kind of relationship<00:26:45.480> uh<00:26:45.560> that<00:26:46.360> uh<00:26:46.560> directly" + }, + { + "start": 1606.91, + "duration": 0.0, + "text": "relationship uh that uh directly" + }, + { + "start": 1606.92, + "duration": 0.0, + "text": "relationship uh that uh directly contradicts<00:26:47.520> our<00:26:47.640> assumption<00:26:48.000> of" + }, + { + "start": 1608.11, + "duration": 0.0, + "text": "contradicts our assumption of" + }, + { + "start": 1608.12, + "duration": 0.0, + "text": "contradicts our assumption of independence." + }, + { + "start": 1609.55, + "duration": 0.0, + "text": "independence." + }, + { + "start": 1609.56, + "duration": 0.0, + "text": "independence. And<00:26:49.840> this<00:26:50.000> This<00:26:50.200> us<00:26:50.280> to<00:26:50.360> a<00:26:50.440> very<00:26:50.680> important" + }, + { + "start": 1611.15, + "duration": 0.0, + "text": "And this This us to a very important" + }, + { + "start": 1611.16, + "duration": 0.0, + "text": "And this This us to a very important point.<00:26:52.160> Uh<00:26:52.720> this<00:26:52.920> is<00:26:53.040> more<00:26:53.160> like<00:26:53.280> a" + }, + { + "start": 1613.31, + "duration": 0.0, + "text": "point. Uh this is more like a" + }, + { + "start": 1613.32, + "duration": 0.0, + "text": "point. Uh this is more like a parenthesis,<00:26:53.840> but<00:26:54.000> it's<00:26:54.120> very<00:26:54.280> important." + }, + { + "start": 1615.51, + "duration": 0.0, + "text": "parenthesis, but it's very important." + }, + { + "start": 1615.52, + "duration": 0.0, + "text": "parenthesis, but it's very important. Independence" + }, + { + "start": 1617.03, + "duration": 0.0, + "text": "Independence" + }, + { + "start": 1617.04, + "duration": 0.0, + "text": "Independence is<00:26:57.920> classically<00:26:58.400> seen<00:26:59.000> or<00:26:59.520> in<00:26:59.640> the<00:26:59.720> orthodox" + }, + { + "start": 1620.03, + "duration": 0.0, + "text": "is classically seen or in the orthodox" + }, + { + "start": 1620.04, + "duration": 0.0, + "text": "is classically seen or in the orthodox methodology<00:27:00.520> somehow<00:27:01.160> a<00:27:01.240> property<00:27:01.920> of<00:27:02.040> the" + }, + { + "start": 1622.15, + "duration": 0.0, + "text": "methodology somehow a property of the" + }, + { + "start": 1622.16, + "duration": 0.0, + "text": "methodology somehow a property of the experiment.<00:27:02.760> You<00:27:02.880> have<00:27:03.200> independent<00:27:03.680> trials," + }, + { + "start": 1624.67, + "duration": 0.0, + "text": "experiment. You have independent trials," + }, + { + "start": 1624.68, + "duration": 0.0, + "text": "experiment. You have independent trials, independent<00:27:05.240> draws<00:27:05.520> from<00:27:05.640> distribution." + }, + { + "start": 1626.75, + "duration": 0.0, + "text": "independent draws from distribution." + }, + { + "start": 1626.76, + "duration": 0.0, + "text": "independent draws from distribution. However,<00:27:07.720> the<00:27:07.800> better<00:27:08.080> way<00:27:08.160> to<00:27:08.280> reason<00:27:08.520> about" + }, + { + "start": 1628.75, + "duration": 0.0, + "text": "However, the better way to reason about" + }, + { + "start": 1628.76, + "duration": 0.0, + "text": "However, the better way to reason about it<00:27:08.840> is<00:27:09.120> independence<00:27:09.680> is<00:27:09.920> epistemological," + }, + { + "start": 1631.07, + "duration": 0.0, + "text": "it is independence is epistemological," + }, + { + "start": 1631.08, + "duration": 0.0, + "text": "it is independence is epistemological, meaning<00:27:11.400> it's<00:27:11.600> just<00:27:11.800> a<00:27:11.840> property<00:27:12.640> of<00:27:12.960> our" + }, + { + "start": 1633.07, + "duration": 0.0, + "text": "meaning it's just a property of our" + }, + { + "start": 1633.08, + "duration": 0.0, + "text": "meaning it's just a property of our knowledge." + }, + { + "start": 1635.84, + "duration": 0.0, + "text": "Formally,<00:27:16.400> independence<00:27:17.000> simply<00:27:17.280> means<00:27:17.560> that" + }, + { + "start": 1637.99, + "duration": 0.0, + "text": "Formally, independence simply means that" + }, + { + "start": 1638.0, + "duration": 0.0, + "text": "Formally, independence simply means that learning<00:27:18.680> about<00:27:19.160> one<00:27:19.400> system<00:27:20.000> doesn't<00:27:20.360> tell" + }, + { + "start": 1640.55, + "duration": 0.0, + "text": "learning about one system doesn't tell" + }, + { + "start": 1640.56, + "duration": 0.0, + "text": "learning about one system doesn't tell us<00:27:20.720> anything<00:27:21.040> about<00:27:21.240> the<00:27:21.360> other." + }, + { + "start": 1642.35, + "duration": 0.0, + "text": "us anything about the other." + }, + { + "start": 1642.36, + "duration": 0.0, + "text": "us anything about the other. Now,<00:27:22.960> to<00:27:23.120> illustrate<00:27:23.960> why<00:27:24.480> in<00:27:24.600> my<00:27:24.720> opinion" + }, + { + "start": 1645.07, + "duration": 0.0, + "text": "Now, to illustrate why in my opinion" + }, + { + "start": 1645.08, + "duration": 0.0, + "text": "Now, to illustrate why in my opinion it's<00:27:25.200> better<00:27:25.400> to<00:27:25.520> say<00:27:25.680> this<00:27:25.840> is" + }, + { + "start": 1646.31, + "duration": 0.0, + "text": "it's better to say this is" + }, + { + "start": 1646.32, + "duration": 0.0, + "text": "it's better to say this is epistemological<00:27:27.240> is<00:27:27.880> imagine<00:27:28.320> the<00:27:28.800> classical" + }, + { + "start": 1649.43, + "duration": 0.0, + "text": "epistemological is imagine the classical" + }, + { + "start": 1649.44, + "duration": 0.0, + "text": "epistemological is imagine the classical Bernoulli<00:27:29.920> urn,<00:27:30.200> right?<00:27:30.440> You<00:27:30.520> have<00:27:30.960> red<00:27:31.200> balls" + }, + { + "start": 1651.51, + "duration": 0.0, + "text": "Bernoulli urn, right? You have red balls" + }, + { + "start": 1651.52, + "duration": 0.0, + "text": "Bernoulli urn, right? You have red balls and<00:27:31.680> white<00:27:31.920> balls." + }, + { + "start": 1653.15, + "duration": 0.0, + "text": "and white balls." + }, + { + "start": 1653.16, + "duration": 0.0, + "text": "and white balls. You<00:27:33.480> draw<00:27:33.880> one<00:27:34.440> and<00:27:34.600> you<00:27:34.840> record<00:27:35.120> the<00:27:35.160> result" + }, + { + "start": 1655.47, + "duration": 0.0, + "text": "You draw one and you record the result" + }, + { + "start": 1655.48, + "duration": 0.0, + "text": "You draw one and you record the result and<00:27:35.560> you<00:27:35.600> put<00:27:35.760> it<00:27:35.840> back<00:27:36.040> in,<00:27:36.200> you<00:27:36.320> shake<00:27:36.600> it<00:27:36.680> and" + }, + { + "start": 1656.83, + "duration": 0.0, + "text": "and you put it back in, you shake it and" + }, + { + "start": 1656.84, + "duration": 0.0, + "text": "and you put it back in, you shake it and whatnot." + }, + { + "start": 1658.27, + "duration": 0.0, + "text": "whatnot." + }, + { + "start": 1658.28, + "duration": 0.0, + "text": "whatnot. Those<00:27:39.160> this<00:27:39.320> kind<00:27:39.440> of<00:27:39.520> experiment<00:27:40.200> is" + }, + { + "start": 1660.31, + "duration": 0.0, + "text": "Those this kind of experiment is" + }, + { + "start": 1660.32, + "duration": 0.0, + "text": "Those this kind of experiment is typically<00:27:40.600> described<00:27:41.160> as<00:27:42.080> IID<00:27:42.560> random" + }, + { + "start": 1662.83, + "duration": 0.0, + "text": "typically described as IID random" + }, + { + "start": 1662.84, + "duration": 0.0, + "text": "typically described as IID random variables,<00:27:43.360> right?<00:27:43.600> You<00:27:43.760> have<00:27:44.560> always<00:27:44.800> the" + }, + { + "start": 1664.87, + "duration": 0.0, + "text": "variables, right? You have always the" + }, + { + "start": 1664.88, + "duration": 0.0, + "text": "variables, right? You have always the same<00:27:45.040> distribution<00:27:45.600> of<00:27:45.760> the<00:27:45.880> same<00:27:46.120> balls<00:27:46.840> and" + }, + { + "start": 1667.03, + "duration": 0.0, + "text": "same distribution of the same balls and" + }, + { + "start": 1667.04, + "duration": 0.0, + "text": "same distribution of the same balls and all<00:27:47.160> the<00:27:47.280> draws<00:27:47.600> are<00:27:47.680> independent,<00:27:48.240> meaning" + }, + { + "start": 1668.91, + "duration": 0.0, + "text": "all the draws are independent, meaning" + }, + { + "start": 1668.92, + "duration": 0.0, + "text": "all the draws are independent, meaning seeing<00:27:49.280> one<00:27:49.600> doesn't<00:27:50.080> tell<00:27:50.120> you<00:27:50.240> anything" + }, + { + "start": 1670.51, + "duration": 0.0, + "text": "seeing one doesn't tell you anything" + }, + { + "start": 1670.52, + "duration": 0.0, + "text": "seeing one doesn't tell you anything about<00:27:50.760> the<00:27:50.840> other.<00:27:51.280> However,<00:27:52.040> physically" + }, + { + "start": 1672.51, + "duration": 0.0, + "text": "about the other. However, physically" + }, + { + "start": 1672.52, + "duration": 0.0, + "text": "about the other. However, physically speaking,<00:27:53.400> they<00:27:53.600> very<00:27:53.920> much<00:27:54.280> depend<00:27:54.640> on<00:27:54.760> each" + }, + { + "start": 1674.87, + "duration": 0.0, + "text": "speaking, they very much depend on each" + }, + { + "start": 1674.88, + "duration": 0.0, + "text": "speaking, they very much depend on each other." + }, + { + "start": 1675.83, + "duration": 0.0, + "text": "other." + }, + { + "start": 1675.84, + "duration": 0.0, + "text": "other. I<00:27:56.400> mean,<00:27:56.840> the<00:27:57.280> the<00:27:57.360> next<00:27:57.600> result<00:27:57.960> definitely" + }, + { + "start": 1678.27, + "duration": 0.0, + "text": "I mean, the the next result definitely" + }, + { + "start": 1678.28, + "duration": 0.0, + "text": "I mean, the the next result definitely depends<00:27:58.680> on<00:27:58.760> which<00:27:58.920> color<00:27:59.160> was<00:27:59.280> previously" + }, + { + "start": 1679.63, + "duration": 0.0, + "text": "depends on which color was previously" + }, + { + "start": 1679.64, + "duration": 0.0, + "text": "depends on which color was previously drawn<00:28:00.000> and<00:28:00.120> then<00:28:00.360> from<00:28:01.040> where<00:28:01.240> this<00:28:01.400> ball<00:28:01.640> was" + }, + { + "start": 1681.79, + "duration": 0.0, + "text": "drawn and then from where this ball was" + }, + { + "start": 1681.8, + "duration": 0.0, + "text": "drawn and then from where this ball was placed<00:28:02.120> again<00:28:02.360> into<00:28:02.560> the<00:28:02.640> ball," + }, + { + "start": 1683.67, + "duration": 0.0, + "text": "placed again into the ball," + }, + { + "start": 1683.68, + "duration": 0.0, + "text": "placed again into the ball, how<00:28:03.800> the<00:28:03.920> ball<00:28:04.200> was<00:28:04.400> shaken<00:28:05.120> and<00:28:05.320> of<00:28:05.440> course" + }, + { + "start": 1685.67, + "duration": 0.0, + "text": "how the ball was shaken and of course" + }, + { + "start": 1685.68, + "duration": 0.0, + "text": "how the ball was shaken and of course and" + }, + { + "start": 1686.35, + "duration": 0.0, + "text": "and" + }, + { + "start": 1686.36, + "duration": 0.0, + "text": "and all<00:28:06.480> the<00:28:06.600> attrition<00:28:07.120> coefficients<00:28:07.800> between" + }, + { + "start": 1688.07, + "duration": 0.0, + "text": "all the attrition coefficients between" + }, + { + "start": 1688.08, + "duration": 0.0, + "text": "all the attrition coefficients between the<00:28:08.200> walls" + }, + { + "start": 1689.43, + "duration": 0.0, + "text": "the walls" + }, + { + "start": 1689.44, + "duration": 0.0, + "text": "the walls of<00:28:09.600> the<00:28:09.960> and<00:28:10.080> the<00:28:10.160> balls<00:28:10.560> and<00:28:10.680> between<00:28:10.920> the" + }, + { + "start": 1690.99, + "duration": 0.0, + "text": "of the and the balls and between the" + }, + { + "start": 1691.0, + "duration": 0.0, + "text": "of the and the balls and between the balls<00:28:11.240> themselves,<00:28:11.640> etc.<00:28:12.200> But" + }, + { + "start": 1692.31, + "duration": 0.0, + "text": "balls themselves, etc. But" + }, + { + "start": 1692.32, + "duration": 0.0, + "text": "balls themselves, etc. But unfortunately,<00:28:12.800> all<00:28:12.960> of<00:28:13.040> these<00:28:13.240> influences" + }, + { + "start": 1693.79, + "duration": 0.0, + "text": "unfortunately, all of these influences" + }, + { + "start": 1693.8, + "duration": 0.0, + "text": "unfortunately, all of these influences are<00:28:13.920> so<00:28:14.080> incredibly<00:28:14.960> complicated<00:28:15.560> to<00:28:15.640> track" + }, + { + "start": 1696.59, + "duration": 0.0, + "text": "are so incredibly complicated to track" + }, + { + "start": 1696.6, + "duration": 0.0, + "text": "are so incredibly complicated to track that" + }, + { + "start": 1697.75, + "duration": 0.0, + "text": "that" + }, + { + "start": 1697.76, + "duration": 0.0, + "text": "that our<00:28:18.000> attempts<00:28:18.440> to<00:28:18.720> extract<00:28:19.200> any<00:28:19.360> kind<00:28:19.520> of" + }, + { + "start": 1699.59, + "duration": 0.0, + "text": "our attempts to extract any kind of" + }, + { + "start": 1699.6, + "duration": 0.0, + "text": "our attempts to extract any kind of information<00:28:20.080> from<00:28:20.240> them<00:28:20.480> would<00:28:20.640> be<00:28:20.800> almost" + }, + { + "start": 1701.11, + "duration": 0.0, + "text": "information from them would be almost" + }, + { + "start": 1701.12, + "duration": 0.0, + "text": "information from them would be almost futile." + }, + { + "start": 1702.31, + "duration": 0.0, + "text": "futile." + }, + { + "start": 1702.32, + "duration": 0.0, + "text": "futile. And<00:28:22.520> so<00:28:22.680> the<00:28:22.800> draws<00:28:23.160> are<00:28:23.280> definitely<00:28:23.680> not" + }, + { + "start": 1704.19, + "duration": 0.0, + "text": "And so the draws are definitely not" + }, + { + "start": 1704.2, + "duration": 0.0, + "text": "And so the draws are definitely not physically<00:28:24.680> independent.<00:28:25.320> However," + }, + { + "start": 1707.19, + "duration": 0.0, + "text": "physically independent. However," + }, + { + "start": 1707.2, + "duration": 0.0, + "text": "physically independent. However, we<00:28:27.920> just<00:28:28.280> because<00:28:28.560> we<00:28:28.680> don't<00:28:28.920> know<00:28:29.080> how<00:28:29.200> to<00:28:29.320> do" + }, + { + "start": 1709.43, + "duration": 0.0, + "text": "we just because we don't know how to do" + }, + { + "start": 1709.44, + "duration": 0.0, + "text": "we just because we don't know how to do any<00:28:29.600> better,<00:28:29.840> we<00:28:30.000> ignore<00:28:30.480> all<00:28:30.640> of<00:28:30.760> these" + }, + { + "start": 1711.15, + "duration": 0.0, + "text": "any better, we ignore all of these" + }, + { + "start": 1711.16, + "duration": 0.0, + "text": "any better, we ignore all of these non-linear<00:28:31.760> dependencies.<00:28:32.640> And<00:28:32.760> in<00:28:32.840> fact," + }, + { + "start": 1713.43, + "duration": 0.0, + "text": "non-linear dependencies. And in fact," + }, + { + "start": 1713.44, + "duration": 0.0, + "text": "non-linear dependencies. And in fact, confusing<00:28:34.240> physical<00:28:34.680> dependence<00:28:35.200> with" + }, + { + "start": 1715.35, + "duration": 0.0, + "text": "confusing physical dependence with" + }, + { + "start": 1715.36, + "duration": 0.0, + "text": "confusing physical dependence with logical<00:28:35.840> independence<00:28:36.800> has<00:28:37.040> led<00:28:37.280> to<00:28:37.640> leads<00:28:37.880> to" + }, + { + "start": 1718.11, + "duration": 0.0, + "text": "logical independence has led to leads to" + }, + { + "start": 1718.12, + "duration": 0.0, + "text": "logical independence has led to leads to all<00:28:38.320> sorts<00:28:38.520> of<00:28:38.640> paradoxes<00:28:39.200> in<00:28:39.280> practice.<00:28:39.960> Um" + }, + { + "start": 1721.47, + "duration": 0.0, + "text": "all sorts of paradoxes in practice. Um" + }, + { + "start": 1721.48, + "duration": 0.0, + "text": "all sorts of paradoxes in practice. Um Famously," + }, + { + "start": 1723.03, + "duration": 0.0, + "text": "Famously," + }, + { + "start": 1723.04, + "duration": 0.0, + "text": "Famously, in<00:28:43.480> quantum<00:28:43.760> mechanics,<00:28:44.480> this<00:28:44.920> this<00:28:45.080> is" + }, + { + "start": 1725.87, + "duration": 0.0, + "text": "in quantum mechanics, this this is" + }, + { + "start": 1725.88, + "duration": 0.0, + "text": "in quantum mechanics, this this is uh" + }, + { + "start": 1726.67, + "duration": 0.0, + "text": "uh" + }, + { + "start": 1726.68, + "duration": 0.0, + "text": "uh you<00:28:46.800> have<00:28:46.960> some<00:28:47.080> paradoxes<00:28:47.520> because<00:28:47.800> of<00:28:47.920> this," + }, + { + "start": 1728.07, + "duration": 0.0, + "text": "you have some paradoxes because of this," + }, + { + "start": 1728.08, + "duration": 0.0, + "text": "you have some paradoxes because of this, but<00:28:48.200> in<00:28:48.320> all<00:28:48.440> sorts<00:28:48.640> of<00:28:48.760> other<00:28:48.920> fields<00:28:49.200> as" + }, + { + "start": 1729.31, + "duration": 0.0, + "text": "but in all sorts of other fields as" + }, + { + "start": 1729.32, + "duration": 0.0, + "text": "but in all sorts of other fields as well." + }, + { + "start": 1729.99, + "duration": 0.0, + "text": "well." + }, + { + "start": 1730.0, + "duration": 0.0, + "text": "well. So,<00:28:50.080> this<00:28:50.240> is<00:28:50.360> distinction<00:28:50.760> is<00:28:50.880> very" + }, + { + "start": 1731.11, + "duration": 0.0, + "text": "So, this is distinction is very" + }, + { + "start": 1731.12, + "duration": 0.0, + "text": "So, this is distinction is very important<00:28:51.440> to<00:28:51.480> keep<00:28:51.680> in<00:28:51.760> mind." + }, + { + "start": 1734.28, + "duration": 0.0, + "text": "Then<00:28:54.400> we<00:28:54.480> have<00:28:54.600> a<00:28:54.640> new<00:28:54.800> symmetry." + }, + { + "start": 1736.07, + "duration": 0.0, + "text": "Then we have a new symmetry." + }, + { + "start": 1736.08, + "duration": 0.0, + "text": "Then we have a new symmetry. Uh<00:28:56.200> we<00:28:56.320> assume<00:28:56.880> or<00:28:57.200> we<00:28:57.360> see<00:28:57.480> that<00:28:57.600> the<00:28:58.000> the<00:28:58.360> the" + }, + { + "start": 1738.43, + "duration": 0.0, + "text": "Uh we assume or we see that the the the" + }, + { + "start": 1738.44, + "duration": 0.0, + "text": "Uh we assume or we see that the the the direct<00:28:58.840> product<00:28:59.240> is<00:28:59.480> associative." + }, + { + "start": 1740.79, + "duration": 0.0, + "text": "direct product is associative." + }, + { + "start": 1740.8, + "duration": 0.0, + "text": "direct product is associative. And<00:29:00.960> therefore<00:29:01.200> also<00:29:01.360> the<00:29:01.480> operator.<00:29:02.280> And<00:29:03.160> um" + }, + { + "start": 1743.27, + "duration": 0.0, + "text": "And therefore also the operator. And um" + }, + { + "start": 1743.28, + "duration": 0.0, + "text": "And therefore also the operator. And um again,<00:29:03.560> this<00:29:03.720> would<00:29:03.920> mean<00:29:04.240> that<00:29:04.400> if<00:29:04.520> this<00:29:04.720> were" + }, + { + "start": 1744.79, + "duration": 0.0, + "text": "again, this would mean that if this were" + }, + { + "start": 1744.8, + "duration": 0.0, + "text": "again, this would mean that if this were not<00:29:05.040> true," + }, + { + "start": 1745.95, + "duration": 0.0, + "text": "not true," + }, + { + "start": 1745.96, + "duration": 0.0, + "text": "not true, then<00:29:06.880> the<00:29:07.000> nature<00:29:07.520> of<00:29:07.680> the<00:29:07.800> elements<00:29:08.160> of<00:29:08.240> a" + }, + { + "start": 1748.27, + "duration": 0.0, + "text": "then the nature of the elements of a" + }, + { + "start": 1748.28, + "duration": 0.0, + "text": "then the nature of the elements of a composite<00:29:08.800> system<00:29:09.160> would<00:29:09.280> somehow<00:29:09.560> depend<00:29:10.760> on" + }, + { + "start": 1750.87, + "duration": 0.0, + "text": "composite system would somehow depend on" + }, + { + "start": 1750.88, + "duration": 0.0, + "text": "composite system would somehow depend on the<00:29:10.960> order<00:29:11.200> in<00:29:11.280> which<00:29:11.400> the<00:29:11.800> independent" + }, + { + "start": 1752.27, + "duration": 0.0, + "text": "the order in which the independent" + }, + { + "start": 1752.28, + "duration": 0.0, + "text": "the order in which the independent contributions<00:29:12.800> were<00:29:12.960> added." + }, + { + "start": 1753.95, + "duration": 0.0, + "text": "contributions were added." + }, + { + "start": 1753.96, + "duration": 0.0, + "text": "contributions were added. Uh<00:29:14.120> meaning,<00:29:14.560> if<00:29:14.760> we<00:29:14.840> somehow<00:29:15.680> combined<00:29:16.600> coins" + }, + { + "start": 1757.15, + "duration": 0.0, + "text": "Uh meaning, if we somehow combined coins" + }, + { + "start": 1757.16, + "duration": 0.0, + "text": "Uh meaning, if we somehow combined coins and<00:29:17.280> die<00:29:17.480> first<00:29:17.880> and<00:29:18.000> then<00:29:18.240> added<00:29:18.480> somehow<00:29:18.760> an" + }, + { + "start": 1758.83, + "duration": 0.0, + "text": "and die first and then added somehow an" + }, + { + "start": 1758.84, + "duration": 0.0, + "text": "and die first and then added somehow an analysis<00:29:19.480> on<00:29:19.880> the<00:29:20.000> weather,<00:29:20.720> just<00:29:20.880> some<00:29:21.280> uh" + }, + { + "start": 1762.35, + "duration": 0.0, + "text": "analysis on the weather, just some uh" + }, + { + "start": 1762.36, + "duration": 0.0, + "text": "analysis on the weather, just some uh random<00:29:22.640> example,<00:29:23.280> would<00:29:23.480> yield<00:29:23.680> a<00:29:23.720> different" + }, + { + "start": 1764.75, + "duration": 0.0, + "text": "random example, would yield a different" + }, + { + "start": 1764.76, + "duration": 0.0, + "text": "random example, would yield a different uh" + }, + { + "start": 1765.79, + "duration": 0.0, + "text": "uh" + }, + { + "start": 1765.8, + "duration": 0.0, + "text": "uh joint<00:29:26.040> space,<00:29:26.320> different<00:29:26.560> probability" + }, + { + "start": 1767.03, + "duration": 0.0, + "text": "joint space, different probability" + }, + { + "start": 1767.04, + "duration": 0.0, + "text": "joint space, different probability space,<00:29:27.360> than<00:29:27.520> if<00:29:27.640> we<00:29:27.720> combined<00:29:28.400> die<00:29:28.640> and<00:29:28.760> the" + }, + { + "start": 1768.87, + "duration": 0.0, + "text": "space, than if we combined die and the" + }, + { + "start": 1768.88, + "duration": 0.0, + "text": "space, than if we combined die and the weather<00:29:29.080> first<00:29:29.400> and<00:29:29.520> then<00:29:29.760> added<00:29:29.960> the<00:29:30.040> coin," + }, + { + "start": 1770.31, + "duration": 0.0, + "text": "weather first and then added the coin," + }, + { + "start": 1770.32, + "duration": 0.0, + "text": "weather first and then added the coin, which<00:29:30.480> makes<00:29:30.680> no<00:29:30.840> sense,<00:29:31.080> of<00:29:31.200> course." + }, + { + "start": 1774.2, + "duration": 0.0, + "text": "And<00:29:34.600> we<00:29:34.720> can<00:29:34.920> also<00:29:35.720> to<00:29:35.880> apply<00:29:36.280> our" + }, + { + "start": 1776.43, + "duration": 0.0, + "text": "And we can also to apply our" + }, + { + "start": 1776.44, + "duration": 0.0, + "text": "And we can also to apply our associativity<00:29:37.040> theorem,<00:29:37.320> we<00:29:37.440> also<00:29:37.720> need" + }, + { + "start": 1779.07, + "duration": 0.0, + "text": "associativity theorem, we also need" + }, + { + "start": 1779.08, + "duration": 0.0, + "text": "associativity theorem, we also need symmetry<00:29:39.440> one.<00:29:39.840> We<00:29:39.920> need<00:29:40.040> to<00:29:40.160> show<00:29:40.320> it<00:29:40.440> for" + }, + { + "start": 1780.67, + "duration": 0.0, + "text": "symmetry one. We need to show it for" + }, + { + "start": 1780.68, + "duration": 0.0, + "text": "symmetry one. We need to show it for this<00:29:40.960> new<00:29:41.520> operator." + }, + { + "start": 1782.79, + "duration": 0.0, + "text": "this new operator." + }, + { + "start": 1782.8, + "duration": 0.0, + "text": "this new operator. The<00:29:42.840> direct<00:29:43.120> product<00:29:43.480> also<00:29:43.760> preserves<00:29:44.160> order." + }, + { + "start": 1785.75, + "duration": 0.0, + "text": "The direct product also preserves order." + }, + { + "start": 1785.76, + "duration": 0.0, + "text": "The direct product also preserves order. Uh<00:29:46.160> meaning" + }, + { + "start": 1787.47, + "duration": 0.0, + "text": "Uh meaning" + }, + { + "start": 1787.48, + "duration": 0.0, + "text": "Uh meaning um" + }, + { + "start": 1788.39, + "duration": 0.0, + "text": "um" + }, + { + "start": 1788.4, + "duration": 0.0, + "text": "um that<00:29:48.560> if<00:29:48.680> we<00:29:48.760> have<00:29:48.880> some<00:29:49.000> kind<00:29:49.160> of<00:29:49.480> uh" + }, + { + "start": 1791.07, + "duration": 0.0, + "text": "that if we have some kind of uh" + }, + { + "start": 1791.08, + "duration": 0.0, + "text": "that if we have some kind of uh uh<00:29:51.480> proposition<00:29:51.960> Y<00:29:52.560> that<00:29:52.760> can<00:29:52.920> be<00:29:53.400> decomposed" + }, + { + "start": 1793.99, + "duration": 0.0, + "text": "uh proposition Y that can be decomposed" + }, + { + "start": 1794.0, + "duration": 0.0, + "text": "uh proposition Y that can be decomposed into" + }, + { + "start": 1795.43, + "duration": 0.0, + "text": "into" + }, + { + "start": 1795.44, + "duration": 0.0, + "text": "into uh<00:29:55.600> X<00:29:55.840> or<00:29:55.960> D," + }, + { + "start": 1797.63, + "duration": 0.0, + "text": "uh X or D," + }, + { + "start": 1797.64, + "duration": 0.0, + "text": "uh X or D, if<00:29:57.720> we<00:29:57.800> apply<00:29:58.320> the<00:29:58.520> direct<00:29:58.880> product,<00:29:59.280> we" + }, + { + "start": 1799.39, + "duration": 0.0, + "text": "if we apply the direct product, we" + }, + { + "start": 1799.4, + "duration": 0.0, + "text": "if we apply the direct product, we obtain<00:30:00.200> uh<00:30:00.360> this.<00:30:01.200> But<00:30:01.360> since<00:30:01.960> neither<00:30:02.440> X" + }, + { + "start": 1803.99, + "duration": 0.0, + "text": "obtain uh this. But since neither X" + }, + { + "start": 1804.0, + "duration": 0.0, + "text": "obtain uh this. But since neither X nor<00:30:04.280> U<00:30:04.840> nor<00:30:05.480> D<00:30:05.600> times<00:30:05.920> X<00:30:06.840> are<00:30:07.080> the<00:30:07.200> bottom" + }, + { + "start": 1807.51, + "duration": 0.0, + "text": "nor U nor D times X are the bottom" + }, + { + "start": 1807.52, + "duration": 0.0, + "text": "nor U nor D times X are the bottom elements<00:30:08.120> here<00:30:08.360> by<00:30:08.480> assumption.<00:30:09.040> Uh<00:30:09.080> this<00:30:09.320> is" + }, + { + "start": 1809.43, + "duration": 0.0, + "text": "elements here by assumption. Uh this is" + }, + { + "start": 1809.44, + "duration": 0.0, + "text": "elements here by assumption. Uh this is definitely<00:30:09.760> possible." + }, + { + "start": 1810.87, + "duration": 0.0, + "text": "definitely possible." + }, + { + "start": 1810.88, + "duration": 0.0, + "text": "definitely possible. Uh<00:30:11.240> then<00:30:11.480> it<00:30:11.600> must<00:30:11.880> mean<00:30:12.320> that<00:30:12.480> somehow<00:30:13.600> Y" + }, + { + "start": 1814.11, + "duration": 0.0, + "text": "Uh then it must mean that somehow Y" + }, + { + "start": 1814.12, + "duration": 0.0, + "text": "Uh then it must mean that somehow Y times<00:30:14.600> U" + }, + { + "start": 1815.59, + "duration": 0.0, + "text": "times U" + }, + { + "start": 1815.6, + "duration": 0.0, + "text": "times U must<00:30:15.880> have<00:30:16.120> a<00:30:16.200> higher<00:30:16.400> valuation" + }, + { + "start": 1817.47, + "duration": 0.0, + "text": "must have a higher valuation" + }, + { + "start": 1817.48, + "duration": 0.0, + "text": "must have a higher valuation than<00:30:17.640> X<00:30:17.880> times<00:30:18.160> U<00:30:18.360> because<00:30:19.000> in<00:30:19.120> order<00:30:19.280> to" + }, + { + "start": 1819.39, + "duration": 0.0, + "text": "than X times U because in order to" + }, + { + "start": 1819.4, + "duration": 0.0, + "text": "than X times U because in order to obtain<00:30:19.880> this<00:30:20.080> one,<00:30:20.240> we<00:30:20.320> must<00:30:20.480> have<00:30:20.880> combined<00:30:21.280> X" + }, + { + "start": 1821.43, + "duration": 0.0, + "text": "obtain this one, we must have combined X" + }, + { + "start": 1821.44, + "duration": 0.0, + "text": "obtain this one, we must have combined X times<00:30:21.680> U<00:30:21.960> with<00:30:22.680> another<00:30:22.920> valuation.<00:30:23.480> And<00:30:23.600> so," + }, + { + "start": 1823.83, + "duration": 0.0, + "text": "times U with another valuation. And so," + }, + { + "start": 1823.84, + "duration": 0.0, + "text": "times U with another valuation. And so, we<00:30:24.040> can" + }, + { + "start": 1825.15, + "duration": 0.0, + "text": "we can" + }, + { + "start": 1825.16, + "duration": 0.0, + "text": "we can write<00:30:25.360> this<00:30:25.480> as<00:30:25.600> such." + }, + { + "start": 1826.71, + "duration": 0.0, + "text": "write this as such." + }, + { + "start": 1826.72, + "duration": 0.0, + "text": "write this as such. To<00:30:26.760> see<00:30:26.880> that<00:30:27.040> X<00:30:27.240> times<00:30:27.480> U<00:30:27.800> is<00:30:28.080> smaller<00:30:28.400> than," + }, + { + "start": 1828.91, + "duration": 0.0, + "text": "To see that X times U is smaller than," + }, + { + "start": 1828.92, + "duration": 0.0, + "text": "To see that X times U is smaller than, well,<00:30:29.240> this<00:30:29.880> uh" + }, + { + "start": 1830.87, + "duration": 0.0, + "text": "well, this uh" + }, + { + "start": 1830.88, + "duration": 0.0, + "text": "well, this uh combined<00:30:31.440> statements,<00:30:32.080> which<00:30:32.320> is<00:30:32.440> smaller" + }, + { + "start": 1832.75, + "duration": 0.0, + "text": "combined statements, which is smaller" + }, + { + "start": 1832.76, + "duration": 0.0, + "text": "combined statements, which is smaller than<00:30:33.560> the<00:30:33.640> top<00:30:33.880> one." + }, + { + "start": 1834.79, + "duration": 0.0, + "text": "than the top one." + }, + { + "start": 1834.8, + "duration": 0.0, + "text": "than the top one. And<00:30:34.880> so,<00:30:35.040> we<00:30:35.160> have<00:30:35.320> symmetry<00:30:35.720> one<00:30:36.480> again<00:30:36.680> also" + }, + { + "start": 1836.87, + "duration": 0.0, + "text": "And so, we have symmetry one again also" + }, + { + "start": 1836.88, + "duration": 0.0, + "text": "And so, we have symmetry one again also for<00:30:37.040> this<00:30:37.640> uh" + }, + { + "start": 1838.19, + "duration": 0.0, + "text": "for this uh" + }, + { + "start": 1838.2, + "duration": 0.0, + "text": "for this uh product." + }, + { + "start": 1839.31, + "duration": 0.0, + "text": "product." + }, + { + "start": 1839.32, + "duration": 0.0, + "text": "product. Now,<00:30:39.840> interestingly,<00:30:40.280> we<00:30:40.440> can<00:30:41.080> uh<00:30:41.160> apply<00:30:41.400> the" + }, + { + "start": 1841.47, + "duration": 0.0, + "text": "Now, interestingly, we can uh apply the" + }, + { + "start": 1841.48, + "duration": 0.0, + "text": "Now, interestingly, we can uh apply the associ-<00:30:42.120> associativity<00:30:42.840> theorem<00:30:43.280> again," + }, + { + "start": 1843.71, + "duration": 0.0, + "text": "associ- associativity theorem again," + }, + { + "start": 1843.72, + "duration": 0.0, + "text": "associ- associativity theorem again, which<00:30:43.920> is<00:30:44.000> very<00:30:44.200> convenient,<00:30:44.560> of<00:30:44.680> course." + }, + { + "start": 1845.51, + "duration": 0.0, + "text": "which is very convenient, of course." + }, + { + "start": 1845.52, + "duration": 0.0, + "text": "which is very convenient, of course. Um<00:30:46.200> so<00:30:46.320> again,<00:30:46.920> we<00:30:47.040> have" + }, + { + "start": 1848.07, + "duration": 0.0, + "text": "Um so again, we have" + }, + { + "start": 1848.08, + "duration": 0.0, + "text": "Um so again, we have two<00:30:48.560> propositions" + }, + { + "start": 1849.83, + "duration": 0.0, + "text": "two propositions" + }, + { + "start": 1849.84, + "duration": 0.0, + "text": "two propositions um<00:30:50.640> of<00:30:50.800> the<00:30:51.440> that<00:30:51.640> are<00:30:51.760> elements<00:30:52.080> of<00:30:52.200> the" + }, + { + "start": 1852.27, + "duration": 0.0, + "text": "um of the that are elements of the" + }, + { + "start": 1852.28, + "duration": 0.0, + "text": "um of the that are elements of the lattice<00:30:53.280> and<00:30:53.400> with<00:30:53.600> the<00:30:53.720> evaluations<00:30:54.520> and<00:30:54.720> we" + }, + { + "start": 1854.83, + "duration": 0.0, + "text": "lattice and with the evaluations and we" + }, + { + "start": 1854.84, + "duration": 0.0, + "text": "lattice and with the evaluations and we must<00:30:55.240> and<00:30:55.400> we<00:30:55.480> know<00:30:55.720> that<00:30:55.880> somehow<00:30:56.160> they<00:30:56.280> must" + }, + { + "start": 1856.67, + "duration": 0.0, + "text": "must and we know that somehow they must" + }, + { + "start": 1856.68, + "duration": 0.0, + "text": "must and we know that somehow they must satisfy<00:30:57.080> this.<00:30:57.360> However," + }, + { + "start": 1858.59, + "duration": 0.0, + "text": "satisfy this. However," + }, + { + "start": 1858.6, + "duration": 0.0, + "text": "satisfy this. However, while<00:30:58.920> before<00:30:59.680> with<00:30:59.840> the<00:30:59.920> combination,<00:31:00.560> we" + }, + { + "start": 1860.67, + "duration": 0.0, + "text": "while before with the combination, we" + }, + { + "start": 1860.68, + "duration": 0.0, + "text": "while before with the combination, we could<00:31:00.800> just<00:31:00.960> simply<00:31:01.160> throw<00:31:01.440> away" + }, + { + "start": 1862.43, + "duration": 0.0, + "text": "could just simply throw away" + }, + { + "start": 1862.44, + "duration": 0.0, + "text": "could just simply throw away the<00:31:02.680> old<00:31:03.120> valuations<00:31:03.800> and<00:31:03.960> simply<00:31:04.520> use<00:31:04.920> the" + }, + { + "start": 1866.23, + "duration": 0.0, + "text": "the old valuations and simply use the" + }, + { + "start": 1866.24, + "duration": 0.0, + "text": "the old valuations and simply use the the<00:31:06.440> the<00:31:06.560> new<00:31:06.800> ones<00:31:07.120> given<00:31:07.320> us<00:31:07.440> by<00:31:07.520> the" + }, + { + "start": 1867.63, + "duration": 0.0, + "text": "the the new ones given us by the" + }, + { + "start": 1867.64, + "duration": 0.0, + "text": "the the new ones given us by the mapping,<00:31:08.000> here<00:31:08.240> we<00:31:08.320> cannot<00:31:08.560> do<00:31:08.640> this<00:31:08.880> because" + }, + { + "start": 1869.15, + "duration": 0.0, + "text": "mapping, here we cannot do this because" + }, + { + "start": 1869.16, + "duration": 0.0, + "text": "mapping, here we cannot do this because the<00:31:09.280> plus<00:31:09.640> operation<00:31:10.120> was<00:31:10.280> already<00:31:11.080> chosen" + }, + { + "start": 1872.31, + "duration": 0.0, + "text": "the plus operation was already chosen" + }, + { + "start": 1872.32, + "duration": 0.0, + "text": "the plus operation was already chosen uh<00:31:12.520> for<00:31:12.760> the<00:31:12.840> combination<00:31:13.280> of<00:31:13.400> the<00:31:13.480> joint" + }, + { + "start": 1873.67, + "duration": 0.0, + "text": "uh for the combination of the joint" + }, + { + "start": 1873.68, + "duration": 0.0, + "text": "uh for the combination of the joint elements.<00:31:14.080> So,<00:31:14.320> we<00:31:14.480> must<00:31:14.960> find<00:31:15.680> uh<00:31:15.800> something" + }, + { + "start": 1876.11, + "duration": 0.0, + "text": "elements. So, we must find uh something" + }, + { + "start": 1876.12, + "duration": 0.0, + "text": "elements. So, we must find uh something else." + }, + { + "start": 1877.03, + "duration": 0.0, + "text": "else." + }, + { + "start": 1877.04, + "duration": 0.0, + "text": "else. And<00:31:17.240> to<00:31:17.320> do<00:31:17.520> that,<00:31:18.240> uh<00:31:18.400> we<00:31:18.520> must<00:31:19.000> find<00:31:19.320> a<00:31:19.360> form" + }, + { + "start": 1879.83, + "duration": 0.0, + "text": "And to do that, uh we must find a form" + }, + { + "start": 1879.84, + "duration": 0.0, + "text": "And to do that, uh we must find a form of<00:31:20.600> theta<00:31:20.920> tilde." + }, + { + "start": 1882.55, + "duration": 0.0, + "text": "of theta tilde." + }, + { + "start": 1882.56, + "duration": 0.0, + "text": "of theta tilde. To<00:31:22.600> do<00:31:22.760> that,<00:31:23.000> we<00:31:23.600> um" + }, + { + "start": 1884.27, + "duration": 0.0, + "text": "To do that, we um" + }, + { + "start": 1884.28, + "duration": 0.0, + "text": "To do that, we um set<00:31:24.520> up<00:31:24.640> some<00:31:24.960> functional<00:31:25.111> [snorts]<00:31:25.280> equation" + }, + { + "start": 1886.31, + "duration": 0.0, + "text": "set up some functional [snorts] equation" + }, + { + "start": 1886.32, + "duration": 0.0, + "text": "set up some functional [snorts] equation uh<00:31:26.560> because" + }, + { + "start": 1888.07, + "duration": 0.0, + "text": "uh because" + }, + { + "start": 1888.08, + "duration": 0.0, + "text": "uh because uh<00:31:28.240> theta<00:31:28.480> tilde<00:31:28.720> needs<00:31:28.880> to<00:31:28.960> be<00:31:29.040> consistent" + }, + { + "start": 1889.67, + "duration": 0.0, + "text": "uh theta tilde needs to be consistent" + }, + { + "start": 1889.68, + "duration": 0.0, + "text": "uh theta tilde needs to be consistent with<00:31:29.960> symmetry<00:31:30.360> three<00:31:31.240> and<00:31:31.360> with<00:31:31.440> the<00:31:31.520> fact" + }, + { + "start": 1891.83, + "duration": 0.0, + "text": "with symmetry three and with the fact" + }, + { + "start": 1891.84, + "duration": 0.0, + "text": "with symmetry three and with the fact that<00:31:31.960> the<00:31:32.040> combination<00:31:32.600> operation<00:31:33.040> is" + }, + { + "start": 1893.15, + "duration": 0.0, + "text": "that the combination operation is" + }, + { + "start": 1893.16, + "duration": 0.0, + "text": "that the combination operation is addition.<00:31:33.960> So,<00:31:34.040> we<00:31:34.120> can<00:31:34.280> set<00:31:34.440> up<00:31:34.560> this<00:31:34.760> kind<00:31:35.000> of" + }, + { + "start": 1895.63, + "duration": 0.0, + "text": "addition. So, we can set up this kind of" + }, + { + "start": 1895.64, + "duration": 0.0, + "text": "addition. So, we can set up this kind of uh<00:31:35.800> equation." + }, + { + "start": 1896.91, + "duration": 0.0, + "text": "uh equation." + }, + { + "start": 1896.92, + "duration": 0.0, + "text": "uh equation. Uh" + }, + { + "start": 1897.67, + "duration": 0.0, + "text": "Uh" + }, + { + "start": 1897.68, + "duration": 0.0, + "text": "Uh and<00:31:37.840> if<00:31:38.000> we<00:31:38.080> define<00:31:38.960> phi<00:31:39.200> tilde<00:31:39.680> as<00:31:40.120> the" + }, + { + "start": 1900.19, + "duration": 0.0, + "text": "and if we define phi tilde as the" + }, + { + "start": 1900.2, + "duration": 0.0, + "text": "and if we define phi tilde as the opposite<00:31:40.760> of<00:31:40.880> theta" + }, + { + "start": 1902.03, + "duration": 0.0, + "text": "opposite of theta" + }, + { + "start": 1902.04, + "duration": 0.0, + "text": "opposite of theta with<00:31:42.160> the<00:31:42.240> inverse<00:31:42.520> mapping,<00:31:43.280> we<00:31:43.400> can<00:31:43.600> write" + }, + { + "start": 1904.23, + "duration": 0.0, + "text": "with the inverse mapping, we can write" + }, + { + "start": 1904.24, + "duration": 0.0, + "text": "with the inverse mapping, we can write uh" + }, + { + "start": 1904.95, + "duration": 0.0, + "text": "uh" + }, + { + "start": 1904.96, + "duration": 0.0, + "text": "uh this<00:31:45.400> equation<00:31:46.360> where" + }, + { + "start": 1907.71, + "duration": 0.0, + "text": "this equation where" + }, + { + "start": 1907.72, + "duration": 0.0, + "text": "this equation where the<00:31:47.840> the<00:31:47.960> plus<00:31:48.200> sign<00:31:48.400> here" + }, + { + "start": 1909.91, + "duration": 0.0, + "text": "the the plus sign here" + }, + { + "start": 1909.92, + "duration": 0.0, + "text": "the the plus sign here uh<00:31:50.120> is<00:31:50.280> used<00:31:51.280> uh" + }, + { + "start": 1912.87, + "duration": 0.0, + "text": "uh is used uh" + }, + { + "start": 1912.88, + "duration": 0.0, + "text": "uh is used uh this<00:31:53.400> because<00:31:53.760> it's<00:31:53.880> between<00:31:54.760> the<00:31:54.840> different" + }, + { + "start": 1915.15, + "duration": 0.0, + "text": "this because it's between the different" + }, + { + "start": 1915.16, + "duration": 0.0, + "text": "this because it's between the different elements<00:31:55.920> of<00:31:56.280> theta<00:31:57.040> and<00:31:57.200> here<00:31:57.520> the<00:31:57.600> different" + }, + { + "start": 1917.91, + "duration": 0.0, + "text": "elements of theta and here the different" + }, + { + "start": 1917.92, + "duration": 0.0, + "text": "elements of theta and here the different valuations<00:31:58.360> are<00:31:58.440> just<00:31:58.600> summed<00:31:58.800> together" + }, + { + "start": 1919.07, + "duration": 0.0, + "text": "valuations are just summed together" + }, + { + "start": 1919.08, + "duration": 0.0, + "text": "valuations are just summed together because<00:31:59.440> those<00:31:59.640> are<00:31:59.960> disjoint<00:32:00.400> elements" + }, + { + "start": 1920.71, + "duration": 0.0, + "text": "because those are disjoint elements" + }, + { + "start": 1920.72, + "duration": 0.0, + "text": "because those are disjoint elements being<00:32:00.960> combined<00:32:02.120> uh<00:32:02.400> by<00:32:02.520> the<00:32:02.600> first<00:32:02.880> operation" + }, + { + "start": 1923.31, + "duration": 0.0, + "text": "being combined uh by the first operation" + }, + { + "start": 1923.32, + "duration": 0.0, + "text": "being combined uh by the first operation that<00:32:03.440> we<00:32:03.560> saw." + }, + { + "start": 1925.92, + "duration": 0.0, + "text": "Um<00:32:06.600> and<00:32:06.760> once<00:32:06.920> again,<00:32:07.280> there<00:32:07.520> is<00:32:07.760> a<00:32:08.040> nice" + }, + { + "start": 1928.31, + "duration": 0.0, + "text": "Um and once again, there is a nice" + }, + { + "start": 1928.32, + "duration": 0.0, + "text": "Um and once again, there is a nice little<00:32:08.480> proof<00:32:08.680> for<00:32:08.800> this" + }, + { + "start": 1929.59, + "duration": 0.0, + "text": "little proof for this" + }, + { + "start": 1929.6, + "duration": 0.0, + "text": "little proof for this and<00:32:10.480> we<00:32:10.800> find<00:32:11.240> that<00:32:11.920> phi<00:32:12.440> is<00:32:12.640> actually<00:32:12.920> c<00:32:13.480> times" + }, + { + "start": 1933.79, + "duration": 0.0, + "text": "and we find that phi is actually c times" + }, + { + "start": 1933.8, + "duration": 0.0, + "text": "and we find that phi is actually c times e<00:32:13.880> to<00:32:14.000> the<00:32:14.120> ax,<00:32:15.200> which<00:32:15.400> in<00:32:15.520> turn<00:32:15.800> means<00:32:16.040> that<00:32:16.120> we" + }, + { + "start": 1936.23, + "duration": 0.0, + "text": "e to the ax, which in turn means that we" + }, + { + "start": 1936.24, + "duration": 0.0, + "text": "e to the ax, which in turn means that we can<00:32:16.400> write<00:32:16.920> uh<00:32:17.240> this<00:32:17.440> beautiful<00:32:17.760> equation" + }, + { + "start": 1938.87, + "duration": 0.0, + "text": "can write uh this beautiful equation" + }, + { + "start": 1938.88, + "duration": 0.0, + "text": "can write uh this beautiful equation where<00:32:19.160> 1<00:32:19.440> over<00:32:19.720> a<00:32:19.800> simplifies<00:32:20.920> and<00:32:21.560> we<00:32:21.720> are" + }, + { + "start": 1941.87, + "duration": 0.0, + "text": "where 1 over a simplifies and we are" + }, + { + "start": 1941.88, + "duration": 0.0, + "text": "where 1 over a simplifies and we are also<00:32:22.160> free<00:32:22.360> to<00:32:22.520> choose<00:32:23.360> c<00:32:23.880> equals<00:32:24.160> to<00:32:24.280> 1<00:32:24.680> with" + }, + { + "start": 1944.83, + "duration": 0.0, + "text": "also free to choose c equals to 1 with" + }, + { + "start": 1944.84, + "duration": 0.0, + "text": "also free to choose c equals to 1 with this<00:32:25.000> just<00:32:25.200> a<00:32:25.240> constant<00:32:26.200> with<00:32:26.360> no<00:32:26.480> loss<00:32:26.680> of" + }, + { + "start": 1946.75, + "duration": 0.0, + "text": "this just a constant with no loss of" + }, + { + "start": 1946.76, + "duration": 0.0, + "text": "this just a constant with no loss of generality." + }, + { + "start": 1948.15, + "duration": 0.0, + "text": "generality." + }, + { + "start": 1948.16, + "duration": 0.0, + "text": "generality. And" + }, + { + "start": 1949.43, + "duration": 0.0, + "text": "And" + }, + { + "start": 1949.44, + "duration": 0.0, + "text": "And this<00:32:29.640> in<00:32:29.760> turn<00:32:30.000> tells<00:32:30.320> us<00:32:30.880> um" + }, + { + "start": 1951.71, + "duration": 0.0, + "text": "this in turn tells us um" + }, + { + "start": 1951.72, + "duration": 0.0, + "text": "this in turn tells us um that<00:32:32.040> this<00:32:32.440> kind<00:32:32.600> of<00:32:32.680> product<00:32:32.960> operator<00:32:33.560> is" + }, + { + "start": 1953.75, + "duration": 0.0, + "text": "that this kind of product operator is" + }, + { + "start": 1953.76, + "duration": 0.0, + "text": "that this kind of product operator is just<00:32:34.160> multiplication," + }, + { + "start": 1955.63, + "duration": 0.0, + "text": "just multiplication," + }, + { + "start": 1955.64, + "duration": 0.0, + "text": "just multiplication, uh<00:32:35.720> which<00:32:35.840> is<00:32:35.960> of<00:32:36.080> course<00:32:36.360> the<00:32:37.040> product<00:32:37.360> rule" + }, + { + "start": 1957.51, + "duration": 0.0, + "text": "uh which is of course the product rule" + }, + { + "start": 1957.52, + "duration": 0.0, + "text": "uh which is of course the product rule that<00:32:37.640> we<00:32:37.760> all<00:32:37.880> know<00:32:38.080> from<00:32:38.240> probability.<00:32:38.920> If<00:32:39.080> we" + }, + { + "start": 1959.19, + "duration": 0.0, + "text": "that we all know from probability. If we" + }, + { + "start": 1959.2, + "duration": 0.0, + "text": "that we all know from probability. If we have" + }, + { + "start": 1960.15, + "duration": 0.0, + "text": "have" + }, + { + "start": 1960.16, + "duration": 0.0, + "text": "have uh,<00:32:40.560> the<00:32:40.680> joint<00:32:40.880> probability<00:32:41.400> of<00:32:41.560> two" + }, + { + "start": 1961.79, + "duration": 0.0, + "text": "uh, the joint probability of two" + }, + { + "start": 1961.8, + "duration": 0.0, + "text": "uh, the joint probability of two independent<00:32:43.200> uh,<00:32:43.280> random<00:32:43.560> variables<00:32:43.920> is<00:32:44.040> just" + }, + { + "start": 1964.35, + "duration": 0.0, + "text": "independent uh, random variables is just" + }, + { + "start": 1964.36, + "duration": 0.0, + "text": "independent uh, random variables is just the<00:32:44.480> product<00:32:44.840> of<00:32:44.960> the<00:32:45.040> individual" + }, + { + "start": 1965.55, + "duration": 0.0, + "text": "the product of the individual" + }, + { + "start": 1965.56, + "duration": 0.0, + "text": "the product of the individual probabilities.<00:32:46.200> This<00:32:46.400> is<00:32:46.520> essentially<00:32:46.880> what" + }, + { + "start": 1966.99, + "duration": 0.0, + "text": "probabilities. This is essentially what" + }, + { + "start": 1967.0, + "duration": 0.0, + "text": "probabilities. This is essentially what we<00:32:47.120> have<00:32:47.240> derived<00:32:47.600> here." + }, + { + "start": 1969.55, + "duration": 0.0, + "text": "we have derived here." + }, + { + "start": 1969.56, + "duration": 0.0, + "text": "we have derived here. Now<00:32:49.840> we're<00:32:49.960> going<00:32:50.080> to<00:32:50.200> talk<00:32:50.720> uh,<00:32:50.840> we're<00:32:50.960> going" + }, + { + "start": 1971.07, + "duration": 0.0, + "text": "Now we're going to talk uh, we're going" + }, + { + "start": 1971.08, + "duration": 0.0, + "text": "Now we're going to talk uh, we're going to<00:32:51.160> go<00:32:51.280> back<00:32:51.600> to<00:32:51.800> our<00:32:52.000> idea<00:32:52.280> of<00:32:52.400> the<00:32:52.520> zeta" + }, + { + "start": 1972.79, + "duration": 0.0, + "text": "to go back to our idea of the zeta" + }, + { + "start": 1972.8, + "duration": 0.0, + "text": "to go back to our idea of the zeta function<00:32:53.160> because<00:32:53.520> if<00:32:53.680> you<00:32:53.760> remember" + }, + { + "start": 1974.87, + "duration": 0.0, + "text": "function because if you remember" + }, + { + "start": 1974.88, + "duration": 0.0, + "text": "function because if you remember zeta<00:32:55.240> function<00:32:55.680> actually<00:32:56.000> always<00:32:56.320> took<00:32:56.560> two" + }, + { + "start": 1976.79, + "duration": 0.0, + "text": "zeta function actually always took two" + }, + { + "start": 1976.8, + "duration": 0.0, + "text": "zeta function actually always took two elements.<00:32:57.840> Uh,<00:32:58.120> it<00:32:58.240> compared<00:32:59.080> sort<00:32:59.320> of<00:32:59.960> it" + }, + { + "start": 1980.27, + "duration": 0.0, + "text": "elements. Uh, it compared sort of it" + }, + { + "start": 1980.28, + "duration": 0.0, + "text": "elements. Uh, it compared sort of it always<00:33:00.840> was<00:33:01.000> evaluation<00:33:01.600> of<00:33:01.760> some<00:33:01.920> kind<00:33:02.080> of" + }, + { + "start": 1982.71, + "duration": 0.0, + "text": "always was evaluation of some kind of" + }, + { + "start": 1982.72, + "duration": 0.0, + "text": "always was evaluation of some kind of range<00:33:03.880> uh,<00:33:04.400> of<00:33:04.520> the<00:33:04.640> lattice.<00:33:05.240> So<00:33:05.400> we" + }, + { + "start": 1985.59, + "duration": 0.0, + "text": "range uh, of the lattice. So we" + }, + { + "start": 1985.6, + "duration": 0.0, + "text": "range uh, of the lattice. So we essentially<00:33:06.040> are<00:33:06.200> trying<00:33:06.520> to<00:33:06.640> build<00:33:06.880> a" + }, + { + "start": 1986.91, + "duration": 0.0, + "text": "essentially are trying to build a" + }, + { + "start": 1986.92, + "duration": 0.0, + "text": "essentially are trying to build a b-valuation,<00:33:07.680> not<00:33:07.840> evaluation<00:33:08.360> of<00:33:08.480> single" + }, + { + "start": 1988.75, + "duration": 0.0, + "text": "b-valuation, not evaluation of single" + }, + { + "start": 1988.76, + "duration": 0.0, + "text": "b-valuation, not evaluation of single elements<00:33:09.120> of<00:33:09.240> the<00:33:09.320> lattice." + }, + { + "start": 1990.55, + "duration": 0.0, + "text": "elements of the lattice." + }, + { + "start": 1990.56, + "duration": 0.0, + "text": "elements of the lattice. Um," + }, + { + "start": 1992.15, + "duration": 0.0, + "text": "Um," + }, + { + "start": 1992.16, + "duration": 0.0, + "text": "Um, in<00:33:12.240> general,<00:33:12.760> we're<00:33:12.880> going<00:33:13.080> to<00:33:13.400> just<00:33:13.800> describe" + }, + { + "start": 1994.15, + "duration": 0.0, + "text": "in general, we're going to just describe" + }, + { + "start": 1994.16, + "duration": 0.0, + "text": "in general, we're going to just describe this<00:33:14.320> as<00:33:14.560> the<00:33:14.920> b-valuation<00:33:15.760> with<00:33:16.000> let<00:33:16.200> b<00:33:16.960> x<00:33:17.200> and" + }, + { + "start": 1997.31, + "duration": 0.0, + "text": "this as the b-valuation with let b x and" + }, + { + "start": 1997.32, + "duration": 0.0, + "text": "this as the b-valuation with let b x and t<00:33:18.120> where<00:33:18.320> the<00:33:18.440> order<00:33:18.840> inside<00:33:19.400> this<00:33:20.000> uh," + }, + { + "start": 2000.11, + "duration": 0.0, + "text": "t where the order inside this uh," + }, + { + "start": 2000.12, + "duration": 0.0, + "text": "t where the order inside this uh, matters<00:33:20.480> a<00:33:20.520> lot.<00:33:21.000> So<00:33:21.360> the<00:33:21.520> right<00:33:21.760> one<00:33:21.920> is<00:33:22.080> the" + }, + { + "start": 2002.23, + "duration": 0.0, + "text": "matters a lot. So the right one is the" + }, + { + "start": 2002.24, + "duration": 0.0, + "text": "matters a lot. So the right one is the top<00:33:22.480> one<00:33:22.600> and<00:33:22.720> the<00:33:22.760> left<00:33:22.960> one<00:33:23.080> is<00:33:23.200> the<00:33:23.280> bottom" + }, + { + "start": 2003.51, + "duration": 0.0, + "text": "top one and the left one is the bottom" + }, + { + "start": 2003.52, + "duration": 0.0, + "text": "top one and the left one is the bottom one." + }, + { + "start": 2004.47, + "duration": 0.0, + "text": "one." + }, + { + "start": 2004.48, + "duration": 0.0, + "text": "one. Um,<00:33:25.080> x<00:33:25.280> is<00:33:25.400> commonly<00:33:25.640> called<00:33:25.960> predicate" + }, + { + "start": 2007.19, + "duration": 0.0, + "text": "Um, x is commonly called predicate" + }, + { + "start": 2007.2, + "duration": 0.0, + "text": "Um, x is commonly called predicate and<00:33:27.440> t<00:33:27.640> is<00:33:27.800> called<00:33:28.160> the<00:33:28.240> context." + }, + { + "start": 2010.27, + "duration": 0.0, + "text": "and t is called the context." + }, + { + "start": 2010.28, + "duration": 0.0, + "text": "and t is called the context. Uh,<00:33:30.720> and<00:33:30.920> note<00:33:31.160> that<00:33:31.400> t<00:33:31.600> does<00:33:31.840> not<00:33:32.640> need<00:33:32.920> to<00:33:33.000> be" + }, + { + "start": 2013.11, + "duration": 0.0, + "text": "Uh, and note that t does not need to be" + }, + { + "start": 2013.12, + "duration": 0.0, + "text": "Uh, and note that t does not need to be the<00:33:33.240> top<00:33:33.480> element<00:33:33.840> of<00:33:33.960> the<00:33:34.040> lattice.<00:33:34.560> Uh,<00:33:34.600> t" + }, + { + "start": 2014.83, + "duration": 0.0, + "text": "the top element of the lattice. Uh, t" + }, + { + "start": 2014.84, + "duration": 0.0, + "text": "the top element of the lattice. Uh, t can<00:33:35.040> just<00:33:35.160> be<00:33:35.320> any<00:33:35.560> element.<00:33:36.240> Uh,<00:33:36.360> this<00:33:36.560> is<00:33:37.080> we" + }, + { + "start": 2017.19, + "duration": 0.0, + "text": "can just be any element. Uh, this is we" + }, + { + "start": 2017.2, + "duration": 0.0, + "text": "can just be any element. Uh, this is we want<00:33:37.400> this<00:33:37.560> to<00:33:37.640> be<00:33:37.760> a<00:33:37.840> very<00:33:38.080> general<00:33:38.480> kind<00:33:38.680> of" + }, + { + "start": 2018.87, + "duration": 0.0, + "text": "want this to be a very general kind of" + }, + { + "start": 2018.88, + "duration": 0.0, + "text": "want this to be a very general kind of operation<00:33:39.360> that<00:33:39.440> we<00:33:39.520> can<00:33:39.640> do." + }, + { + "start": 2022.4, + "duration": 0.0, + "text": "Now<00:33:43.000> these<00:33:43.480> b-valuations,<00:33:44.720> because<00:33:45.280> they<00:33:45.480> are" + }, + { + "start": 2025.63, + "duration": 0.0, + "text": "Now these b-valuations, because they are" + }, + { + "start": 2025.64, + "duration": 0.0, + "text": "Now these b-valuations, because they are essentially<00:33:46.080> just<00:33:46.840> ranges<00:33:47.720> o-<00:33:47.920> over<00:33:48.480> sort<00:33:48.680> of" + }, + { + "start": 2028.99, + "duration": 0.0, + "text": "essentially just ranges o- over sort of" + }, + { + "start": 2029.0, + "duration": 0.0, + "text": "essentially just ranges o- over sort of connect<00:33:49.480> arrays<00:33:50.120> of<00:33:50.280> the<00:33:50.400> elements<00:33:50.760> of<00:33:50.880> the" + }, + { + "start": 2030.95, + "duration": 0.0, + "text": "connect arrays of the elements of the" + }, + { + "start": 2030.96, + "duration": 0.0, + "text": "connect arrays of the elements of the lattice," + }, + { + "start": 2032.15, + "duration": 0.0, + "text": "lattice," + }, + { + "start": 2032.16, + "duration": 0.0, + "text": "lattice, um,<00:33:52.720> they<00:33:52.920> can<00:33:53.160> be<00:33:53.360> ordered<00:33:53.760> by<00:33:53.960> set" + }, + { + "start": 2034.23, + "duration": 0.0, + "text": "um, they can be ordered by set" + }, + { + "start": 2034.24, + "duration": 0.0, + "text": "um, they can be ordered by set inclusion.<00:33:55.080> And<00:33:55.400> consistency<00:33:56.000> requires<00:33:56.520> that" + }, + { + "start": 2037.39, + "duration": 0.0, + "text": "inclusion. And consistency requires that" + }, + { + "start": 2037.4, + "duration": 0.0, + "text": "inclusion. And consistency requires that um," + }, + { + "start": 2038.75, + "duration": 0.0, + "text": "um," + }, + { + "start": 2038.76, + "duration": 0.0, + "text": "um, smaller" + }, + { + "start": 2040.15, + "duration": 0.0, + "text": "smaller" + }, + { + "start": 2040.16, + "duration": 0.0, + "text": "smaller uh,<00:34:00.320> smaller<00:34:00.760> ranges<00:34:01.320> have<00:34:02.120> uh,<00:34:02.560> higher" + }, + { + "start": 2042.83, + "duration": 0.0, + "text": "uh, smaller ranges have uh, higher" + }, + { + "start": 2042.84, + "duration": 0.0, + "text": "uh, smaller ranges have uh, higher valuations<00:34:03.720> and<00:34:03.960> sort<00:34:04.160> of<00:34:04.400> valuations<00:34:04.920> of<00:34:05.040> two" + }, + { + "start": 2045.19, + "duration": 0.0, + "text": "valuations and sort of valuations of two" + }, + { + "start": 2045.2, + "duration": 0.0, + "text": "valuations and sort of valuations of two elements<00:34:05.520> that<00:34:05.640> are<00:34:05.760> further<00:34:06.120> apart<00:34:07.200> uh,<00:34:07.440> have" + }, + { + "start": 2047.67, + "duration": 0.0, + "text": "elements that are further apart uh, have" + }, + { + "start": 2047.68, + "duration": 0.0, + "text": "elements that are further apart uh, have smaller<00:34:08.040> valuations." + }, + { + "start": 2049.35, + "duration": 0.0, + "text": "smaller valuations." + }, + { + "start": 2049.36, + "duration": 0.0, + "text": "smaller valuations. The<00:34:09.440> reason<00:34:10.200> is<00:34:10.480> that<00:34:11.200> if<00:34:11.399> we<00:34:11.520> have<00:34:11.679> the<00:34:11.800> same" + }, + { + "start": 2052.07, + "duration": 0.0, + "text": "The reason is that if we have the same" + }, + { + "start": 2052.08, + "duration": 0.0, + "text": "The reason is that if we have the same context,<00:34:13.320> then<00:34:13.879> an<00:34:14.040> element<00:34:14.320> that<00:34:14.440> is<00:34:14.560> further" + }, + { + "start": 2054.79, + "duration": 0.0, + "text": "context, then an element that is further" + }, + { + "start": 2054.8, + "duration": 0.0, + "text": "context, then an element that is further down<00:34:15.120> the<00:34:15.200> lattice" + }, + { + "start": 2056.389, + "duration": 0.0, + "text": "down the lattice" + }, + { + "start": 2056.399, + "duration": 0.0, + "text": "down the lattice um," + }, + { + "start": 2057.59, + "duration": 0.0, + "text": "um," + }, + { + "start": 2057.6, + "duration": 0.0, + "text": "um, needs<00:34:17.960> to<00:34:18.080> have<00:34:18.560> uh,<00:34:18.840> a<00:34:18.960> lower<00:34:19.520> valuation<00:34:20.080> than" + }, + { + "start": 2060.23, + "duration": 0.0, + "text": "needs to have uh, a lower valuation than" + }, + { + "start": 2060.24, + "duration": 0.0, + "text": "needs to have uh, a lower valuation than an<00:34:20.399> element<00:34:20.720> that<00:34:20.840> is<00:34:20.960> implied<00:34:21.679> by<00:34:21.840> the<00:34:21.919> same" + }, + { + "start": 2062.19, + "duration": 0.0, + "text": "an element that is implied by the same" + }, + { + "start": 2062.2, + "duration": 0.0, + "text": "an element that is implied by the same element<00:34:22.639> x,<00:34:23.040> right?<00:34:23.879> Uh," + }, + { + "start": 2066.07, + "duration": 0.0, + "text": "element x, right? Uh," + }, + { + "start": 2066.08, + "duration": 0.0, + "text": "element x, right? Uh, for<00:34:26.159> example,<00:34:26.639> if<00:34:27.320> uh,<00:34:27.399> what's<00:34:27.600> a<00:34:27.639> good" + }, + { + "start": 2067.75, + "duration": 0.0, + "text": "for example, if uh, what's a good" + }, + { + "start": 2067.76, + "duration": 0.0, + "text": "for example, if uh, what's a good example<00:34:28.120> to<00:34:28.240> illustrate<00:34:28.679> this?" + }, + { + "start": 2069.869, + "duration": 0.0, + "text": "example to illustrate this?" + }, + { + "start": 2069.879, + "duration": 0.0, + "text": "example to illustrate this? Uh,<00:34:30.159> well<00:34:30.240> I<00:34:30.280> think<00:34:30.440> this<00:34:30.560> one's<00:34:30.720> kind<00:34:30.879> of" + }, + { + "start": 2070.99, + "duration": 0.0, + "text": "Uh, well I think this one's kind of" + }, + { + "start": 2071.0, + "duration": 0.0, + "text": "Uh, well I think this one's kind of obvious.<00:34:31.280> it's<00:34:31.399> the<00:34:31.480> opposite<00:34:31.879> that<00:34:31.960> needs<00:34:32.120> to" + }, + { + "start": 2072.19, + "duration": 0.0, + "text": "obvious. it's the opposite that needs to" + }, + { + "start": 2072.2, + "duration": 0.0, + "text": "obvious. it's the opposite that needs to be<00:34:32.320> explained.<00:34:33.240> Um," + }, + { + "start": 2075.11, + "duration": 0.0, + "text": "be explained. Um," + }, + { + "start": 2075.12, + "duration": 0.0, + "text": "be explained. Um, but<00:34:35.240> given<00:34:35.480> the<00:34:35.560> same<00:34:35.800> predicate,<00:34:36.399> also<00:34:36.679> we" + }, + { + "start": 2076.79, + "duration": 0.0, + "text": "but given the same predicate, also we" + }, + { + "start": 2076.8, + "duration": 0.0, + "text": "but given the same predicate, also we know<00:34:37.040> that<00:34:37.240> a<00:34:37.280> more<00:34:37.679> diluted<00:34:38.200> context<00:34:38.679> gives<00:34:38.919> a" + }, + { + "start": 2078.95, + "duration": 0.0, + "text": "know that a more diluted context gives a" + }, + { + "start": 2078.96, + "duration": 0.0, + "text": "know that a more diluted context gives a lower<00:34:39.200> valuation.<00:34:39.760> So,<00:34:39.879> this<00:34:40.080> one<00:34:40.399> needs<00:34:40.600> an" + }, + { + "start": 2080.669, + "duration": 0.0, + "text": "lower valuation. So, this one needs an" + }, + { + "start": 2080.679, + "duration": 0.0, + "text": "lower valuation. So, this one needs an example.<00:34:41.200> So,<00:34:41.360> imagine<00:34:41.840> that<00:34:42.000> you<00:34:42.120> want<00:34:42.360> to" + }, + { + "start": 2082.43, + "duration": 0.0, + "text": "example. So, imagine that you want to" + }, + { + "start": 2082.44, + "duration": 0.0, + "text": "example. So, imagine that you want to define<00:34:42.840> the<00:34:42.919> probability<00:34:43.480> that<00:34:43.639> a<00:34:43.679> person<00:34:44.080> is" + }, + { + "start": 2084.31, + "duration": 0.0, + "text": "define the probability that a person is" + }, + { + "start": 2084.32, + "duration": 0.0, + "text": "define the probability that a person is in<00:34:44.480> a<00:34:44.679> city,<00:34:45.240> let's<00:34:45.480> say<00:34:45.679> Paris<00:34:46.040> for<00:34:46.159> example," + }, + { + "start": 2086.51, + "duration": 0.0, + "text": "in a city, let's say Paris for example," + }, + { + "start": 2086.52, + "duration": 0.0, + "text": "in a city, let's say Paris for example, right?" + }, + { + "start": 2087.389, + "duration": 0.0, + "text": "right?" + }, + { + "start": 2087.399, + "duration": 0.0, + "text": "right? So,<00:34:47.840> the<00:34:48.000> probability<00:34:48.560> that<00:34:48.639> the<00:34:48.760> person<00:34:49.080> is" + }, + { + "start": 2089.19, + "duration": 0.0, + "text": "So, the probability that the person is" + }, + { + "start": 2089.2, + "duration": 0.0, + "text": "So, the probability that the person is in<00:34:49.360> Paris<00:34:50.000> given<00:34:50.360> that<00:34:50.520> you<00:34:50.639> know<00:34:50.800> the<00:34:50.879> person" + }, + { + "start": 2091.19, + "duration": 0.0, + "text": "in Paris given that you know the person" + }, + { + "start": 2091.2, + "duration": 0.0, + "text": "in Paris given that you know the person is<00:34:51.320> in<00:34:51.440> France<00:34:52.480> is<00:34:52.679> definitely<00:34:53.040> much<00:34:53.280> bigger" + }, + { + "start": 2093.55, + "duration": 0.0, + "text": "is in France is definitely much bigger" + }, + { + "start": 2093.56, + "duration": 0.0, + "text": "is in France is definitely much bigger than<00:34:53.679> the<00:34:53.760> probability<00:34:54.360> that<00:34:54.520> they<00:34:54.639> are<00:34:54.800> in<00:34:55.040> in" + }, + { + "start": 2095.27, + "duration": 0.0, + "text": "than the probability that they are in in" + }, + { + "start": 2095.28, + "duration": 0.0, + "text": "than the probability that they are in in Paris<00:34:56.159> knowing<00:34:56.639> only<00:34:56.840> that<00:34:57.000> they<00:34:57.120> are" + }, + { + "start": 2097.23, + "duration": 0.0, + "text": "Paris knowing only that they are" + }, + { + "start": 2097.24, + "duration": 0.0, + "text": "Paris knowing only that they are somewhere<00:34:58.040> in<00:34:58.200> Europe.<00:34:59.200> Uh,<00:34:59.560> the<00:35:00.040> the<00:35:00.120> second" + }, + { + "start": 2100.43, + "duration": 0.0, + "text": "somewhere in Europe. Uh, the the second" + }, + { + "start": 2100.44, + "duration": 0.0, + "text": "somewhere in Europe. Uh, the the second context<00:35:00.920> is<00:35:01.080> much<00:35:01.440> much<00:35:02.120> uh<00:35:02.240> more<00:35:02.400> diluted." + }, + { + "start": 2102.87, + "duration": 0.0, + "text": "context is much much uh more diluted." + }, + { + "start": 2102.88, + "duration": 0.0, + "text": "context is much much uh more diluted. It's<00:35:03.000> much<00:35:03.560> Being<00:35:03.800> in<00:35:03.880> France<00:35:04.240> implies<00:35:04.680> being" + }, + { + "start": 2104.83, + "duration": 0.0, + "text": "It's much Being in France implies being" + }, + { + "start": 2104.84, + "duration": 0.0, + "text": "It's much Being in France implies being in<00:35:04.920> Europe,<00:35:05.720> hence<00:35:06.440> uh<00:35:06.600> France<00:35:06.920> is<00:35:07.040> strictly" + }, + { + "start": 2107.51, + "duration": 0.0, + "text": "in Europe, hence uh France is strictly" + }, + { + "start": 2107.52, + "duration": 0.0, + "text": "in Europe, hence uh France is strictly below<00:35:07.880> Europe<00:35:08.240> in<00:35:08.320> kind<00:35:08.480> of<00:35:08.560> this<00:35:08.760> ordering<00:35:09.120> by" + }, + { + "start": 2109.23, + "duration": 0.0, + "text": "below Europe in kind of this ordering by" + }, + { + "start": 2109.24, + "duration": 0.0, + "text": "below Europe in kind of this ordering by implication.<00:35:10.320> Uh,<00:35:10.480> and<00:35:10.640> Europe<00:35:10.840> is<00:35:10.920> just<00:35:11.240> so" + }, + { + "start": 2111.47, + "duration": 0.0, + "text": "implication. Uh, and Europe is just so" + }, + { + "start": 2111.48, + "duration": 0.0, + "text": "implication. Uh, and Europe is just so much<00:35:11.680> more<00:35:11.800> diluted.<00:35:12.160> It<00:35:12.520> allows<00:35:12.800> for<00:35:12.920> so<00:35:13.160> many" + }, + { + "start": 2113.39, + "duration": 0.0, + "text": "much more diluted. It allows for so many" + }, + { + "start": 2113.4, + "duration": 0.0, + "text": "much more diluted. It allows for so many more<00:35:13.600> possibilities." + }, + { + "start": 2115.15, + "duration": 0.0, + "text": "more possibilities." + }, + { + "start": 2115.16, + "duration": 0.0, + "text": "more possibilities. Uh,<00:35:15.480> and<00:35:15.560> this<00:35:15.720> is<00:35:15.960> what<00:35:16.120> we're<00:35:16.520> essentially" + }, + { + "start": 2116.99, + "duration": 0.0, + "text": "Uh, and this is what we're essentially" + }, + { + "start": 2117.0, + "duration": 0.0, + "text": "Uh, and this is what we're essentially reconstructing<00:35:17.600> here,<00:35:17.760> what<00:35:17.920> we're" + }, + { + "start": 2118.27, + "duration": 0.0, + "text": "reconstructing here, what we're" + }, + { + "start": 2118.28, + "duration": 0.0, + "text": "reconstructing here, what we're obtaining." + }, + { + "start": 2120.55, + "duration": 0.0, + "text": "obtaining." + }, + { + "start": 2120.56, + "duration": 0.0, + "text": "obtaining. And<00:35:20.640> this<00:35:20.880> B-valuation<00:35:21.680> is<00:35:21.880> then<00:35:22.080> what<00:35:22.240> we<00:35:22.720> uh" + }, + { + "start": 2122.83, + "duration": 0.0, + "text": "And this B-valuation is then what we uh" + }, + { + "start": 2122.84, + "duration": 0.0, + "text": "And this B-valuation is then what we uh call<00:35:23.080> probability,<00:35:23.840> where<00:35:24.000> again<00:35:24.240> here<00:35:24.840> this" + }, + { + "start": 2124.99, + "duration": 0.0, + "text": "call probability, where again here this" + }, + { + "start": 2125.0, + "duration": 0.0, + "text": "call probability, where again here this B-valuation<00:35:25.560> of<00:35:25.640> the<00:35:25.720> range<00:35:26.040> X<00:35:26.840> uh<00:35:26.920> and<00:35:27.080> T" + }, + { + "start": 2127.87, + "duration": 0.0, + "text": "B-valuation of the range X uh and T" + }, + { + "start": 2127.88, + "duration": 0.0, + "text": "B-valuation of the range X uh and T is<00:35:28.280> the<00:35:28.360> probability<00:35:28.800> of<00:35:28.960> X<00:35:29.560> given<00:35:29.880> T." + }, + { + "start": 2133.44, + "duration": 0.0, + "text": "Um,<00:35:34.200> and<00:35:34.400> the<00:35:34.480> interesting<00:35:34.800> thing<00:35:35.000> is" + }, + { + "start": 2135.91, + "duration": 0.0, + "text": "Um, and the interesting thing is" + }, + { + "start": 2135.92, + "duration": 0.0, + "text": "Um, and the interesting thing is uh<00:35:36.440> given<00:35:37.480> a<00:35:37.520> context,<00:35:38.680> then<00:35:38.840> the<00:35:38.960> valuation" + }, + { + "start": 2139.43, + "duration": 0.0, + "text": "uh given a context, then the valuation" + }, + { + "start": 2139.44, + "duration": 0.0, + "text": "uh given a context, then the valuation of<00:35:39.560> all<00:35:39.720> the<00:35:39.840> elements<00:35:40.200> sort<00:35:40.320> of<00:35:40.440> below<00:35:40.840> it" + }, + { + "start": 2140.99, + "duration": 0.0, + "text": "of all the elements sort of below it" + }, + { + "start": 2141.0, + "duration": 0.0, + "text": "of all the elements sort of below it just<00:35:41.200> reduces<00:35:41.680> to<00:35:41.840> the<00:35:42.560> valuation<00:35:43.160> of<00:35:43.480> those" + }, + { + "start": 2143.71, + "duration": 0.0, + "text": "just reduces to the valuation of those" + }, + { + "start": 2143.72, + "duration": 0.0, + "text": "just reduces to the valuation of those single<00:35:44.040> elements,<00:35:44.920> which<00:35:45.120> is<00:35:45.240> why<00:35:45.520> we<00:35:45.640> can" + }, + { + "start": 2145.79, + "duration": 0.0, + "text": "single elements, which is why we can" + }, + { + "start": 2145.8, + "duration": 0.0, + "text": "single elements, which is why we can apply<00:35:46.160> those<00:35:46.760> uh<00:35:46.840> results<00:35:47.240> that<00:35:47.320> we<00:35:47.400> saw" + }, + { + "start": 2147.55, + "duration": 0.0, + "text": "apply those uh results that we saw" + }, + { + "start": 2147.56, + "duration": 0.0, + "text": "apply those uh results that we saw before<00:35:47.920> is<00:35:48.000> only<00:35:48.200> caring<00:35:48.520> about<00:35:49.120> valuation<00:35:49.600> of" + }, + { + "start": 2149.71, + "duration": 0.0, + "text": "before is only caring about valuation of" + }, + { + "start": 2149.72, + "duration": 0.0, + "text": "before is only caring about valuation of single<00:35:49.960> elements." + }, + { + "start": 2151.19, + "duration": 0.0, + "text": "single elements." + }, + { + "start": 2151.2, + "duration": 0.0, + "text": "single elements. And<00:35:51.520> this<00:35:51.680> allows<00:35:52.040> us<00:35:52.160> to<00:35:52.640> write<00:35:53.080> the<00:35:53.320> the<00:35:53.400> sum" + }, + { + "start": 2153.63, + "duration": 0.0, + "text": "And this allows us to write the the sum" + }, + { + "start": 2153.64, + "duration": 0.0, + "text": "And this allows us to write the the sum rule,<00:35:53.960> which<00:35:54.120> is<00:35:54.240> very<00:35:54.400> nice." + }, + { + "start": 2155.47, + "duration": 0.0, + "text": "rule, which is very nice." + }, + { + "start": 2155.48, + "duration": 0.0, + "text": "rule, which is very nice. Uh,<00:35:55.600> so<00:35:55.720> we<00:35:55.760> already<00:35:56.000> have<00:35:56.160> this.<00:35:56.960> And<00:35:57.480> we<00:35:57.600> can" + }, + { + "start": 2157.79, + "duration": 0.0, + "text": "Uh, so we already have this. And we can" + }, + { + "start": 2157.8, + "duration": 0.0, + "text": "Uh, so we already have this. And we can also<00:35:58.000> use<00:35:58.160> the<00:35:58.280> product<00:35:58.600> rule<00:35:58.760> for" + }, + { + "start": 2158.95, + "duration": 0.0, + "text": "also use the product rule for" + }, + { + "start": 2158.96, + "duration": 0.0, + "text": "also use the product rule for independently<00:35:59.600> treated<00:35:59.960> systems<00:36:00.880> uh<00:36:01.200> like" + }, + { + "start": 2161.39, + "duration": 0.0, + "text": "independently treated systems uh like" + }, + { + "start": 2161.4, + "duration": 0.0, + "text": "independently treated systems uh like so," + }, + { + "start": 2162.31, + "duration": 0.0, + "text": "so," + }, + { + "start": 2162.32, + "duration": 0.0, + "text": "so, uh<00:36:02.600> where<00:36:03.080> the<00:36:03.240> top<00:36:03.480> element<00:36:04.360> uh<00:36:04.760> so<00:36:04.880> that<00:36:05.040> the" + }, + { + "start": 2165.11, + "duration": 0.0, + "text": "uh where the top element uh so that the" + }, + { + "start": 2165.12, + "duration": 0.0, + "text": "uh where the top element uh so that the combined<00:36:05.600> context<00:36:06.080> is<00:36:06.280> again<00:36:06.840> T<00:36:07.840> uh<00:36:07.920> context" + }, + { + "start": 2168.27, + "duration": 0.0, + "text": "combined context is again T uh context" + }, + { + "start": 2168.28, + "duration": 0.0, + "text": "combined context is again T uh context one<00:36:08.480> times<00:36:08.680> context<00:36:09.040> two." + }, + { + "start": 2169.87, + "duration": 0.0, + "text": "one times context two." + }, + { + "start": 2169.88, + "duration": 0.0, + "text": "one times context two. And<00:36:10.240> just<00:36:10.400> to<00:36:10.520> illustrate<00:36:11.000> what<00:36:11.120> this<00:36:11.280> would" + }, + { + "start": 2171.43, + "duration": 0.0, + "text": "And just to illustrate what this would" + }, + { + "start": 2171.44, + "duration": 0.0, + "text": "And just to illustrate what this would look<00:36:11.640> like<00:36:12.000> with<00:36:12.160> some<00:36:12.320> kind<00:36:12.520> of<00:36:12.920> uh" + }, + { + "start": 2173.59, + "duration": 0.0, + "text": "look like with some kind of uh" + }, + { + "start": 2173.6, + "duration": 0.0, + "text": "look like with some kind of uh with<00:36:13.720> the<00:36:13.800> lattice<00:36:14.120> case," + }, + { + "start": 2175.03, + "duration": 0.0, + "text": "with the lattice case," + }, + { + "start": 2175.04, + "duration": 0.0, + "text": "with the lattice case, uh<00:36:15.360> so<00:36:15.480> a<00:36:15.520> quick<00:36:15.720> example,<00:36:16.200> let's<00:36:16.440> take<00:36:16.760> these" + }, + { + "start": 2177.03, + "duration": 0.0, + "text": "uh so a quick example, let's take these" + }, + { + "start": 2177.04, + "duration": 0.0, + "text": "uh so a quick example, let's take these two<00:36:17.560> simple<00:36:17.920> lattices<00:36:18.360> with<00:36:18.520> just<00:36:18.800> two<00:36:19.080> atoms" + }, + { + "start": 2179.91, + "duration": 0.0, + "text": "two simple lattices with just two atoms" + }, + { + "start": 2179.92, + "duration": 0.0, + "text": "two simple lattices with just two atoms on<00:36:20.000> top<00:36:20.200> element<00:36:20.480> and<00:36:20.560> bottom<00:36:20.840> element,<00:36:21.680> and" + }, + { + "start": 2181.91, + "duration": 0.0, + "text": "on top element and bottom element, and" + }, + { + "start": 2181.92, + "duration": 0.0, + "text": "on top element and bottom element, and we<00:36:22.000> want<00:36:22.280> to<00:36:22.800> say<00:36:23.400> find<00:36:24.120> the<00:36:24.240> valuation<00:36:24.880> of<00:36:25.720> uh" + }, + { + "start": 2186.03, + "duration": 0.0, + "text": "we want to say find the valuation of uh" + }, + { + "start": 2186.04, + "duration": 0.0, + "text": "we want to say find the valuation of uh A<00:36:26.200> times<00:36:26.520> X.<00:36:27.280> Um,<00:36:27.800> then<00:36:27.960> the<00:36:28.080> top<00:36:28.360> element<00:36:28.800> here" + }, + { + "start": 2189.39, + "duration": 0.0, + "text": "A times X. Um, then the top element here" + }, + { + "start": 2189.4, + "duration": 0.0, + "text": "A times X. Um, then the top element here becomes<00:36:29.920> t1<00:36:30.200> times<00:36:30.360> t2.<00:36:31.200> We<00:36:31.320> can<00:36:31.520> use<00:36:31.880> the" + }, + { + "start": 2192.11, + "duration": 0.0, + "text": "becomes t1 times t2. We can use the" + }, + { + "start": 2192.12, + "duration": 0.0, + "text": "becomes t1 times t2. We can use the distributivity<00:36:32.840> property<00:36:33.560> to<00:36:33.720> obtain<00:36:34.080> that" + }, + { + "start": 2194.19, + "duration": 0.0, + "text": "distributivity property to obtain that" + }, + { + "start": 2194.2, + "duration": 0.0, + "text": "distributivity property to obtain that this<00:36:34.360> is<00:36:34.480> the<00:36:34.720> top<00:36:35.000> element<00:36:35.360> of<00:36:35.480> the<00:36:35.600> new" + }, + { + "start": 2195.79, + "duration": 0.0, + "text": "this is the top element of the new" + }, + { + "start": 2195.8, + "duration": 0.0, + "text": "this is the top element of the new lattice<00:36:36.120> that<00:36:36.240> we're<00:36:36.320> going<00:36:36.480> to<00:36:37.080> uh<00:36:37.120> get." + }, + { + "start": 2198.35, + "duration": 0.0, + "text": "lattice that we're going to uh get." + }, + { + "start": 2198.36, + "duration": 0.0, + "text": "lattice that we're going to uh get. And<00:36:39.000> again,<00:36:39.360> note<00:36:39.520> that<00:36:39.680> neither<00:36:40.160> t1<00:36:40.480> or<00:36:40.560> t2" + }, + { + "start": 2200.91, + "duration": 0.0, + "text": "And again, note that neither t1 or t2" + }, + { + "start": 2200.92, + "duration": 0.0, + "text": "And again, note that neither t1 or t2 need<00:36:41.160> to<00:36:41.200> be<00:36:41.320> the<00:36:41.440> top<00:36:41.680> element<00:36:42.120> of<00:36:42.200> their" + }, + { + "start": 2202.31, + "duration": 0.0, + "text": "need to be the top element of their" + }, + { + "start": 2202.32, + "duration": 0.0, + "text": "need to be the top element of their respective<00:36:42.800> lattices.<00:36:43.120> This<00:36:43.320> could<00:36:43.480> just<00:36:43.680> be" + }, + { + "start": 2204.15, + "duration": 0.0, + "text": "respective lattices. This could just be" + }, + { + "start": 2204.16, + "duration": 0.0, + "text": "respective lattices. This could just be These<00:36:44.400> two<00:36:44.600> could<00:36:44.760> just<00:36:44.920> be<00:36:45.480> sub-lattices<00:36:46.160> of" + }, + { + "start": 2206.27, + "duration": 0.0, + "text": "These two could just be sub-lattices of" + }, + { + "start": 2206.28, + "duration": 0.0, + "text": "These two could just be sub-lattices of some<00:36:46.480> kind<00:36:46.640> of<00:36:46.720> bigger<00:36:46.960> structure.<00:36:47.400> It" + }, + { + "start": 2207.47, + "duration": 0.0, + "text": "some kind of bigger structure. It" + }, + { + "start": 2207.48, + "duration": 0.0, + "text": "some kind of bigger structure. It doesn't<00:36:47.760> matter." + }, + { + "start": 2208.91, + "duration": 0.0, + "text": "doesn't matter." + }, + { + "start": 2208.92, + "duration": 0.0, + "text": "doesn't matter. And<00:36:49.160> the<00:36:49.280> combination<00:36:49.880> of<00:36:49.960> them<00:36:50.720> uh<00:36:50.800> results" + }, + { + "start": 2211.11, + "duration": 0.0, + "text": "And the combination of them uh results" + }, + { + "start": 2211.12, + "duration": 0.0, + "text": "And the combination of them uh results in<00:36:51.200> this.<00:36:51.800> Um" + }, + { + "start": 2212.75, + "duration": 0.0, + "text": "in this. Um" + }, + { + "start": 2212.76, + "duration": 0.0, + "text": "in this. Um And<00:36:52.800> this<00:36:52.960> is<00:36:53.400> what<00:36:53.560> we're<00:36:53.640> doing<00:36:54.000> when<00:36:54.160> we<00:36:54.320> are" + }, + { + "start": 2214.55, + "duration": 0.0, + "text": "And this is what we're doing when we are" + }, + { + "start": 2214.56, + "duration": 0.0, + "text": "And this is what we're doing when we are combining<00:36:55.080> two<00:36:55.960> uh<00:36:56.040> systems<00:36:56.400> that<00:36:56.520> we<00:36:56.600> treat" + }, + { + "start": 2216.87, + "duration": 0.0, + "text": "combining two uh systems that we treat" + }, + { + "start": 2216.88, + "duration": 0.0, + "text": "combining two uh systems that we treat independently.<00:36:57.640> We<00:36:58.040> kind<00:36:58.240> of<00:36:58.320> create<00:36:58.560> this" + }, + { + "start": 2218.67, + "duration": 0.0, + "text": "independently. We kind of create this" + }, + { + "start": 2218.68, + "duration": 0.0, + "text": "independently. We kind of create this new<00:36:59.120> bigger<00:36:59.400> structure" + }, + { + "start": 2221.03, + "duration": 0.0, + "text": "new bigger structure" + }, + { + "start": 2221.04, + "duration": 0.0, + "text": "new bigger structure that<00:37:01.160> has<00:37:01.400> all<00:37:01.520> these<00:37:02.040> uh<00:37:02.360> cross<00:37:02.600> product" + }, + { + "start": 2223.83, + "duration": 0.0, + "text": "that has all these uh cross product" + }, + { + "start": 2223.84, + "duration": 0.0, + "text": "that has all these uh cross product points." + }, + { + "start": 2226.64, + "duration": 0.0, + "text": "Um<00:37:07.440> yeah." + }, + { + "start": 2229.03, + "duration": 0.0, + "text": "Um yeah." + }, + { + "start": 2229.04, + "duration": 0.0, + "text": "Um yeah. Now,<00:37:09.240> the<00:37:09.360> next<00:37:09.600> thing<00:37:09.760> that<00:37:09.880> we<00:37:10.000> would<00:37:10.240> that" + }, + { + "start": 2230.35, + "duration": 0.0, + "text": "Now, the next thing that we would that" + }, + { + "start": 2230.36, + "duration": 0.0, + "text": "Now, the next thing that we would that we<00:37:10.480> need<00:37:10.680> to<00:37:10.760> do<00:37:10.960> to<00:37:11.080> have<00:37:11.240> some<00:37:11.360> kind<00:37:11.560> of" + }, + { + "start": 2231.63, + "duration": 0.0, + "text": "we need to do to have some kind of" + }, + { + "start": 2231.64, + "duration": 0.0, + "text": "we need to do to have some kind of complete<00:37:12.480> uh" + }, + { + "start": 2233.15, + "duration": 0.0, + "text": "complete uh" + }, + { + "start": 2233.16, + "duration": 0.0, + "text": "complete uh reasoning<00:37:13.520> apparatus<00:37:14.000> is<00:37:14.200> that<00:37:14.320> we<00:37:14.400> need<00:37:14.560> to" + }, + { + "start": 2234.63, + "duration": 0.0, + "text": "reasoning apparatus is that we need to" + }, + { + "start": 2234.64, + "duration": 0.0, + "text": "reasoning apparatus is that we need to somehow<00:37:14.920> quantify<00:37:15.440> the<00:37:15.560> degree<00:37:15.800> of" + }, + { + "start": 2235.91, + "duration": 0.0, + "text": "somehow quantify the degree of" + }, + { + "start": 2235.92, + "duration": 0.0, + "text": "somehow quantify the degree of implication<00:37:16.440> between<00:37:16.760> two<00:37:16.880> elements<00:37:17.960> that" + }, + { + "start": 2238.31, + "duration": 0.0, + "text": "implication between two elements that" + }, + { + "start": 2238.32, + "duration": 0.0, + "text": "implication between two elements that are<00:37:18.840> not<00:37:19.160> directly<00:37:19.560> one<00:37:19.680> above<00:37:20.040> each<00:37:20.240> other." + }, + { + "start": 2241.19, + "duration": 0.0, + "text": "are not directly one above each other." + }, + { + "start": 2241.2, + "duration": 0.0, + "text": "are not directly one above each other. Because<00:37:21.440> if<00:37:21.560> they<00:37:21.720> are<00:37:21.800> above<00:37:22.120> each<00:37:22.320> other,<00:37:23.040> uh" + }, + { + "start": 2243.11, + "duration": 0.0, + "text": "Because if they are above each other, uh" + }, + { + "start": 2243.12, + "duration": 0.0, + "text": "Because if they are above each other, uh you<00:37:23.240> can<00:37:23.440> somehow<00:37:23.800> just<00:37:24.560> uh<00:37:24.640> combine<00:37:25.200> the" + }, + { + "start": 2246.23, + "duration": 0.0, + "text": "you can somehow just uh combine the" + }, + { + "start": 2246.24, + "duration": 0.0, + "text": "you can somehow just uh combine the the<00:37:26.440> all<00:37:26.600> the<00:37:26.720> elements<00:37:27.800> uh<00:37:27.880> with<00:37:28.040> the<00:37:28.120> join" + }, + { + "start": 2248.43, + "duration": 0.0, + "text": "the all the elements uh with the join" + }, + { + "start": 2248.44, + "duration": 0.0, + "text": "the all the elements uh with the join operation,<00:37:29.200> with<00:37:29.360> the<00:37:29.720> sum.<00:37:30.120> But<00:37:30.320> if<00:37:30.480> they're" + }, + { + "start": 2250.63, + "duration": 0.0, + "text": "operation, with the sum. But if they're" + }, + { + "start": 2250.64, + "duration": 0.0, + "text": "operation, with the sum. But if they're not<00:37:30.800> directly<00:37:31.080> above<00:37:31.320> each<00:37:31.440> other,<00:37:31.600> what<00:37:31.760> do" + }, + { + "start": 2251.83, + "duration": 0.0, + "text": "not directly above each other, what do" + }, + { + "start": 2251.84, + "duration": 0.0, + "text": "not directly above each other, what do you<00:37:31.920> do?" + }, + { + "start": 2252.79, + "duration": 0.0, + "text": "you do?" + }, + { + "start": 2252.8, + "duration": 0.0, + "text": "you do? Um" + }, + { + "start": 2254.51, + "duration": 0.0, + "text": "Um" + }, + { + "start": 2254.52, + "duration": 0.0, + "text": "Um So,<00:37:34.640> for<00:37:34.760> example,<00:37:35.040> imagine<00:37:35.360> the<00:37:35.480> chain<00:37:36.040> where" + }, + { + "start": 2256.51, + "duration": 0.0, + "text": "So, for example, imagine the chain where" + }, + { + "start": 2256.52, + "duration": 0.0, + "text": "So, for example, imagine the chain where all<00:37:36.640> these<00:37:36.800> elements<00:37:37.160> are<00:37:37.280> directly<00:37:38.240> one<00:37:38.440> the" + }, + { + "start": 2258.51, + "duration": 0.0, + "text": "all these elements are directly one the" + }, + { + "start": 2258.52, + "duration": 0.0, + "text": "all these elements are directly one the superior<00:37:39.000> of<00:37:39.120> the<00:37:39.240> other.<00:37:40.080> Then<00:37:40.240> we<00:37:40.360> somehow" + }, + { + "start": 2260.79, + "duration": 0.0, + "text": "superior of the other. Then we somehow" + }, + { + "start": 2260.8, + "duration": 0.0, + "text": "superior of the other. Then we somehow need<00:37:41.040> to<00:37:41.280> obtain" + }, + { + "start": 2262.79, + "duration": 0.0, + "text": "need to obtain" + }, + { + "start": 2262.8, + "duration": 0.0, + "text": "need to obtain the<00:37:42.840> valuation<00:37:43.400> of<00:37:43.600> the<00:37:44.000> in<00:37:44.440> of<00:37:44.520> this" + }, + { + "start": 2264.71, + "duration": 0.0, + "text": "the valuation of the in of this" + }, + { + "start": 2264.72, + "duration": 0.0, + "text": "the valuation of the in of this generalization<00:37:45.280> of<00:37:45.400> the<00:37:45.480> inverse<00:37:45.800> zeta" + }, + { + "start": 2266.03, + "duration": 0.0, + "text": "generalization of the inverse zeta" + }, + { + "start": 2266.04, + "duration": 0.0, + "text": "generalization of the inverse zeta function" + }, + { + "start": 2267.07, + "duration": 0.0, + "text": "function" + }, + { + "start": 2267.08, + "duration": 0.0, + "text": "function uh<00:37:47.360> of" + }, + { + "start": 2268.55, + "duration": 0.0, + "text": "uh of" + }, + { + "start": 2268.56, + "duration": 0.0, + "text": "uh of over<00:37:48.680> the<00:37:48.760> whole<00:37:49.040> range<00:37:49.320> of<00:37:49.520> x<00:37:50.200> to<00:37:50.360> t." + }, + { + "start": 2271.39, + "duration": 0.0, + "text": "over the whole range of x to t." + }, + { + "start": 2271.4, + "duration": 0.0, + "text": "over the whole range of x to t. Um<00:37:52.160> and<00:37:52.320> we<00:37:52.440> can<00:37:52.640> have<00:37:52.880> to<00:37:53.000> find<00:37:53.320> this<00:37:53.480> from<00:37:53.680> all" + }, + { + "start": 2273.87, + "duration": 0.0, + "text": "Um and we can have to find this from all" + }, + { + "start": 2273.88, + "duration": 0.0, + "text": "Um and we can have to find this from all the<00:37:54.000> sub-intervals,<00:37:55.040> x<00:37:55.280> to<00:37:55.400> y,<00:37:55.880> y<00:37:56.120> to<00:37:56.280> z,<00:37:56.640> and" + }, + { + "start": 2276.75, + "duration": 0.0, + "text": "the sub-intervals, x to y, y to z, and" + }, + { + "start": 2276.76, + "duration": 0.0, + "text": "the sub-intervals, x to y, y to z, and then<00:37:56.920> finally<00:37:57.120> z<00:37:57.320> to<00:37:57.400> t.<00:37:57.560> We<00:37:57.640> need<00:37:57.760> to<00:37:57.840> somehow" + }, + { + "start": 2278.87, + "duration": 0.0, + "text": "then finally z to t. We need to somehow" + }, + { + "start": 2278.88, + "duration": 0.0, + "text": "then finally z to t. We need to somehow be<00:37:59.120> able<00:37:59.360> to<00:37:59.480> combine<00:38:00.000> those<00:38:00.360> to<00:38:00.440> get<00:38:00.760> the<00:38:01.520> the" + }, + { + "start": 2281.63, + "duration": 0.0, + "text": "be able to combine those to get the the" + }, + { + "start": 2281.64, + "duration": 0.0, + "text": "be able to combine those to get the the bigger<00:38:01.880> valuation.<00:38:03.000> And<00:38:03.240> this,<00:38:03.640> mind<00:38:04.000> you,<00:38:04.120> is" + }, + { + "start": 2284.27, + "duration": 0.0, + "text": "bigger valuation. And this, mind you, is" + }, + { + "start": 2284.28, + "duration": 0.0, + "text": "bigger valuation. And this, mind you, is an<00:38:04.400> entirely<00:38:04.960> different<00:38:05.320> operation" + }, + { + "start": 2286.75, + "duration": 0.0, + "text": "an entirely different operation" + }, + { + "start": 2286.76, + "duration": 0.0, + "text": "an entirely different operation than<00:38:07.920> adding<00:38:08.320> independent<00:38:08.800> systems" + }, + { + "start": 2289.15, + "duration": 0.0, + "text": "than adding independent systems" + }, + { + "start": 2289.16, + "duration": 0.0, + "text": "than adding independent systems together,<00:38:10.080> but<00:38:10.320> it<00:38:10.440> turns<00:38:10.760> out<00:38:10.880> that<00:38:11.000> this" + }, + { + "start": 2291.19, + "duration": 0.0, + "text": "together, but it turns out that this" + }, + { + "start": 2291.2, + "duration": 0.0, + "text": "together, but it turns out that this will<00:38:11.400> also<00:38:11.680> be<00:38:12.320> uh<00:38:12.520> a<00:38:12.600> product<00:38:13.080> rule." + }, + { + "start": 2295.44, + "duration": 0.0, + "text": "Now,<00:38:15.640> we<00:38:15.760> have<00:38:16.000> the<00:38:16.440> fifth<00:38:16.720> and<00:38:16.840> last<00:38:17.080> symmetry" + }, + { + "start": 2297.67, + "duration": 0.0, + "text": "Now, we have the fifth and last symmetry" + }, + { + "start": 2297.68, + "duration": 0.0, + "text": "Now, we have the fifth and last symmetry that<00:38:17.840> we<00:38:17.920> are<00:38:18.040> going<00:38:18.200> to<00:38:18.440> look<00:38:18.640> at." + }, + { + "start": 2299.75, + "duration": 0.0, + "text": "that we are going to look at." + }, + { + "start": 2299.76, + "duration": 0.0, + "text": "that we are going to look at. The<00:38:19.840> chaining<00:38:20.480> of<00:38:20.680> these<00:38:21.360> uh<00:38:21.760> intervals<00:38:22.320> in" + }, + { + "start": 2302.47, + "duration": 0.0, + "text": "The chaining of these uh intervals in" + }, + { + "start": 2302.48, + "duration": 0.0, + "text": "The chaining of these uh intervals in the<00:38:22.560> lattice<00:38:22.960> is<00:38:23.280> associative." + }, + { + "start": 2304.63, + "duration": 0.0, + "text": "the lattice is associative." + }, + { + "start": 2304.64, + "duration": 0.0, + "text": "the lattice is associative. Meaning<00:38:25.160> it<00:38:25.400> doesn't<00:38:25.720> really<00:38:25.880> matter<00:38:26.240> in<00:38:26.800> what" + }, + { + "start": 2307.07, + "duration": 0.0, + "text": "Meaning it doesn't really matter in what" + }, + { + "start": 2307.08, + "duration": 0.0, + "text": "Meaning it doesn't really matter in what order" + }, + { + "start": 2308.15, + "duration": 0.0, + "text": "order" + }, + { + "start": 2308.16, + "duration": 0.0, + "text": "order we<00:38:28.280> do<00:38:28.400> the<00:38:28.520> chaining<00:38:28.840> operation." + }, + { + "start": 2309.99, + "duration": 0.0, + "text": "we do the chaining operation." + }, + { + "start": 2310.0, + "duration": 0.0, + "text": "we do the chaining operation. This<00:38:30.160> simply<00:38:30.400> follows<00:38:30.640> from<00:38:30.760> the<00:38:30.840> fact<00:38:31.160> that" + }, + { + "start": 2311.47, + "duration": 0.0, + "text": "This simply follows from the fact that" + }, + { + "start": 2311.48, + "duration": 0.0, + "text": "This simply follows from the fact that implication<00:38:32.280> is<00:38:32.400> a<00:38:32.440> property<00:38:33.000> of<00:38:33.200> the" + }, + { + "start": 2313.31, + "duration": 0.0, + "text": "implication is a property of the" + }, + { + "start": 2313.32, + "duration": 0.0, + "text": "implication is a property of the intervals<00:38:33.800> as<00:38:33.920> a<00:38:33.960> whole.<00:38:34.680> And<00:38:34.840> so<00:38:35.080> it's<00:38:35.280> absurd" + }, + { + "start": 2315.67, + "duration": 0.0, + "text": "intervals as a whole. And so it's absurd" + }, + { + "start": 2315.68, + "duration": 0.0, + "text": "intervals as a whole. And so it's absurd to<00:38:35.760> make<00:38:35.920> it<00:38:36.040> somehow<00:38:36.320> depend<00:38:36.800> on<00:38:37.000> our<00:38:37.440> way<00:38:37.600> of" + }, + { + "start": 2317.75, + "duration": 0.0, + "text": "to make it somehow depend on our way of" + }, + { + "start": 2317.76, + "duration": 0.0, + "text": "to make it somehow depend on our way of chaining<00:38:38.080> them<00:38:38.200> together." + }, + { + "start": 2319.27, + "duration": 0.0, + "text": "chaining them together." + }, + { + "start": 2319.28, + "duration": 0.0, + "text": "chaining them together. As<00:38:39.480> long<00:38:39.640> as<00:38:39.760> of<00:38:39.840> course" + }, + { + "start": 2321.11, + "duration": 0.0, + "text": "As long as of course" + }, + { + "start": 2321.12, + "duration": 0.0, + "text": "As long as of course we<00:38:41.240> don't<00:38:41.520> just<00:38:42.080> skip<00:38:42.400> elements<00:38:42.760> and<00:38:42.880> won't<00:38:43.040> do" + }, + { + "start": 2323.11, + "duration": 0.0, + "text": "we don't just skip elements and won't do" + }, + { + "start": 2323.12, + "duration": 0.0, + "text": "we don't just skip elements and won't do anything<00:38:43.480> do<00:38:43.640> anything<00:38:43.920> stupid.<00:38:44.280> But<00:38:44.640> given" + }, + { + "start": 2324.87, + "duration": 0.0, + "text": "anything do anything stupid. But given" + }, + { + "start": 2324.88, + "duration": 0.0, + "text": "anything do anything stupid. But given that<00:38:45.000> this<00:38:45.120> is<00:38:45.880> this<00:38:46.000> is<00:38:46.080> a<00:38:46.120> valid<00:38:46.720> you<00:38:46.800> know" + }, + { + "start": 2328.67, + "duration": 0.0, + "text": "that this is this is a valid you know" + }, + { + "start": 2328.68, + "duration": 0.0, + "text": "that this is this is a valid you know choice<00:38:49.040> of<00:38:49.480> elements" + }, + { + "start": 2330.79, + "duration": 0.0, + "text": "choice of elements" + }, + { + "start": 2330.8, + "duration": 0.0, + "text": "choice of elements it<00:38:50.920> doesn't<00:38:51.120> matter<00:38:51.320> how<00:38:51.440> we<00:38:51.520> combine<00:38:51.840> them." + }, + { + "start": 2333.59, + "duration": 0.0, + "text": "it doesn't matter how we combine them." + }, + { + "start": 2333.6, + "duration": 0.0, + "text": "it doesn't matter how we combine them. And<00:38:53.720> then<00:38:53.840> of<00:38:53.920> course<00:38:54.400> the<00:38:54.600> operator<00:38:55.080> which<00:38:55.240> is" + }, + { + "start": 2335.31, + "duration": 0.0, + "text": "And then of course the operator which is" + }, + { + "start": 2335.32, + "duration": 0.0, + "text": "And then of course the operator which is kind<00:38:55.440> of<00:38:55.520> this<00:38:56.040> bull's-eye<00:38:57.080> bull's-eye" + }, + { + "start": 2338.23, + "duration": 0.0, + "text": "kind of this bull's-eye bull's-eye" + }, + { + "start": 2338.24, + "duration": 0.0, + "text": "kind of this bull's-eye bull's-eye symbol" + }, + { + "start": 2340.55, + "duration": 0.0, + "text": "symbol" + }, + { + "start": 2340.56, + "duration": 0.0, + "text": "symbol must<00:39:01.080> also<00:39:01.560> satisfy<00:39:02.040> this.<00:39:02.320> This<00:39:02.520> is<00:39:02.640> just" + }, + { + "start": 2343.07, + "duration": 0.0, + "text": "must also satisfy this. This is just" + }, + { + "start": 2343.08, + "duration": 0.0, + "text": "must also satisfy this. This is just written<00:39:03.400> down<00:39:03.640> with<00:39:03.800> the<00:39:04.120> probability" + }, + { + "start": 2344.59, + "duration": 0.0, + "text": "written down with the probability" + }, + { + "start": 2344.6, + "duration": 0.0, + "text": "written down with the probability notation." + }, + { + "start": 2346.87, + "duration": 0.0, + "text": "notation." + }, + { + "start": 2346.88, + "duration": 0.0, + "text": "notation. And<00:39:07.480> we<00:39:08.080> in<00:39:08.240> order<00:39:08.400> to<00:39:08.920> use<00:39:09.240> our<00:39:09.360> beautiful" + }, + { + "start": 2349.95, + "duration": 0.0, + "text": "And we in order to use our beautiful" + }, + { + "start": 2349.96, + "duration": 0.0, + "text": "And we in order to use our beautiful associativity<00:39:10.640> theorem<00:39:10.960> once<00:39:11.160> again<00:39:11.440> we" + }, + { + "start": 2351.59, + "duration": 0.0, + "text": "associativity theorem once again we" + }, + { + "start": 2351.6, + "duration": 0.0, + "text": "associativity theorem once again we unfortunately<00:39:12.080> need<00:39:12.400> to<00:39:12.600> show<00:39:12.800> that<00:39:12.960> it" + }, + { + "start": 2353.03, + "duration": 0.0, + "text": "unfortunately need to show that it" + }, + { + "start": 2353.04, + "duration": 0.0, + "text": "unfortunately need to show that it satisfies<00:39:13.560> symmetry<00:39:13.960> one<00:39:14.800> yet<00:39:14.920> again.<00:39:15.480> I'm" + }, + { + "start": 2355.59, + "duration": 0.0, + "text": "satisfies symmetry one yet again. I'm" + }, + { + "start": 2355.6, + "duration": 0.0, + "text": "satisfies symmetry one yet again. I'm sorry." + }, + { + "start": 2357.23, + "duration": 0.0, + "text": "sorry." + }, + { + "start": 2357.24, + "duration": 0.0, + "text": "sorry. And<00:39:17.560> we<00:39:17.640> show<00:39:17.840> that<00:39:18.120> chaining<00:39:18.800> of<00:39:18.960> intervals" + }, + { + "start": 2359.51, + "duration": 0.0, + "text": "And we show that chaining of intervals" + }, + { + "start": 2359.52, + "duration": 0.0, + "text": "And we show that chaining of intervals preserves<00:39:20.040> the<00:39:20.160> order<00:39:21.120> from<00:39:21.760> the<00:39:21.880> left<00:39:22.320> and" + }, + { + "start": 2362.43, + "duration": 0.0, + "text": "preserves the order from the left and" + }, + { + "start": 2362.44, + "duration": 0.0, + "text": "preserves the order from the left and the<00:39:22.520> right.<00:39:23.160> So<00:39:23.400> again<00:39:23.680> if<00:39:23.800> you<00:39:23.880> have<00:39:24.040> this" + }, + { + "start": 2364.19, + "duration": 0.0, + "text": "the right. So again if you have this" + }, + { + "start": 2364.2, + "duration": 0.0, + "text": "the right. So again if you have this element<00:39:24.640> XYZT" + }, + { + "start": 2366.19, + "duration": 0.0, + "text": "element XYZT" + }, + { + "start": 2366.2, + "duration": 0.0, + "text": "element XYZT then<00:39:26.880> given<00:39:27.280> the<00:39:27.400> same<00:39:27.640> context" + }, + { + "start": 2369.15, + "duration": 0.0, + "text": "then given the same context" + }, + { + "start": 2369.16, + "duration": 0.0, + "text": "then given the same context the<00:39:29.280> probability<00:39:29.680> of<00:39:29.840> X<00:39:30.240> is<00:39:30.600> strictly<00:39:30.880> lower" + }, + { + "start": 2370.99, + "duration": 0.0, + "text": "the probability of X is strictly lower" + }, + { + "start": 2371.0, + "duration": 0.0, + "text": "the probability of X is strictly lower than<00:39:31.080> probability<00:39:31.440> of<00:39:31.560> Y<00:39:32.360> because<00:39:33.080> it's" + }, + { + "start": 2373.23, + "duration": 0.0, + "text": "than probability of Y because it's" + }, + { + "start": 2373.24, + "duration": 0.0, + "text": "than probability of Y because it's further<00:39:33.480> apart<00:39:33.720> from<00:39:33.840> the<00:39:33.920> context.<00:39:34.440> It's<00:39:34.600> a" + }, + { + "start": 2374.63, + "duration": 0.0, + "text": "further apart from the context. It's a" + }, + { + "start": 2374.64, + "duration": 0.0, + "text": "further apart from the context. It's a smaller<00:39:35.520> it's<00:39:35.720> a<00:39:35.760> smaller<00:39:36.040> thing.<00:39:36.240> It's<00:39:36.320> a" + }, + { + "start": 2376.35, + "duration": 0.0, + "text": "smaller it's a smaller thing. It's a" + }, + { + "start": 2376.36, + "duration": 0.0, + "text": "smaller it's a smaller thing. It's a smaller<00:39:36.640> subset<00:39:36.960> of<00:39:37.040> the<00:39:37.120> possibilities." + }, + { + "start": 2378.75, + "duration": 0.0, + "text": "smaller subset of the possibilities." + }, + { + "start": 2378.76, + "duration": 0.0, + "text": "smaller subset of the possibilities. Um" + }, + { + "start": 2379.75, + "duration": 0.0, + "text": "Um" + }, + { + "start": 2379.76, + "duration": 0.0, + "text": "Um then<00:39:40.200> if<00:39:40.440> we<00:39:40.560> kind<00:39:40.720> of<00:39:41.040> chain<00:39:41.440> them<00:39:42.160> so<00:39:42.320> that" + }, + { + "start": 2382.51, + "duration": 0.0, + "text": "then if we kind of chain them so that" + }, + { + "start": 2382.52, + "duration": 0.0, + "text": "then if we kind of chain them so that both<00:39:43.120> both<00:39:43.360> have<00:39:43.520> kind<00:39:43.680> of<00:39:44.080> a<00:39:44.120> larger<00:39:44.440> context" + }, + { + "start": 2385.31, + "duration": 0.0, + "text": "both both have kind of a larger context" + }, + { + "start": 2385.32, + "duration": 0.0, + "text": "both both have kind of a larger context we<00:39:45.480> know<00:39:45.800> from<00:39:46.000> the<00:39:46.080> property<00:39:46.840> that<00:39:47.000> these<00:39:47.160> B" + }, + { + "start": 2387.31, + "duration": 0.0, + "text": "we know from the property that these B" + }, + { + "start": 2387.32, + "duration": 0.0, + "text": "we know from the property that these B valuations<00:39:48.000> are<00:39:48.160> just<00:39:49.120> um" + }, + { + "start": 2389.87, + "duration": 0.0, + "text": "valuations are just um" + }, + { + "start": 2389.88, + "duration": 0.0, + "text": "valuations are just um sort<00:39:50.040> of<00:39:50.160> these<00:39:50.400> these<00:39:50.640> these<00:39:50.920> ranges" + }, + { + "start": 2392.15, + "duration": 0.0, + "text": "sort of these these these ranges" + }, + { + "start": 2392.16, + "duration": 0.0, + "text": "sort of these these these ranges we<00:39:52.320> know<00:39:52.520> that<00:39:52.680> this<00:39:53.280> still<00:39:53.520> preserves<00:39:53.880> the" + }, + { + "start": 2393.99, + "duration": 0.0, + "text": "we know that this still preserves the" + }, + { + "start": 2394.0, + "duration": 0.0, + "text": "we know that this still preserves the order.<00:39:54.400> And<00:39:54.960> dually<00:39:55.320> for<00:39:55.440> the<00:39:55.560> other<00:39:55.760> case<00:39:56.560> if" + }, + { + "start": 2396.91, + "duration": 0.0, + "text": "order. And dually for the other case if" + }, + { + "start": 2396.92, + "duration": 0.0, + "text": "order. And dually for the other case if we<00:39:57.040> have<00:39:57.200> the<00:39:57.840> two<00:39:58.120> propositions<00:39:58.920> that<00:39:59.440> have" + }, + { + "start": 2399.71, + "duration": 0.0, + "text": "we have the two propositions that have" + }, + { + "start": 2399.72, + "duration": 0.0, + "text": "we have the two propositions that have different<00:40:00.000> contexts" + }, + { + "start": 2401.31, + "duration": 0.0, + "text": "different contexts" + }, + { + "start": 2401.32, + "duration": 0.0, + "text": "different contexts if<00:40:01.560> we<00:40:01.680> somehow<00:40:02.840> um" + }, + { + "start": 2403.99, + "duration": 0.0, + "text": "if we somehow um" + }, + { + "start": 2404.0, + "duration": 0.0, + "text": "if we somehow um chain<00:40:04.200> them<00:40:04.280> together<00:40:04.560> such<00:40:04.840> that<00:40:05.000> they<00:40:05.120> have" + }, + { + "start": 2405.83, + "duration": 0.0, + "text": "chain them together such that they have" + }, + { + "start": 2405.84, + "duration": 0.0, + "text": "chain them together such that they have another<00:40:06.200> new<00:40:06.480> common<00:40:06.760> proposition<00:40:07.160> that<00:40:07.320> is" + }, + { + "start": 2407.83, + "duration": 0.0, + "text": "another new common proposition that is" + }, + { + "start": 2407.84, + "duration": 0.0, + "text": "another new common proposition that is further<00:40:08.160> up<00:40:08.360> the<00:40:08.480> order<00:40:09.240> then<00:40:09.440> this<00:40:09.720> also" + }, + { + "start": 2410.95, + "duration": 0.0, + "text": "further up the order then this also" + }, + { + "start": 2410.96, + "duration": 0.0, + "text": "further up the order then this also preserves" + }, + { + "start": 2412.39, + "duration": 0.0, + "text": "preserves" + }, + { + "start": 2412.4, + "duration": 0.0, + "text": "preserves the<00:40:12.440> original<00:40:12.840> order." + }, + { + "start": 2416.64, + "duration": 0.0, + "text": "And<00:40:16.760> now<00:40:16.880> we<00:40:16.960> can<00:40:17.120> once<00:40:17.360> again<00:40:17.760> apply<00:40:18.160> a" + }, + { + "start": 2418.23, + "duration": 0.0, + "text": "And now we can once again apply a" + }, + { + "start": 2418.24, + "duration": 0.0, + "text": "And now we can once again apply a beautiful<00:40:18.760> associativity<00:40:19.480> theorem<00:40:19.880> yet" + }, + { + "start": 2420.03, + "duration": 0.0, + "text": "beautiful associativity theorem yet" + }, + { + "start": 2420.04, + "duration": 0.0, + "text": "beautiful associativity theorem yet again." + }, + { + "start": 2421.03, + "duration": 0.0, + "text": "again." + }, + { + "start": 2421.04, + "duration": 0.0, + "text": "again. Um<00:40:21.320> I<00:40:21.400> just<00:40:21.600> gave<00:40:21.760> it<00:40:21.840> a<00:40:21.920> slightly<00:40:22.200> different" + }, + { + "start": 2422.47, + "duration": 0.0, + "text": "Um I just gave it a slightly different" + }, + { + "start": 2422.48, + "duration": 0.0, + "text": "Um I just gave it a slightly different name<00:40:22.840> with<00:40:23.240> theta<00:40:23.560> hat<00:40:23.840> because<00:40:24.160> it<00:40:24.520> me<00:40:24.840> it" + }, + { + "start": 2424.95, + "duration": 0.0, + "text": "name with theta hat because it me it" + }, + { + "start": 2424.96, + "duration": 0.0, + "text": "name with theta hat because it me it always<00:40:25.200> pisses<00:40:25.480> me<00:40:25.640> off<00:40:25.920> when<00:40:26.440> authors<00:40:26.840> reuse" + }, + { + "start": 2427.15, + "duration": 0.0, + "text": "always pisses me off when authors reuse" + }, + { + "start": 2427.16, + "duration": 0.0, + "text": "always pisses me off when authors reuse the<00:40:27.240> symbols<00:40:27.800> in<00:40:27.920> different<00:40:28.160> contexts,<00:40:28.600> so" + }, + { + "start": 2429.59, + "duration": 0.0, + "text": "the symbols in different contexts, so" + }, + { + "start": 2429.6, + "duration": 0.0, + "text": "the symbols in different contexts, so I<00:40:29.640> thought<00:40:29.800> I'd<00:40:29.880> do<00:40:29.960> the<00:40:30.040> courtesy<00:40:30.440> of<00:40:30.600> not" + }, + { + "start": 2430.79, + "duration": 0.0, + "text": "I thought I'd do the courtesy of not" + }, + { + "start": 2430.8, + "duration": 0.0, + "text": "I thought I'd do the courtesy of not doing<00:40:31.040> it.<00:40:31.680> And<00:40:31.800> once<00:40:31.920> again,<00:40:32.200> we<00:40:32.360> know<00:40:32.600> that" + }, + { + "start": 2432.83, + "duration": 0.0, + "text": "doing it. And once again, we know that" + }, + { + "start": 2432.84, + "duration": 0.0, + "text": "doing it. And once again, we know that somehow<00:40:33.480> some<00:40:34.360> ins-<00:40:34.720> for<00:40:34.880> some<00:40:35.920> uh" + }, + { + "start": 2437.55, + "duration": 0.0, + "text": "somehow some ins- for some uh" + }, + { + "start": 2437.56, + "duration": 0.0, + "text": "somehow some ins- for some uh theta<00:40:37.760> hat<00:40:37.920> for<00:40:38.000> some<00:40:38.360> mapping,<00:40:39.160> we<00:40:39.280> have" + }, + { + "start": 2439.51, + "duration": 0.0, + "text": "theta hat for some mapping, we have" + }, + { + "start": 2439.52, + "duration": 0.0, + "text": "theta hat for some mapping, we have addition<00:40:39.960> with" + }, + { + "start": 2441.47, + "duration": 0.0, + "text": "addition with" + }, + { + "start": 2441.48, + "duration": 0.0, + "text": "addition with this<00:40:41.640> operation<00:40:42.040> is<00:40:42.120> going<00:40:42.240> to<00:40:42.320> be<00:40:42.400> addition" + }, + { + "start": 2442.75, + "duration": 0.0, + "text": "this operation is going to be addition" + }, + { + "start": 2442.76, + "duration": 0.0, + "text": "this operation is going to be addition again." + }, + { + "start": 2444.15, + "duration": 0.0, + "text": "again." + }, + { + "start": 2444.16, + "duration": 0.0, + "text": "again. Um" + }, + { + "start": 2445.19, + "duration": 0.0, + "text": "Um" + }, + { + "start": 2445.2, + "duration": 0.0, + "text": "Um once<00:40:45.520> again,<00:40:45.880> we<00:40:46.040> have<00:40:46.240> already<00:40:46.480> chosen" + }, + { + "start": 2446.75, + "duration": 0.0, + "text": "once again, we have already chosen" + }, + { + "start": 2446.76, + "duration": 0.0, + "text": "once again, we have already chosen addition<00:40:47.120> for<00:40:47.480> our<00:40:48.040> first<00:40:48.320> operator,<00:40:49.400> and<00:40:49.560> so" + }, + { + "start": 2449.79, + "duration": 0.0, + "text": "addition for our first operator, and so" + }, + { + "start": 2449.8, + "duration": 0.0, + "text": "addition for our first operator, and so we<00:40:50.040> can<00:40:50.360> just<00:40:50.520> simply<00:40:50.800> recycle<00:40:51.200> the<00:40:51.280> proof" + }, + { + "start": 2451.91, + "duration": 0.0, + "text": "we can just simply recycle the proof" + }, + { + "start": 2451.92, + "duration": 0.0, + "text": "we can just simply recycle the proof from<00:40:52.360> our<00:40:53.080> previous<00:40:53.440> case.<00:40:54.200> And<00:40:54.880> so<00:40:55.520> this" + }, + { + "start": 2455.79, + "duration": 0.0, + "text": "from our previous case. And so this" + }, + { + "start": 2455.8, + "duration": 0.0, + "text": "from our previous case. And so this obviously<00:40:56.400> gave<00:40:56.640> you<00:40:56.720> already<00:40:56.960> the<00:40:57.040> hint<00:40:57.280> that" + }, + { + "start": 2457.35, + "duration": 0.0, + "text": "obviously gave you already the hint that" + }, + { + "start": 2457.36, + "duration": 0.0, + "text": "obviously gave you already the hint that it's<00:40:57.480> going<00:40:57.600> to<00:40:57.640> be<00:40:57.720> another<00:40:57.920> product<00:40:58.280> rule." + }, + { + "start": 2459.15, + "duration": 0.0, + "text": "it's going to be another product rule." + }, + { + "start": 2459.16, + "duration": 0.0, + "text": "it's going to be another product rule. Uh<00:40:59.680> we<00:41:00.080> set<00:41:00.320> up<00:41:00.440> kind<00:41:00.680> of<00:41:00.800> the<00:41:01.600> very<00:41:02.040> similar" + }, + { + "start": 2462.51, + "duration": 0.0, + "text": "Uh we set up kind of the very similar" + }, + { + "start": 2462.52, + "duration": 0.0, + "text": "Uh we set up kind of the very similar kind<00:41:02.880> of" + }, + { + "start": 2463.67, + "duration": 0.0, + "text": "kind of" + }, + { + "start": 2463.68, + "duration": 0.0, + "text": "kind of um" + }, + { + "start": 2466.24, + "duration": 0.0, + "text": "consistency<00:41:06.760> check<00:41:07.320> that<00:41:07.480> must<00:41:07.640> be" + }, + { + "start": 2467.75, + "duration": 0.0, + "text": "consistency check that must be" + }, + { + "start": 2467.76, + "duration": 0.0, + "text": "consistency check that must be consistent<00:41:08.200> with<00:41:08.360> the<00:41:08.920> operation<00:41:09.360> of<00:41:09.480> joining" + }, + { + "start": 2469.99, + "duration": 0.0, + "text": "consistent with the operation of joining" + }, + { + "start": 2470.0, + "duration": 0.0, + "text": "consistent with the operation of joining to<00:41:10.400> disjoint<00:41:10.920> elements<00:41:11.720> that<00:41:11.880> we<00:41:11.960> just<00:41:12.680> for" + }, + { + "start": 2472.79, + "duration": 0.0, + "text": "to disjoint elements that we just for" + }, + { + "start": 2472.8, + "duration": 0.0, + "text": "to disjoint elements that we just for which<00:41:12.960> we<00:41:13.080> know<00:41:13.720> for<00:41:13.840> which<00:41:14.000> we<00:41:14.080> have<00:41:14.240> chosen" + }, + { + "start": 2474.79, + "duration": 0.0, + "text": "which we know for which we have chosen" + }, + { + "start": 2474.8, + "duration": 0.0, + "text": "which we know for which we have chosen addition." + }, + { + "start": 2476.03, + "duration": 0.0, + "text": "addition." + }, + { + "start": 2476.04, + "duration": 0.0, + "text": "addition. Um<00:41:16.520> we<00:41:16.640> again<00:41:16.880> define<00:41:17.160> the<00:41:17.280> opposite.<00:41:18.320> We<00:41:18.440> find" + }, + { + "start": 2478.99, + "duration": 0.0, + "text": "Um we again define the opposite. We find" + }, + { + "start": 2479.0, + "duration": 0.0, + "text": "Um we again define the opposite. We find this<00:41:19.160> very<00:41:19.400> similar<00:41:19.720> structure<00:41:20.000> to<00:41:20.120> before," + }, + { + "start": 2480.75, + "duration": 0.0, + "text": "this very similar structure to before," + }, + { + "start": 2480.76, + "duration": 0.0, + "text": "this very similar structure to before, and<00:41:20.920> through<00:41:21.120> the<00:41:21.200> same<00:41:21.480> proof<00:41:21.800> in<00:41:21.920> fact,<00:41:22.200> we" + }, + { + "start": 2482.31, + "duration": 0.0, + "text": "and through the same proof in fact, we" + }, + { + "start": 2482.32, + "duration": 0.0, + "text": "and through the same proof in fact, we find<00:41:22.640> that<00:41:23.040> V<00:41:23.240> is<00:41:23.400> once<00:41:23.560> again<00:41:23.840> C<00:41:24.040> to<00:41:24.160> the<00:41:24.360> AX." + }, + { + "start": 2485.51, + "duration": 0.0, + "text": "find that V is once again C to the AX." + }, + { + "start": 2485.52, + "duration": 0.0, + "text": "find that V is once again C to the AX. And<00:41:25.880> meaning<00:41:26.120> once<00:41:26.320> again,<00:41:26.880> we<00:41:27.120> see<00:41:27.360> that<00:41:27.600> this" + }, + { + "start": 2488.39, + "duration": 0.0, + "text": "And meaning once again, we see that this" + }, + { + "start": 2488.4, + "duration": 0.0, + "text": "And meaning once again, we see that this uh<00:41:28.520> chaining<00:41:28.920> operator<00:41:29.400> is<00:41:29.640> also" + }, + { + "start": 2490.11, + "duration": 0.0, + "text": "uh chaining operator is also" + }, + { + "start": 2490.12, + "duration": 0.0, + "text": "uh chaining operator is also multiplication.<00:41:30.840> So<00:41:31.040> the<00:41:31.160> probability<00:41:31.840> of<00:41:32.120> A" + }, + { + "start": 2492.31, + "duration": 0.0, + "text": "multiplication. So the probability of A" + }, + { + "start": 2492.32, + "duration": 0.0, + "text": "multiplication. So the probability of A given<00:41:32.600> C" + }, + { + "start": 2493.63, + "duration": 0.0, + "text": "given C" + }, + { + "start": 2493.64, + "duration": 0.0, + "text": "given C is<00:41:33.760> equal<00:41:33.920> to<00:41:34.000> the<00:41:34.080> probability<00:41:34.600> of<00:41:34.880> A<00:41:35.160> given<00:41:35.480> B" + }, + { + "start": 2495.83, + "duration": 0.0, + "text": "is equal to the probability of A given B" + }, + { + "start": 2495.84, + "duration": 0.0, + "text": "is equal to the probability of A given B and<00:41:36.320> times<00:41:36.600> the<00:41:36.680> probability<00:41:37.120> of<00:41:37.360> B<00:41:37.640> given<00:41:37.960> C." + }, + { + "start": 2499.55, + "duration": 0.0, + "text": "and times the probability of B given C." + }, + { + "start": 2499.56, + "duration": 0.0, + "text": "and times the probability of B given C. Um" + }, + { + "start": 2500.23, + "duration": 0.0, + "text": "Um" + }, + { + "start": 2500.24, + "duration": 0.0, + "text": "Um and<00:41:40.400> there<00:41:40.520> we<00:41:40.600> go.<00:41:40.680> Now<00:41:40.840> we<00:41:40.960> have<00:41:41.160> derived" + }, + { + "start": 2501.63, + "duration": 0.0, + "text": "and there we go. Now we have derived" + }, + { + "start": 2501.64, + "duration": 0.0, + "text": "and there we go. Now we have derived basically<00:41:42.440> uh<00:41:42.960> all<00:41:43.120> the<00:41:43.200> probability" + }, + { + "start": 2503.71, + "duration": 0.0, + "text": "basically uh all the probability" + }, + { + "start": 2503.72, + "duration": 0.0, + "text": "basically uh all the probability calculus<00:41:44.280> of<00:41:44.480> the<00:41:44.720> Bayesian<00:41:45.160> inference" + }, + { + "start": 2505.47, + "duration": 0.0, + "text": "calculus of the Bayesian inference" + }, + { + "start": 2505.48, + "duration": 0.0, + "text": "calculus of the Bayesian inference machine." + }, + { + "start": 2506.75, + "duration": 0.0, + "text": "machine." + }, + { + "start": 2506.76, + "duration": 0.0, + "text": "machine. And<00:41:47.160> we<00:41:47.400> just<00:41:48.040> need<00:41:48.280> to<00:41:48.400> find<00:41:48.840> the<00:41:49.120> kind<00:41:49.360> of" + }, + { + "start": 2509.67, + "duration": 0.0, + "text": "And we just need to find the kind of" + }, + { + "start": 2509.68, + "duration": 0.0, + "text": "And we just need to find the kind of more<00:41:50.160> general<00:41:50.680> form<00:41:51.400> of<00:41:51.560> the<00:41:51.640> product<00:41:52.040> rule." + }, + { + "start": 2513.63, + "duration": 0.0, + "text": "more general form of the product rule." + }, + { + "start": 2513.64, + "duration": 0.0, + "text": "more general form of the product rule. Uh<00:41:54.040> and<00:41:54.200> we<00:41:54.320> can<00:41:54.600> do<00:41:54.760> that<00:41:54.960> by<00:41:55.080> simply<00:41:55.400> noting" + }, + { + "start": 2515.83, + "duration": 0.0, + "text": "Uh and we can do that by simply noting" + }, + { + "start": 2515.84, + "duration": 0.0, + "text": "Uh and we can do that by simply noting that<00:41:56.400> probability<00:41:57.160> of<00:41:57.560> X<00:41:57.800> and<00:41:58.000> Y<00:41:58.480> given<00:41:58.880> Y<00:41:59.320> is" + }, + { + "start": 2519.47, + "duration": 0.0, + "text": "that probability of X and Y given Y is" + }, + { + "start": 2519.48, + "duration": 0.0, + "text": "that probability of X and Y given Y is equal<00:41:59.680> to<00:41:59.760> the<00:41:59.840> probability<00:42:00.320> of<00:42:00.520> X<00:42:00.800> given<00:42:01.080> Y<00:42:01.560> if" + }, + { + "start": 2521.87, + "duration": 0.0, + "text": "equal to the probability of X given Y if" + }, + { + "start": 2521.88, + "duration": 0.0, + "text": "equal to the probability of X given Y if X<00:42:02.240> is<00:42:02.640> strictly<00:42:03.000> below<00:42:03.240> Y.<00:42:04.160> Uh<00:42:04.400> if<00:42:04.560> X<00:42:04.680> implies" + }, + { + "start": 2525.03, + "duration": 0.0, + "text": "X is strictly below Y. Uh if X implies" + }, + { + "start": 2525.04, + "duration": 0.0, + "text": "X is strictly below Y. Uh if X implies Y.<00:42:05.840> Uh<00:42:05.920> that's<00:42:06.200> simply<00:42:06.480> because<00:42:07.360> you<00:42:07.480> have<00:42:07.600> to" + }, + { + "start": 2527.67, + "duration": 0.0, + "text": "Y. Uh that's simply because you have to" + }, + { + "start": 2527.68, + "duration": 0.0, + "text": "Y. Uh that's simply because you have to remember<00:42:08.000> that<00:42:08.240> these<00:42:08.520> valuations<00:42:09.360> are<00:42:09.480> just" + }, + { + "start": 2529.99, + "duration": 0.0, + "text": "remember that these valuations are just" + }, + { + "start": 2530.0, + "duration": 0.0, + "text": "remember that these valuations are just uh,<00:42:10.360> these<00:42:10.960> sort<00:42:11.160> of<00:42:11.400> ranges<00:42:12.200> on<00:42:12.400> on<00:42:12.480> the" + }, + { + "start": 2532.59, + "duration": 0.0, + "text": "uh, these sort of ranges on on the" + }, + { + "start": 2532.6, + "duration": 0.0, + "text": "uh, these sort of ranges on on the lattice<00:42:13.000> and<00:42:13.320> obviously<00:42:14.240> these<00:42:14.520> two" + }, + { + "start": 2535.59, + "duration": 0.0, + "text": "lattice and obviously these two" + }, + { + "start": 2535.6, + "duration": 0.0, + "text": "lattice and obviously these two describe<00:42:16.600> the<00:42:16.680> same<00:42:17.160> because<00:42:18.000> X<00:42:18.320> implies<00:42:18.800> Y" + }, + { + "start": 2538.99, + "duration": 0.0, + "text": "describe the same because X implies Y" + }, + { + "start": 2539.0, + "duration": 0.0, + "text": "describe the same because X implies Y already." + }, + { + "start": 2540.63, + "duration": 0.0, + "text": "already." + }, + { + "start": 2540.64, + "duration": 0.0, + "text": "already. So,<00:42:20.720> these<00:42:20.880> two<00:42:21.000> are<00:42:21.080> equivalent.<00:42:22.000> And<00:42:22.160> so,<00:42:22.560> if" + }, + { + "start": 2542.75, + "duration": 0.0, + "text": "So, these two are equivalent. And so, if" + }, + { + "start": 2542.76, + "duration": 0.0, + "text": "So, these two are equivalent. And so, if we<00:42:22.840> have<00:42:22.960> a<00:42:23.040> probability<00:42:23.480> of<00:42:23.680> A<00:42:23.840> and<00:42:23.960> B<00:42:24.240> given" + }, + { + "start": 2544.51, + "duration": 0.0, + "text": "we have a probability of A and B given" + }, + { + "start": 2544.52, + "duration": 0.0, + "text": "we have a probability of A and B given C,<00:42:25.240> we<00:42:25.360> can<00:42:25.560> kind<00:42:25.760> of<00:42:26.000> expand<00:42:26.520> that<00:42:26.800> by<00:42:27.000> adding" + }, + { + "start": 2547.35, + "duration": 0.0, + "text": "C, we can kind of expand that by adding" + }, + { + "start": 2547.36, + "duration": 0.0, + "text": "C, we can kind of expand that by adding another<00:42:27.840> NC<00:42:28.320> at<00:42:28.440> the<00:42:28.560> end<00:42:29.000> because<00:42:29.640> this" + }, + { + "start": 2549.79, + "duration": 0.0, + "text": "another NC at the end because this" + }, + { + "start": 2549.8, + "duration": 0.0, + "text": "another NC at the end because this doesn't<00:42:30.040> change<00:42:30.240> our<00:42:30.320> valuation.<00:42:31.480> And<00:42:31.640> then" + }, + { + "start": 2551.83, + "duration": 0.0, + "text": "doesn't change our valuation. And then" + }, + { + "start": 2551.84, + "duration": 0.0, + "text": "doesn't change our valuation. And then we<00:42:31.960> can<00:42:32.120> kind<00:42:32.360> of<00:42:32.480> split<00:42:33.040> it<00:42:33.160> up<00:42:33.280> into<00:42:33.480> two" + }, + { + "start": 2553.59, + "duration": 0.0, + "text": "we can kind of split it up into two" + }, + { + "start": 2553.6, + "duration": 0.0, + "text": "we can kind of split it up into two parts<00:42:34.440> and<00:42:34.680> we<00:42:34.800> use<00:42:35.160> our<00:42:35.280> chaining" + }, + { + "start": 2556.55, + "duration": 0.0, + "text": "parts and we use our chaining" + }, + { + "start": 2556.56, + "duration": 0.0, + "text": "parts and we use our chaining uh,<00:42:37.040> like<00:42:37.200> this.<00:42:37.720> Here<00:42:37.960> we<00:42:38.280> sort<00:42:38.480> of<00:42:38.600> condition" + }, + { + "start": 2559.07, + "duration": 0.0, + "text": "uh, like this. Here we sort of condition" + }, + { + "start": 2559.08, + "duration": 0.0, + "text": "uh, like this. Here we sort of condition on<00:42:39.520> B<00:42:39.760> and<00:42:39.920> C<00:42:40.280> and<00:42:40.440> then<00:42:40.640> we<00:42:41.280> go<00:42:41.440> from<00:42:41.640> B<00:42:41.840> and<00:42:41.960> C" + }, + { + "start": 2562.67, + "duration": 0.0, + "text": "on B and C and then we go from B and C" + }, + { + "start": 2562.68, + "duration": 0.0, + "text": "on B and C and then we go from B and C to<00:42:43.040> just<00:42:43.360> C." + }, + { + "start": 2564.31, + "duration": 0.0, + "text": "to just C." + }, + { + "start": 2564.32, + "duration": 0.0, + "text": "to just C. But<00:42:44.720> these<00:42:45.040> two,<00:42:45.760> uh,<00:42:45.960> can<00:42:46.120> again<00:42:46.360> be" + }, + { + "start": 2566.47, + "duration": 0.0, + "text": "But these two, uh, can again be" + }, + { + "start": 2566.48, + "duration": 0.0, + "text": "But these two, uh, can again be simplified.<00:42:47.560> The<00:42:47.680> right<00:42:47.960> one<00:42:48.120> just<00:42:48.320> becomes" + }, + { + "start": 2568.63, + "duration": 0.0, + "text": "simplified. The right one just becomes" + }, + { + "start": 2568.64, + "duration": 0.0, + "text": "simplified. The right one just becomes probability<00:42:49.160> of<00:42:49.280> B<00:42:49.480> given<00:42:49.760> C<00:42:50.120> and<00:42:50.320> the<00:42:50.400> first" + }, + { + "start": 2570.71, + "duration": 0.0, + "text": "probability of B given C and the first" + }, + { + "start": 2570.72, + "duration": 0.0, + "text": "probability of B given C and the first one<00:42:51.280> just<00:42:51.400> becomes<00:42:51.760> probability<00:42:52.160> of<00:42:52.400> A<00:42:52.680> given" + }, + { + "start": 2572.95, + "duration": 0.0, + "text": "one just becomes probability of A given" + }, + { + "start": 2572.96, + "duration": 0.0, + "text": "one just becomes probability of A given B<00:42:53.080> and<00:42:53.200> C,<00:42:53.400> which<00:42:53.480> is<00:42:53.600> always<00:42:54.280> referring<00:42:54.720> to<00:42:55.080> to" + }, + { + "start": 2575.19, + "duration": 0.0, + "text": "B and C, which is always referring to to" + }, + { + "start": 2575.2, + "duration": 0.0, + "text": "B and C, which is always referring to to this<00:42:55.440> property<00:42:55.880> here,<00:42:56.320> right?" + }, + { + "start": 2577.87, + "duration": 0.0, + "text": "this property here, right?" + }, + { + "start": 2577.88, + "duration": 0.0, + "text": "this property here, right? And<00:42:59.040> very<00:42:59.320> interestingly,<00:42:59.800> we<00:42:59.920> can<00:43:00.120> also" + }, + { + "start": 2580.55, + "duration": 0.0, + "text": "And very interestingly, we can also" + }, + { + "start": 2580.56, + "duration": 0.0, + "text": "And very interestingly, we can also write<00:43:00.880> this<00:43:01.360> as<00:43:02.000> probability<00:43:02.560> of<00:43:02.840> B<00:43:03.080> given<00:43:03.400> AC" + }, + { + "start": 2583.75, + "duration": 0.0, + "text": "write this as probability of B given AC" + }, + { + "start": 2583.76, + "duration": 0.0, + "text": "write this as probability of B given AC times<00:43:04.000> probability<00:43:04.360> of<00:43:04.600> A<00:43:04.760> given<00:43:05.040> C.<00:43:05.240> So," + }, + { + "start": 2586.23, + "duration": 0.0, + "text": "times probability of A given C. So," + }, + { + "start": 2586.24, + "duration": 0.0, + "text": "times probability of A given C. So, this<00:43:06.440> probability" + }, + { + "start": 2587.91, + "duration": 0.0, + "text": "this probability" + }, + { + "start": 2587.92, + "duration": 0.0, + "text": "this probability of<00:43:08.080> A<00:43:08.200> and<00:43:08.320> B<00:43:08.480> given<00:43:08.760> C<00:43:09.560> can<00:43:09.760> be<00:43:09.880> written<00:43:10.080> in<00:43:10.160> two" + }, + { + "start": 2590.39, + "duration": 0.0, + "text": "of A and B given C can be written in two" + }, + { + "start": 2590.4, + "duration": 0.0, + "text": "of A and B given C can be written in two different<00:43:10.760> ways<00:43:11.320> and<00:43:11.440> we<00:43:11.560> can<00:43:11.720> exploit<00:43:12.280> this" + }, + { + "start": 2592.47, + "duration": 0.0, + "text": "different ways and we can exploit this" + }, + { + "start": 2592.48, + "duration": 0.0, + "text": "different ways and we can exploit this fact" + }, + { + "start": 2593.47, + "duration": 0.0, + "text": "fact" + }, + { + "start": 2593.48, + "duration": 0.0, + "text": "fact to<00:43:13.760> write<00:43:14.000> down<00:43:14.320> this<00:43:14.560> equivalence,<00:43:15.160> which<00:43:15.480> is" + }, + { + "start": 2595.63, + "duration": 0.0, + "text": "to write down this equivalence, which is" + }, + { + "start": 2595.64, + "duration": 0.0, + "text": "to write down this equivalence, which is just<00:43:16.280> Bayes'<00:43:16.600> rule.<00:43:17.200> So,<00:43:17.280> Bayes'<00:43:17.600> rule<00:43:17.760> is" + }, + { + "start": 2597.83, + "duration": 0.0, + "text": "just Bayes' rule. So, Bayes' rule is" + }, + { + "start": 2597.84, + "duration": 0.0, + "text": "just Bayes' rule. So, Bayes' rule is just<00:43:18.000> a<00:43:18.040> consequence" + }, + { + "start": 2599.35, + "duration": 0.0, + "text": "just a consequence" + }, + { + "start": 2599.36, + "duration": 0.0, + "text": "just a consequence of<00:43:19.720> this,<00:43:20.280> uh,<00:43:20.400> product<00:43:20.800> rule<00:43:20.920> for<00:43:21.040> chaining" + }, + { + "start": 2602.31, + "duration": 0.0, + "text": "of this, uh, product rule for chaining" + }, + { + "start": 2602.32, + "duration": 0.0, + "text": "of this, uh, product rule for chaining uh,<00:43:22.680> different<00:43:22.960> valuations." + }, + { + "start": 2604.87, + "duration": 0.0, + "text": "uh, different valuations." + }, + { + "start": 2604.88, + "duration": 0.0, + "text": "uh, different valuations. And<00:43:25.080> I'm<00:43:25.200> sure<00:43:25.400> you<00:43:25.480> all<00:43:25.640> know<00:43:26.000> just<00:43:26.200> how" + }, + { + "start": 2606.35, + "duration": 0.0, + "text": "And I'm sure you all know just how" + }, + { + "start": 2606.36, + "duration": 0.0, + "text": "And I'm sure you all know just how powerful<00:43:26.760> this<00:43:26.960> rule<00:43:27.160> is." + }, + { + "start": 2608.43, + "duration": 0.0, + "text": "powerful this rule is." + }, + { + "start": 2608.44, + "duration": 0.0, + "text": "powerful this rule is. Um," + }, + { + "start": 2609.79, + "duration": 0.0, + "text": "Um," + }, + { + "start": 2609.8, + "duration": 0.0, + "text": "Um, one<00:43:30.160> last<00:43:30.320> thing<00:43:30.480> that<00:43:30.600> we<00:43:30.680> need<00:43:30.800> to<00:43:30.920> show<00:43:31.480> is" + }, + { + "start": 2611.99, + "duration": 0.0, + "text": "one last thing that we need to show is" + }, + { + "start": 2612.0, + "duration": 0.0, + "text": "one last thing that we need to show is what<00:43:32.200> happens<00:43:32.680> when<00:43:33.480> we<00:43:33.640> choose<00:43:34.080> a<00:43:34.160> context," + }, + { + "start": 2615.23, + "duration": 0.0, + "text": "what happens when we choose a context," + }, + { + "start": 2615.24, + "duration": 0.0, + "text": "what happens when we choose a context, some<00:43:35.480> an<00:43:35.640> element<00:43:35.960> T<00:43:36.080> that<00:43:36.240> is<00:43:36.360> not<00:43:36.640> the<00:43:36.760> top" + }, + { + "start": 2617.03, + "duration": 0.0, + "text": "some an element T that is not the top" + }, + { + "start": 2617.04, + "duration": 0.0, + "text": "some an element T that is not the top element<00:43:37.440> of<00:43:37.560> its<00:43:37.720> lattice.<00:43:38.120> What<00:43:38.440> happens" + }, + { + "start": 2618.75, + "duration": 0.0, + "text": "element of its lattice. What happens" + }, + { + "start": 2618.76, + "duration": 0.0, + "text": "element of its lattice. What happens when<00:43:38.880> we<00:43:38.920> somehow<00:43:39.200> choose<00:43:39.440> something<00:43:39.760> else?" + }, + { + "start": 2620.59, + "duration": 0.0, + "text": "when we somehow choose something else?" + }, + { + "start": 2620.6, + "duration": 0.0, + "text": "when we somehow choose something else? Cuz<00:43:40.840> I<00:43:40.960> just<00:43:41.400> uh,<00:43:41.560> kept<00:43:41.880> saying<00:43:42.120> the<00:43:42.240> whole" + }, + { + "start": 2622.39, + "duration": 0.0, + "text": "Cuz I just uh, kept saying the whole" + }, + { + "start": 2622.4, + "duration": 0.0, + "text": "Cuz I just uh, kept saying the whole time,<00:43:42.680> \"Well,<00:43:43.000> well,<00:43:43.200> it<00:43:43.320> doesn't<00:43:43.600> really" + }, + { + "start": 2623.79, + "duration": 0.0, + "text": "time, \"Well, well, it doesn't really" + }, + { + "start": 2623.8, + "duration": 0.0, + "text": "time, \"Well, well, it doesn't really matter<00:43:44.320> what<00:43:44.480> it<00:43:44.640> is,<00:43:44.960> you<00:43:45.040> know," + }, + { + "start": 2626.03, + "duration": 0.0, + "text": "matter what it is, you know," + }, + { + "start": 2626.04, + "duration": 0.0, + "text": "matter what it is, you know, you<00:43:46.120> can<00:43:46.240> just<00:43:46.400> use<00:43:46.520> whatever.\"<00:43:47.320> But<00:43:47.480> how<00:43:47.640> does" + }, + { + "start": 2627.79, + "duration": 0.0, + "text": "you can just use whatever.\" But how does" + }, + { + "start": 2627.8, + "duration": 0.0, + "text": "you can just use whatever.\" But how does our<00:43:47.880> valuation<00:43:48.320> change?<00:43:49.160> Um," + }, + { + "start": 2631.68, + "duration": 0.0, + "text": "we<00:43:51.840> can<00:43:52.000> just,<00:43:52.480> uh," + }, + { + "start": 2633.11, + "duration": 0.0, + "text": "we can just, uh," + }, + { + "start": 2633.12, + "duration": 0.0, + "text": "we can just, uh, go<00:43:54.120> evaluate<00:43:54.520> the<00:43:54.600> extreme<00:43:55.040> case<00:43:55.920> uh,<00:43:56.400> where" + }, + { + "start": 2636.55, + "duration": 0.0, + "text": "go evaluate the extreme case uh, where" + }, + { + "start": 2636.56, + "duration": 0.0, + "text": "go evaluate the extreme case uh, where we<00:43:56.680> have<00:43:56.840> this<00:43:57.120> this<00:43:57.280> is<00:43:57.360> obviously<00:43:57.880> the<00:43:58.280> valid" + }, + { + "start": 2638.59, + "duration": 0.0, + "text": "we have this this is obviously the valid" + }, + { + "start": 2638.6, + "duration": 0.0, + "text": "we have this this is obviously the valid form<00:43:58.760> that<00:43:58.880> we<00:43:58.960> explored<00:43:59.320> before.<00:44:00.160> But<00:44:00.320> what" + }, + { + "start": 2640.59, + "duration": 0.0, + "text": "form that we explored before. But what" + }, + { + "start": 2640.6, + "duration": 0.0, + "text": "form that we explored before. But what if<00:44:00.880> we<00:44:01.080> have<00:44:01.240> the<00:44:01.360> probability<00:44:02.000> of<00:44:02.240> the" + }, + { + "start": 2643.51, + "duration": 0.0, + "text": "if we have the probability of the" + }, + { + "start": 2643.52, + "duration": 0.0, + "text": "if we have the probability of the sort<00:44:03.640> of<00:44:03.720> the<00:44:03.800> context<00:44:04.480> being<00:44:04.720> itself?<00:44:05.800> The" + }, + { + "start": 2645.99, + "duration": 0.0, + "text": "sort of the context being itself? The" + }, + { + "start": 2646.0, + "duration": 0.0, + "text": "sort of the context being itself? The the<00:44:06.360> given<00:44:06.600> itself<00:44:07.240> and<00:44:07.880> obviously<00:44:08.280> the" + }, + { + "start": 2648.39, + "duration": 0.0, + "text": "the given itself and obviously the" + }, + { + "start": 2648.4, + "duration": 0.0, + "text": "the given itself and obviously the product<00:44:08.720> rule<00:44:08.840> must<00:44:09.040> still<00:44:09.240> apply<00:44:10.000> because" + }, + { + "start": 2650.23, + "duration": 0.0, + "text": "product rule must still apply because" + }, + { + "start": 2650.24, + "duration": 0.0, + "text": "product rule must still apply because it's<00:44:10.360> true<00:44:10.520> for<00:44:10.680> any<00:44:11.320> any<00:44:11.480> kind<00:44:11.680> of<00:44:11.800> range<00:44:12.680> or" + }, + { + "start": 2652.75, + "duration": 0.0, + "text": "it's true for any any kind of range or" + }, + { + "start": 2652.76, + "duration": 0.0, + "text": "it's true for any any kind of range or any<00:44:13.480> section<00:44:13.840> of<00:44:13.920> the<00:44:14.000> lattice." + }, + { + "start": 2655.15, + "duration": 0.0, + "text": "any section of the lattice." + }, + { + "start": 2655.16, + "duration": 0.0, + "text": "any section of the lattice. And<00:44:15.520> this<00:44:15.680> means<00:44:16.000> that<00:44:16.280> the<00:44:16.400> probability<00:44:17.040> of" + }, + { + "start": 2657.19, + "duration": 0.0, + "text": "And this means that the probability of" + }, + { + "start": 2657.2, + "duration": 0.0, + "text": "And this means that the probability of the<00:44:17.280> context<00:44:17.640> given<00:44:17.840> itself<00:44:18.200> must<00:44:18.480> be<00:44:18.640> equal" + }, + { + "start": 2658.87, + "duration": 0.0, + "text": "the context given itself must be equal" + }, + { + "start": 2658.88, + "duration": 0.0, + "text": "the context given itself must be equal to<00:44:19.000> itself<00:44:19.320> squared,<00:44:19.720> so<00:44:19.920> it<00:44:20.080> must<00:44:20.280> be<00:44:20.400> equal" + }, + { + "start": 2660.59, + "duration": 0.0, + "text": "to itself squared, so it must be equal" + }, + { + "start": 2660.6, + "duration": 0.0, + "text": "to itself squared, so it must be equal to<00:44:20.760> one." + }, + { + "start": 2661.75, + "duration": 0.0, + "text": "to one." + }, + { + "start": 2661.76, + "duration": 0.0, + "text": "to one. And<00:44:21.880> this<00:44:22.080> means<00:44:22.400> that<00:44:22.600> very<00:44:22.840> conveniently" + }, + { + "start": 2663.55, + "duration": 0.0, + "text": "And this means that very conveniently" + }, + { + "start": 2663.56, + "duration": 0.0, + "text": "And this means that very conveniently our<00:44:23.720> measure<00:44:24.080> is<00:44:24.360> just<00:44:24.720> always<00:44:25.080> normalized." + }, + { + "start": 2665.59, + "duration": 0.0, + "text": "our measure is just always normalized." + }, + { + "start": 2665.6, + "duration": 0.0, + "text": "our measure is just always normalized. It<00:44:25.960> uh" + }, + { + "start": 2667.27, + "duration": 0.0, + "text": "It uh" + }, + { + "start": 2667.28, + "duration": 0.0, + "text": "It uh it<00:44:27.480> always<00:44:27.760> sums<00:44:28.040> up<00:44:28.160> to<00:44:28.280> one." + }, + { + "start": 2671.88, + "duration": 0.0, + "text": "And<00:44:32.240> here<00:44:32.440> we<00:44:32.520> have<00:44:32.720> Bayes'<00:44:32.960> theorem<00:44:33.320> in<00:44:33.400> its" + }, + { + "start": 2673.55, + "duration": 0.0, + "text": "And here we have Bayes' theorem in its" + }, + { + "start": 2673.56, + "duration": 0.0, + "text": "And here we have Bayes' theorem in its full<00:44:33.760> power<00:44:34.640> um" + }, + { + "start": 2675.31, + "duration": 0.0, + "text": "full power um" + }, + { + "start": 2675.32, + "duration": 0.0, + "text": "full power um because<00:44:35.880> it<00:44:36.280> allows<00:44:36.720> us<00:44:36.920> to<00:44:37.040> do<00:44:37.240> something" + }, + { + "start": 2677.55, + "duration": 0.0, + "text": "because it allows us to do something" + }, + { + "start": 2677.56, + "duration": 0.0, + "text": "because it allows us to do something that<00:44:37.720> a<00:44:37.800> frequentist<00:44:38.320> methodology<00:44:38.800> just<00:44:39.120> not" + }, + { + "start": 2679.39, + "duration": 0.0, + "text": "that a frequentist methodology just not" + }, + { + "start": 2679.4, + "duration": 0.0, + "text": "that a frequentist methodology just not allow<00:44:39.640> us<00:44:39.760> to<00:44:39.840> do,<00:44:40.040> meaning<00:44:40.840> we<00:44:41.000> can<00:44:41.640> update" + }, + { + "start": 2682.11, + "duration": 0.0, + "text": "allow us to do, meaning we can update" + }, + { + "start": 2682.12, + "duration": 0.0, + "text": "allow us to do, meaning we can update the<00:44:42.240> probability<00:44:42.800> of<00:44:42.920> a<00:44:43.000> parameter" + }, + { + "start": 2683.83, + "duration": 0.0, + "text": "the probability of a parameter" + }, + { + "start": 2683.84, + "duration": 0.0, + "text": "the probability of a parameter rigorously." + }, + { + "start": 2685.11, + "duration": 0.0, + "text": "rigorously." + }, + { + "start": 2685.12, + "duration": 0.0, + "text": "rigorously. So,<00:44:45.200> imagine<00:44:45.600> you<00:44:45.680> have<00:44:45.800> some<00:44:46.000> context<00:44:46.760> for" + }, + { + "start": 2686.95, + "duration": 0.0, + "text": "So, imagine you have some context for" + }, + { + "start": 2686.96, + "duration": 0.0, + "text": "So, imagine you have some context for your<00:44:47.120> investigation,<00:44:47.840> your<00:44:47.960> experiment," + }, + { + "start": 2688.55, + "duration": 0.0, + "text": "your investigation, your experiment," + }, + { + "start": 2688.56, + "duration": 0.0, + "text": "your investigation, your experiment, whatever<00:44:48.880> it<00:44:48.960> is,<00:44:49.200> and<00:44:49.320> we<00:44:49.400> call<00:44:49.560> it<00:44:49.720> T." + }, + { + "start": 2690.71, + "duration": 0.0, + "text": "whatever it is, and we call it T." + }, + { + "start": 2690.72, + "duration": 0.0, + "text": "whatever it is, and we call it T. And<00:44:50.800> you<00:44:50.880> want<00:44:51.080> to<00:44:51.160> learn<00:44:51.320> the<00:44:51.440> parameter" + }, + { + "start": 2691.95, + "duration": 0.0, + "text": "And you want to learn the parameter" + }, + { + "start": 2691.96, + "duration": 0.0, + "text": "And you want to learn the parameter theta<00:44:52.960> from<00:44:53.200> data<00:44:53.880> D." + }, + { + "start": 2694.91, + "duration": 0.0, + "text": "theta from data D." + }, + { + "start": 2694.92, + "duration": 0.0, + "text": "theta from data D. Then<00:44:55.160> by<00:44:55.280> simply<00:44:55.560> applying<00:44:55.880> the<00:44:56.000> product<00:44:56.320> and" + }, + { + "start": 2696.43, + "duration": 0.0, + "text": "Then by simply applying the product and" + }, + { + "start": 2696.44, + "duration": 0.0, + "text": "Then by simply applying the product and the<00:44:56.520> sum<00:44:56.800> rules" + }, + { + "start": 2697.91, + "duration": 0.0, + "text": "the sum rules" + }, + { + "start": 2697.92, + "duration": 0.0, + "text": "the sum rules uh" + }, + { + "start": 2698.47, + "duration": 0.0, + "text": "uh" + }, + { + "start": 2698.48, + "duration": 0.0, + "text": "uh you<00:44:58.800> obtain<00:44:59.160> Bayes'<00:45:00.000> uh<00:45:00.080> theorem<00:45:00.560> and<00:45:00.720> you<00:45:00.800> can" + }, + { + "start": 2700.99, + "duration": 0.0, + "text": "you obtain Bayes' uh theorem and you can" + }, + { + "start": 2701.0, + "duration": 0.0, + "text": "you obtain Bayes' uh theorem and you can simply<00:45:01.600> write<00:45:01.800> it<00:45:01.880> down<00:45:02.040> as<00:45:02.160> this." + }, + { + "start": 2703.07, + "duration": 0.0, + "text": "simply write it down as this." + }, + { + "start": 2703.08, + "duration": 0.0, + "text": "simply write it down as this. Uh<00:45:03.160> you<00:45:03.320> have<00:45:03.600> the<00:45:03.680> probability<00:45:04.520> which<00:45:04.680> is" + }, + { + "start": 2704.79, + "duration": 0.0, + "text": "Uh you have the probability which is" + }, + { + "start": 2704.8, + "duration": 0.0, + "text": "Uh you have the probability which is called<00:45:05.040> the<00:45:05.120> posterior," + }, + { + "start": 2706.67, + "duration": 0.0, + "text": "called the posterior," + }, + { + "start": 2706.68, + "duration": 0.0, + "text": "called the posterior, the<00:45:06.720> probability<00:45:07.400> of<00:45:07.600> the<00:45:07.680> parameter<00:45:08.200> given" + }, + { + "start": 2708.43, + "duration": 0.0, + "text": "the probability of the parameter given" + }, + { + "start": 2708.44, + "duration": 0.0, + "text": "the probability of the parameter given the<00:45:08.520> data<00:45:08.760> and<00:45:08.880> the<00:45:09.000> context,<00:45:09.720> which<00:45:09.920> is<00:45:10.080> equal" + }, + { + "start": 2710.35, + "duration": 0.0, + "text": "the data and the context, which is equal" + }, + { + "start": 2710.36, + "duration": 0.0, + "text": "the data and the context, which is equal to<00:45:10.720> on<00:45:10.840> the<00:45:10.920> numerator<00:45:11.440> you<00:45:11.560> have<00:45:12.400> the<00:45:12.560> prior" + }, + { + "start": 2712.95, + "duration": 0.0, + "text": "to on the numerator you have the prior" + }, + { + "start": 2712.96, + "duration": 0.0, + "text": "to on the numerator you have the prior probability<00:45:13.760> of<00:45:13.920> the<00:45:14.040> parameter." + }, + { + "start": 2716.31, + "duration": 0.0, + "text": "probability of the parameter." + }, + { + "start": 2716.32, + "duration": 0.0, + "text": "probability of the parameter. Why<00:45:16.440> can't<00:45:16.720> I<00:45:16.760> highlight<00:45:17.080> anymore?" + }, + { + "start": 2718.43, + "duration": 0.0, + "text": "Why can't I highlight anymore?" + }, + { + "start": 2718.44, + "duration": 0.0, + "text": "Why can't I highlight anymore? Probability<00:45:18.680> of<00:45:18.760> the<00:45:18.840> parameter<00:45:19.320> given<00:45:19.600> the" + }, + { + "start": 2719.67, + "duration": 0.0, + "text": "Probability of the parameter given the" + }, + { + "start": 2719.68, + "duration": 0.0, + "text": "Probability of the parameter given the context<00:45:20.640> times<00:45:21.080> the<00:45:21.160> likelihood,<00:45:21.640> which<00:45:21.760> is" + }, + { + "start": 2721.83, + "duration": 0.0, + "text": "context times the likelihood, which is" + }, + { + "start": 2721.84, + "duration": 0.0, + "text": "context times the likelihood, which is the<00:45:21.920> probability<00:45:22.480> of<00:45:22.600> the<00:45:22.680> data<00:45:23.160> given<00:45:23.560> your" + }, + { + "start": 2724.87, + "duration": 0.0, + "text": "the probability of the data given your" + }, + { + "start": 2724.88, + "duration": 0.0, + "text": "the probability of the data given your uh<00:45:24.960> parameter." + }, + { + "start": 2726.27, + "duration": 0.0, + "text": "uh parameter." + }, + { + "start": 2726.28, + "duration": 0.0, + "text": "uh parameter. And<00:45:26.400> on<00:45:26.480> the<00:45:26.560> bottom<00:45:26.840> you<00:45:26.920> have<00:45:27.040> some<00:45:27.160> kind<00:45:27.360> of" + }, + { + "start": 2727.83, + "duration": 0.0, + "text": "And on the bottom you have some kind of" + }, + { + "start": 2727.84, + "duration": 0.0, + "text": "And on the bottom you have some kind of normalizing<00:45:28.440> constant<00:45:29.360> which<00:45:29.680> makes<00:45:29.920> it<00:45:30.040> so" + }, + { + "start": 2730.31, + "duration": 0.0, + "text": "normalizing constant which makes it so" + }, + { + "start": 2730.32, + "duration": 0.0, + "text": "normalizing constant which makes it so that<00:45:30.880> this<00:45:31.160> all<00:45:31.520> sums<00:45:31.840> up<00:45:31.960> to<00:45:32.080> one<00:45:32.760> and<00:45:32.920> it<00:45:33.080> is" + }, + { + "start": 2733.15, + "duration": 0.0, + "text": "that this all sums up to one and it is" + }, + { + "start": 2733.16, + "duration": 0.0, + "text": "that this all sums up to one and it is simply<00:45:33.480> obtained<00:45:33.960> by<00:45:34.600> um" + }, + { + "start": 2736.19, + "duration": 0.0, + "text": "simply obtained by um" + }, + { + "start": 2736.2, + "duration": 0.0, + "text": "simply obtained by um summing<00:45:36.520> over<00:45:36.680> all<00:45:36.880> possible<00:45:37.240> values<00:45:38.000> of" + }, + { + "start": 2738.15, + "duration": 0.0, + "text": "summing over all possible values of" + }, + { + "start": 2738.16, + "duration": 0.0, + "text": "summing over all possible values of theta" + }, + { + "start": 2739.11, + "duration": 0.0, + "text": "theta" + }, + { + "start": 2739.12, + "duration": 0.0, + "text": "theta and<00:45:39.240> then<00:45:39.440> having<00:45:39.720> all<00:45:39.880> possible<00:45:40.440> for<00:45:40.600> all" + }, + { + "start": 2740.71, + "duration": 0.0, + "text": "and then having all possible for all" + }, + { + "start": 2740.72, + "duration": 0.0, + "text": "and then having all possible for all possible<00:45:40.960> values<00:45:41.360> summing<00:45:41.760> over<00:45:41.960> this" + }, + { + "start": 2742.31, + "duration": 0.0, + "text": "possible values summing over this" + }, + { + "start": 2742.32, + "duration": 0.0, + "text": "possible values summing over this expression<00:45:42.920> in<00:45:43.040> the<00:45:43.120> numerator." + }, + { + "start": 2746.92, + "duration": 0.0, + "text": "And<00:45:47.440> again,<00:45:47.720> usually<00:45:48.080> for<00:45:48.240> gravity" + }, + { + "start": 2749.63, + "duration": 0.0, + "text": "And again, usually for gravity" + }, + { + "start": 2749.64, + "duration": 0.0, + "text": "And again, usually for gravity the<00:45:50.280> context<00:45:50.800> T<00:45:51.160> because<00:45:51.480> it's<00:45:51.680> common<00:45:52.000> in<00:45:52.120> our" + }, + { + "start": 2752.27, + "duration": 0.0, + "text": "the context T because it's common in our" + }, + { + "start": 2752.28, + "duration": 0.0, + "text": "the context T because it's common in our expression<00:45:52.760> is<00:45:52.920> omitted.<00:45:53.760> However,<00:45:54.080> remember" + }, + { + "start": 2754.39, + "duration": 0.0, + "text": "expression is omitted. However, remember" + }, + { + "start": 2754.4, + "duration": 0.0, + "text": "expression is omitted. However, remember that<00:45:54.600> in<00:45:54.800> this<00:45:55.160> view<00:45:55.680> of<00:45:56.160> probability" + }, + { + "start": 2757.51, + "duration": 0.0, + "text": "that in this view of probability" + }, + { + "start": 2757.52, + "duration": 0.0, + "text": "that in this view of probability we<00:45:57.760> all<00:45:58.080> it<00:45:58.160> is<00:45:58.360> always<00:45:59.400> a<00:45:59.480> measure<00:45:59.960> of<00:46:00.120> a" + }, + { + "start": 2760.15, + "duration": 0.0, + "text": "we all it is always a measure of a" + }, + { + "start": 2760.16, + "duration": 0.0, + "text": "we all it is always a measure of a degree<00:46:00.480> of<00:46:00.600> implication,<00:46:01.320> meaning<00:46:01.680> it's" + }, + { + "start": 2761.95, + "duration": 0.0, + "text": "degree of implication, meaning it's" + }, + { + "start": 2761.96, + "duration": 0.0, + "text": "degree of implication, meaning it's always<00:46:02.480> by<00:46:02.640> necessity<00:46:03.200> a<00:46:03.280> B<00:46:03.480> valuation." + }, + { + "start": 2765.99, + "duration": 0.0, + "text": "always by necessity a B valuation." + }, + { + "start": 2766.0, + "duration": 0.0, + "text": "always by necessity a B valuation. There's<00:46:06.120> no<00:46:06.240> such<00:46:06.400> thing<00:46:06.560> as<00:46:06.720> a<00:46:06.840> properly" + }, + { + "start": 2768.07, + "duration": 0.0, + "text": "There's no such thing as a properly" + }, + { + "start": 2768.08, + "duration": 0.0, + "text": "There's no such thing as a properly truly<00:46:08.400> in<00:46:08.520> conditional<00:46:08.920> probability." + }, + { + "start": 2769.51, + "duration": 0.0, + "text": "truly in conditional probability." + }, + { + "start": 2769.52, + "duration": 0.0, + "text": "truly in conditional probability. Somehow<00:46:09.880> floats<00:46:10.320> on<00:46:10.400> its<00:46:10.560> own<00:46:10.920> in<00:46:11.080> the" + }, + { + "start": 2772.23, + "duration": 0.0, + "text": "Somehow floats on its own in the" + }, + { + "start": 2772.24, + "duration": 0.0, + "text": "Somehow floats on its own in the no<00:46:12.360> kind<00:46:12.480> of<00:46:12.560> platforming<00:46:12.960> world.<00:46:13.280> That's<00:46:13.440> not" + }, + { + "start": 2773.59, + "duration": 0.0, + "text": "no kind of platforming world. That's not" + }, + { + "start": 2773.6, + "duration": 0.0, + "text": "no kind of platforming world. That's not a<00:46:13.640> thing." + }, + { + "start": 2775.47, + "duration": 0.0, + "text": "a thing." + }, + { + "start": 2775.48, + "duration": 0.0, + "text": "a thing. Um<00:46:16.280> then<00:46:16.680> we<00:46:16.840> of<00:46:17.000> course<00:46:17.320> need<00:46:17.480> to<00:46:17.560> spend<00:46:17.880> a" + }, + { + "start": 2777.95, + "duration": 0.0, + "text": "Um then we of course need to spend a" + }, + { + "start": 2777.96, + "duration": 0.0, + "text": "Um then we of course need to spend a little<00:46:18.200> bit<00:46:18.840> we<00:46:18.920> need<00:46:19.080> to<00:46:19.160> say<00:46:19.360> a<00:46:19.440> word<00:46:19.920> on" + }, + { + "start": 2780.83, + "duration": 0.0, + "text": "little bit we need to say a word on" + }, + { + "start": 2780.84, + "duration": 0.0, + "text": "little bit we need to say a word on prior<00:46:21.080> probabilities<00:46:21.600> because<00:46:21.840> that<00:46:22.000> is<00:46:22.120> a" + }, + { + "start": 2782.19, + "duration": 0.0, + "text": "prior probabilities because that is a" + }, + { + "start": 2782.2, + "duration": 0.0, + "text": "prior probabilities because that is a major" + }, + { + "start": 2783.59, + "duration": 0.0, + "text": "major" + }, + { + "start": 2783.6, + "duration": 0.0, + "text": "major uh<00:46:23.760> point<00:46:24.000> of<00:46:24.080> contention<00:46:25.120> and<00:46:25.240> is<00:46:25.400> also<00:46:25.680> why" + }, + { + "start": 2785.99, + "duration": 0.0, + "text": "uh point of contention and is also why" + }, + { + "start": 2786.0, + "duration": 0.0, + "text": "uh point of contention and is also why the<00:46:26.160> Bayesian<00:46:26.760> approach<00:46:27.240> gets<00:46:27.480> such<00:46:27.680> a<00:46:27.720> bad" + }, + { + "start": 2787.95, + "duration": 0.0, + "text": "the Bayesian approach gets such a bad" + }, + { + "start": 2787.96, + "duration": 0.0, + "text": "the Bayesian approach gets such a bad reputation" + }, + { + "start": 2789.35, + "duration": 0.0, + "text": "reputation" + }, + { + "start": 2789.36, + "duration": 0.0, + "text": "reputation uh<00:46:29.440> because" + }, + { + "start": 2790.59, + "duration": 0.0, + "text": "uh because" + }, + { + "start": 2790.6, + "duration": 0.0, + "text": "uh because where<00:46:30.840> do<00:46:30.920> you<00:46:31.000> get<00:46:31.160> those<00:46:31.320> prior" + }, + { + "start": 2791.59, + "duration": 0.0, + "text": "where do you get those prior" + }, + { + "start": 2791.6, + "duration": 0.0, + "text": "where do you get those prior probabilities?<00:46:32.160> I<00:46:32.200> mean<00:46:32.400> you<00:46:32.480> don't<00:46:32.600> you" + }, + { + "start": 2792.67, + "duration": 0.0, + "text": "probabilities? I mean you don't you" + }, + { + "start": 2792.68, + "duration": 0.0, + "text": "probabilities? I mean you don't you haven't<00:46:32.920> done<00:46:33.040> the<00:46:33.120> experiment<00:46:33.640> yet<00:46:33.800> by" + }, + { + "start": 2793.91, + "duration": 0.0, + "text": "haven't done the experiment yet by" + }, + { + "start": 2793.92, + "duration": 0.0, + "text": "haven't done the experiment yet by definition.<00:46:34.720> So,<00:46:35.040> what<00:46:35.200> the<00:46:35.280> hell<00:46:35.480> do<00:46:35.640> you" + }, + { + "start": 2795.71, + "duration": 0.0, + "text": "definition. So, what the hell do you" + }, + { + "start": 2795.72, + "duration": 0.0, + "text": "definition. So, what the hell do you know?<00:46:36.440> Um" + }, + { + "start": 2797.35, + "duration": 0.0, + "text": "know? Um" + }, + { + "start": 2797.36, + "duration": 0.0, + "text": "know? Um and" + }, + { + "start": 2798.71, + "duration": 0.0, + "text": "and" + }, + { + "start": 2798.72, + "duration": 0.0, + "text": "and unfortunately,<00:46:39.480> the<00:46:39.600> language<00:46:39.960> has<00:46:40.080> caught" + }, + { + "start": 2800.39, + "duration": 0.0, + "text": "unfortunately, the language has caught" + }, + { + "start": 2800.4, + "duration": 0.0, + "text": "unfortunately, the language has caught on<00:46:40.680> that<00:46:41.360> somehow<00:46:42.400> the<00:46:42.520> probability<00:46:43.120> the" + }, + { + "start": 2803.19, + "duration": 0.0, + "text": "on that somehow the probability the" + }, + { + "start": 2803.2, + "duration": 0.0, + "text": "on that somehow the probability the prior<00:46:43.440> probability<00:46:44.200> represents<00:46:44.760> the<00:46:44.880> belief" + }, + { + "start": 2805.67, + "duration": 0.0, + "text": "prior probability represents the belief" + }, + { + "start": 2805.68, + "duration": 0.0, + "text": "prior probability represents the belief of<00:46:45.840> the<00:46:45.960> researcher<00:46:46.520> and<00:46:46.640> this<00:46:46.760> makes<00:46:46.960> it" + }, + { + "start": 2807.19, + "duration": 0.0, + "text": "of the researcher and this makes it" + }, + { + "start": 2807.2, + "duration": 0.0, + "text": "of the researcher and this makes it sound<00:46:47.520> like<00:46:47.680> you<00:46:47.800> can<00:46:47.920> just<00:46:48.480> somehow<00:46:49.240> listen" + }, + { + "start": 2809.59, + "duration": 0.0, + "text": "sound like you can just somehow listen" + }, + { + "start": 2809.6, + "duration": 0.0, + "text": "sound like you can just somehow listen into<00:46:49.800> your<00:46:49.960> gut<00:46:50.280> and<00:46:50.480> and<00:46:50.600> choose<00:46:50.840> the<00:46:50.920> numbers" + }, + { + "start": 2811.23, + "duration": 0.0, + "text": "into your gut and and choose the numbers" + }, + { + "start": 2811.24, + "duration": 0.0, + "text": "into your gut and and choose the numbers yourself.<00:46:51.960> Uh<00:46:52.000> and<00:46:52.120> in<00:46:52.200> fact,<00:46:52.480> there<00:46:52.640> is<00:46:53.400> a" + }, + { + "start": 2813.51, + "duration": 0.0, + "text": "yourself. Uh and in fact, there is a" + }, + { + "start": 2813.52, + "duration": 0.0, + "text": "yourself. Uh and in fact, there is a school<00:46:53.920> of<00:46:54.040> probability<00:46:54.480> which<00:46:54.800> does" + }, + { + "start": 2815.19, + "duration": 0.0, + "text": "school of probability which does" + }, + { + "start": 2815.2, + "duration": 0.0, + "text": "school of probability which does something<00:46:55.640> like<00:46:55.880> this,<00:46:56.640> the<00:46:56.720> school<00:46:57.000> of<00:46:57.120> the" + }, + { + "start": 2817.19, + "duration": 0.0, + "text": "something like this, the school of the" + }, + { + "start": 2817.2, + "duration": 0.0, + "text": "something like this, the school of the de<00:46:57.320> Finetti.<00:46:58.240> But<00:46:58.960> uh<00:46:59.040> I<00:46:59.240> and<00:46:59.400> many<00:46:59.680> others," + }, + { + "start": 2820.07, + "duration": 0.0, + "text": "de Finetti. But uh I and many others," + }, + { + "start": 2820.08, + "duration": 0.0, + "text": "de Finetti. But uh I and many others, including<00:47:00.800> Edwin<00:47:01.080> Thompson<00:47:01.360> James<00:47:01.760> famously," + }, + { + "start": 2822.59, + "duration": 0.0, + "text": "including Edwin Thompson James famously," + }, + { + "start": 2822.6, + "duration": 0.0, + "text": "including Edwin Thompson James famously, uh<00:47:02.960> reject<00:47:03.360> this<00:47:03.520> and<00:47:03.640> find<00:47:03.840> it<00:47:03.920> totally" + }, + { + "start": 2824.19, + "duration": 0.0, + "text": "uh reject this and find it totally" + }, + { + "start": 2824.2, + "duration": 0.0, + "text": "uh reject this and find it totally absurd." + }, + { + "start": 2825.31, + "duration": 0.0, + "text": "absurd." + }, + { + "start": 2825.32, + "duration": 0.0, + "text": "absurd. And<00:47:05.720> uh<00:47:06.200> it<00:47:06.320> has<00:47:06.440> to<00:47:06.520> be<00:47:06.640> pointed<00:47:07.000> out<00:47:07.160> that" + }, + { + "start": 2827.79, + "duration": 0.0, + "text": "And uh it has to be pointed out that" + }, + { + "start": 2827.8, + "duration": 0.0, + "text": "And uh it has to be pointed out that while<00:47:08.080> the<00:47:08.160> word<00:47:08.360> subjective<00:47:09.040> is<00:47:09.240> usually<00:47:10.080> uh" + }, + { + "start": 2830.27, + "duration": 0.0, + "text": "while the word subjective is usually uh" + }, + { + "start": 2830.28, + "duration": 0.0, + "text": "while the word subjective is usually uh used<00:47:10.640> for<00:47:10.760> something<00:47:11.120> that<00:47:11.840> it<00:47:11.960> is<00:47:12.120> not<00:47:12.360> valid," + }, + { + "start": 2833.15, + "duration": 0.0, + "text": "used for something that it is not valid," + }, + { + "start": 2833.16, + "duration": 0.0, + "text": "used for something that it is not valid, actually<00:47:13.840> subjective<00:47:14.440> is<00:47:14.600> not<00:47:14.800> quite<00:47:15.080> the" + }, + { + "start": 2835.15, + "duration": 0.0, + "text": "actually subjective is not quite the" + }, + { + "start": 2835.16, + "duration": 0.0, + "text": "actually subjective is not quite the same<00:47:15.440> thing<00:47:15.680> as<00:47:16.000> arbitrary." + }, + { + "start": 2837.35, + "duration": 0.0, + "text": "same thing as arbitrary." + }, + { + "start": 2837.36, + "duration": 0.0, + "text": "same thing as arbitrary. Um" + }, + { + "start": 2838.03, + "duration": 0.0, + "text": "Um" + }, + { + "start": 2838.04, + "duration": 0.0, + "text": "Um because<00:47:18.280> probability<00:47:18.960> is<00:47:19.320> in<00:47:19.480> this<00:47:19.720> sense" + }, + { + "start": 2840.19, + "duration": 0.0, + "text": "because probability is in this sense" + }, + { + "start": 2840.2, + "duration": 0.0, + "text": "because probability is in this sense necessarily<00:47:20.720> subjective<00:47:21.200> because<00:47:21.560> it<00:47:21.760> always" + }, + { + "start": 2842.03, + "duration": 0.0, + "text": "necessarily subjective because it always" + }, + { + "start": 2842.04, + "duration": 0.0, + "text": "necessarily subjective because it always depends<00:47:22.440> on<00:47:22.560> what<00:47:22.760> knowledge<00:47:23.040> that<00:47:23.160> you<00:47:23.240> have." + }, + { + "start": 2843.83, + "duration": 0.0, + "text": "depends on what knowledge that you have." + }, + { + "start": 2843.84, + "duration": 0.0, + "text": "depends on what knowledge that you have. So,<00:47:23.920> people<00:47:24.160> with<00:47:24.360> very<00:47:24.600> different<00:47:24.960> knowledge" + }, + { + "start": 2845.39, + "duration": 0.0, + "text": "So, people with very different knowledge" + }, + { + "start": 2845.4, + "duration": 0.0, + "text": "So, people with very different knowledge might<00:47:25.640> assign" + }, + { + "start": 2846.87, + "duration": 0.0, + "text": "might assign" + }, + { + "start": 2846.88, + "duration": 0.0, + "text": "might assign very<00:47:27.080> different<00:47:27.280> probabilities<00:47:27.920> to<00:47:28.000> the<00:47:28.120> same" + }, + { + "start": 2848.95, + "duration": 0.0, + "text": "very different probabilities to the same" + }, + { + "start": 2848.96, + "duration": 0.0, + "text": "very different probabilities to the same uh<00:47:29.440> to<00:47:29.520> the<00:47:29.600> same<00:47:29.880> notion<00:47:30.240> to<00:47:30.360> the<00:47:30.440> same<00:47:31.080> uh" + }, + { + "start": 2851.15, + "duration": 0.0, + "text": "uh to the same notion to the same uh" + }, + { + "start": 2851.16, + "duration": 0.0, + "text": "uh to the same notion to the same uh proposition<00:47:31.640> to<00:47:31.720> the<00:47:31.800> same<00:47:31.960> event,<00:47:32.840> right?<00:47:33.520> Uh" + }, + { + "start": 2853.59, + "duration": 0.0, + "text": "proposition to the same event, right? Uh" + }, + { + "start": 2853.6, + "duration": 0.0, + "text": "proposition to the same event, right? Uh but<00:47:33.920> what<00:47:34.160> we<00:47:34.400> require<00:47:35.080> of<00:47:35.280> this<00:47:36.040> uh" + }, + { + "start": 2856.11, + "duration": 0.0, + "text": "but what we require of this uh" + }, + { + "start": 2856.12, + "duration": 0.0, + "text": "but what we require of this uh probability<00:47:36.560> calculus<00:47:37.240> is<00:47:37.400> simply<00:47:37.760> that" + }, + { + "start": 2858.67, + "duration": 0.0, + "text": "probability calculus is simply that" + }, + { + "start": 2858.68, + "duration": 0.0, + "text": "probability calculus is simply that two<00:47:38.880> different<00:47:39.400> researchers<00:47:40.080> with<00:47:41.000> uh<00:47:41.080> the" + }, + { + "start": 2861.19, + "duration": 0.0, + "text": "two different researchers with uh the" + }, + { + "start": 2861.2, + "duration": 0.0, + "text": "two different researchers with uh the same<00:47:41.560> information<00:47:42.640> will<00:47:42.760> give<00:47:42.960> the<00:47:43.040> exact" + }, + { + "start": 2863.55, + "duration": 0.0, + "text": "same information will give the exact" + }, + { + "start": 2863.56, + "duration": 0.0, + "text": "same information will give the exact same<00:47:43.840> prior<00:47:44.080> probability<00:47:44.640> and<00:47:44.760> this<00:47:44.920> is<00:47:45.040> what" + }, + { + "start": 2865.15, + "duration": 0.0, + "text": "same prior probability and this is what" + }, + { + "start": 2865.16, + "duration": 0.0, + "text": "same prior probability and this is what is<00:47:45.240> called<00:47:45.560> the<00:47:45.920> objective<00:47:46.880> uh<00:47:47.120> Bayesian" + }, + { + "start": 2867.47, + "duration": 0.0, + "text": "is called the objective uh Bayesian" + }, + { + "start": 2867.48, + "duration": 0.0, + "text": "is called the objective uh Bayesian school." + }, + { + "start": 2871.72, + "duration": 0.0, + "text": "And<00:47:51.960> now<00:47:52.200> we're<00:47:52.320> going<00:47:52.440> to<00:47:52.520> look<00:47:52.680> at<00:47:52.800> some<00:47:52.960> of" + }, + { + "start": 2873.07, + "duration": 0.0, + "text": "And now we're going to look at some of" + }, + { + "start": 2873.08, + "duration": 0.0, + "text": "And now we're going to look at some of the<00:47:53.160> amazing<00:47:53.600> things<00:47:53.880> you<00:47:53.960> can<00:47:54.120> do<00:47:54.360> with<00:47:54.560> this" + }, + { + "start": 2875.43, + "duration": 0.0, + "text": "the amazing things you can do with this" + }, + { + "start": 2875.44, + "duration": 0.0, + "text": "the amazing things you can do with this uh<00:47:55.520> with<00:47:55.640> this<00:47:56.040> cal-<00:47:56.400> of<00:47:56.800> this<00:47:56.960> reasoning" + }, + { + "start": 2877.27, + "duration": 0.0, + "text": "uh with this cal- of this reasoning" + }, + { + "start": 2877.28, + "duration": 0.0, + "text": "uh with this cal- of this reasoning apparatus,<00:47:57.760> this<00:47:58.000> common" + }, + { + "start": 2878.95, + "duration": 0.0, + "text": "apparatus, this common" + }, + { + "start": 2878.96, + "duration": 0.0, + "text": "apparatus, this common this<00:47:59.160> quantified<00:47:59.680> common<00:48:00.000> sense." + }, + { + "start": 2880.95, + "duration": 0.0, + "text": "this quantified common sense." + }, + { + "start": 2880.96, + "duration": 0.0, + "text": "this quantified common sense. Um,<00:48:01.440> if<00:48:01.600> you<00:48:01.680> want<00:48:01.840> to<00:48:02.200> know<00:48:02.440> more<00:48:02.800> as<00:48:02.960> to<00:48:03.120> why" + }, + { + "start": 2883.47, + "duration": 0.0, + "text": "Um, if you want to know more as to why" + }, + { + "start": 2883.48, + "duration": 0.0, + "text": "Um, if you want to know more as to why it's<00:48:03.720> quantified<00:48:04.160> common<00:48:04.400> sense,<00:48:04.760> once" + }, + { + "start": 2884.95, + "duration": 0.0, + "text": "it's quantified common sense, once" + }, + { + "start": 2884.96, + "duration": 0.0, + "text": "it's quantified common sense, once again,<00:48:05.240> I<00:48:05.320> recommend<00:48:05.840> you<00:48:05.920> the<00:48:06.040> book<00:48:06.920> um,<00:48:07.520> by" + }, + { + "start": 2888.11, + "duration": 0.0, + "text": "again, I recommend you the book um, by" + }, + { + "start": 2888.12, + "duration": 0.0, + "text": "again, I recommend you the book um, by Edwin<00:48:08.400> Thompson<00:48:08.680> Jaynes,<00:48:09.040> which<00:48:09.200> is<00:48:09.320> called" + }, + { + "start": 2889.79, + "duration": 0.0, + "text": "Edwin Thompson Jaynes, which is called" + }, + { + "start": 2889.8, + "duration": 0.0, + "text": "Edwin Thompson Jaynes, which is called Probability:<00:48:10.400> The<00:48:10.520> Logic<00:48:10.840> of<00:48:10.960> Science." + }, + { + "start": 2891.63, + "duration": 0.0, + "text": "Probability: The Logic of Science." + }, + { + "start": 2891.64, + "duration": 0.0, + "text": "Probability: The Logic of Science. Amazing<00:48:12.120> book.<00:48:12.360> Just<00:48:12.600> read<00:48:12.720> the<00:48:12.800> first<00:48:13.040> two" + }, + { + "start": 2893.15, + "duration": 0.0, + "text": "Amazing book. Just read the first two" + }, + { + "start": 2893.16, + "duration": 0.0, + "text": "Amazing book. Just read the first two chapters<00:48:13.560> and<00:48:13.760> you<00:48:13.920> immediately<00:48:14.440> see<00:48:14.800> what<00:48:14.920> he" + }, + { + "start": 2895.03, + "duration": 0.0, + "text": "chapters and you immediately see what he" + }, + { + "start": 2895.04, + "duration": 0.0, + "text": "chapters and you immediately see what he means.<00:48:15.360> Very<00:48:15.680> easy<00:48:15.920> and<00:48:16.200> quick<00:48:16.440> to<00:48:16.520> read<00:48:16.720> and" + }, + { + "start": 2896.79, + "duration": 0.0, + "text": "means. Very easy and quick to read and" + }, + { + "start": 2896.8, + "duration": 0.0, + "text": "means. Very easy and quick to read and super<00:48:17.040> interesting." + }, + { + "start": 2899.8, + "duration": 0.0, + "text": "But<00:48:20.000> now,<00:48:20.160> assume<00:48:20.520> that<00:48:20.640> we<00:48:20.760> have<00:48:21.080> again<00:48:21.320> some" + }, + { + "start": 2901.47, + "duration": 0.0, + "text": "But now, assume that we have again some" + }, + { + "start": 2901.48, + "duration": 0.0, + "text": "But now, assume that we have again some context.<00:48:22.000> Oops." + }, + { + "start": 2903.11, + "duration": 0.0, + "text": "context. Oops." + }, + { + "start": 2903.12, + "duration": 0.0, + "text": "context. Oops. A<00:48:23.160> context<00:48:23.720> with<00:48:23.880> a<00:48:23.920> parameter<00:48:24.560> A" + }, + { + "start": 2905.43, + "duration": 0.0, + "text": "A context with a parameter A" + }, + { + "start": 2905.44, + "duration": 0.0, + "text": "A context with a parameter A uh," + }, + { + "start": 2906.15, + "duration": 0.0, + "text": "uh," + }, + { + "start": 2906.16, + "duration": 0.0, + "text": "uh, that<00:48:26.400> we<00:48:26.520> will<00:48:26.760> for<00:48:26.920> now,<00:48:27.200> just<00:48:27.360> for" + }, + { + "start": 2907.47, + "duration": 0.0, + "text": "that we will for now, just for" + }, + { + "start": 2907.48, + "duration": 0.0, + "text": "that we will for now, just for simplicity,<00:48:28.040> assume<00:48:28.360> to<00:48:28.480> be<00:48:28.560> discrete.<00:48:29.320> And" + }, + { + "start": 2909.51, + "duration": 0.0, + "text": "simplicity, assume to be discrete. And" + }, + { + "start": 2909.52, + "duration": 0.0, + "text": "simplicity, assume to be discrete. And this<00:48:29.680> parameter<00:48:30.160> can<00:48:30.320> take<00:48:30.600> values<00:48:31.360> 1,<00:48:31.800> A1," + }, + { + "start": 2912.27, + "duration": 0.0, + "text": "this parameter can take values 1, A1," + }, + { + "start": 2912.28, + "duration": 0.0, + "text": "this parameter can take values 1, A1, A2,<00:48:33.040> up<00:48:33.200> to<00:48:33.360> AN." + }, + { + "start": 2915.83, + "duration": 0.0, + "text": "A2, up to AN." + }, + { + "start": 2915.84, + "duration": 0.0, + "text": "A2, up to AN. Since<00:48:36.040> these<00:48:36.160> values<00:48:36.480> are<00:48:36.560> mutually" + }, + { + "start": 2916.83, + "duration": 0.0, + "text": "Since these values are mutually" + }, + { + "start": 2916.84, + "duration": 0.0, + "text": "Since these values are mutually exclusive<00:48:37.360> and<00:48:37.480> disjoint,<00:48:38.760> um,<00:48:39.400> and" + }, + { + "start": 2919.51, + "duration": 0.0, + "text": "exclusive and disjoint, um, and" + }, + { + "start": 2919.52, + "duration": 0.0, + "text": "exclusive and disjoint, um, and exhaustive,<00:48:40.240> we<00:48:40.440> know<00:48:40.760> that<00:48:41.440> the<00:48:41.840> the" + }, + { + "start": 2921.91, + "duration": 0.0, + "text": "exhaustive, we know that the the" + }, + { + "start": 2921.92, + "duration": 0.0, + "text": "exhaustive, we know that the the combination<00:48:42.480> of<00:48:42.600> all<00:48:42.760> of<00:48:42.880> them<00:48:43.720> uh," + }, + { + "start": 2924.83, + "duration": 0.0, + "text": "combination of all of them uh," + }, + { + "start": 2924.84, + "duration": 0.0, + "text": "combination of all of them uh, must<00:48:45.200> sum<00:48:45.400> up<00:48:45.520> to<00:48:45.640> one.<00:48:45.800> So,<00:48:46.200> the<00:48:46.320> the" + }, + { + "start": 2926.43, + "duration": 0.0, + "text": "must sum up to one. So, the the" + }, + { + "start": 2926.44, + "duration": 0.0, + "text": "must sum up to one. So, the the probability<00:48:46.920> of<00:48:47.040> their<00:48:47.160> combination<00:48:47.720> must" + }, + { + "start": 2927.99, + "duration": 0.0, + "text": "probability of their combination must" + }, + { + "start": 2928.0, + "duration": 0.0, + "text": "probability of their combination must sum<00:48:48.200> must<00:48:48.440> be<00:48:48.560> one.<00:48:49.320> Uh,<00:48:49.760> because<00:48:50.320> otherwise" + }, + { + "start": 2930.79, + "duration": 0.0, + "text": "sum must be one. Uh, because otherwise" + }, + { + "start": 2930.8, + "duration": 0.0, + "text": "sum must be one. Uh, because otherwise there's<00:48:51.080> possibilities<00:48:51.600> that<00:48:51.760> we<00:48:51.880> have<00:48:52.040> not" + }, + { + "start": 2932.19, + "duration": 0.0, + "text": "there's possibilities that we have not" + }, + { + "start": 2932.2, + "duration": 0.0, + "text": "there's possibilities that we have not considered." + }, + { + "start": 2933.43, + "duration": 0.0, + "text": "considered." + }, + { + "start": 2933.44, + "duration": 0.0, + "text": "considered. Uh,<00:48:53.680> and<00:48:53.840> since<00:48:54.160> this<00:48:54.360> notion<00:48:54.800> is<00:48:54.920> already<00:48:55.200> a" + }, + { + "start": 2935.27, + "duration": 0.0, + "text": "Uh, and since this notion is already a" + }, + { + "start": 2935.28, + "duration": 0.0, + "text": "Uh, and since this notion is already a consequence<00:48:55.760> of<00:48:55.840> the<00:48:55.920> context,<00:48:56.560> because<00:48:56.800> the" + }, + { + "start": 2936.83, + "duration": 0.0, + "text": "consequence of the context, because the" + }, + { + "start": 2936.84, + "duration": 0.0, + "text": "consequence of the context, because the context<00:48:57.320> tells<00:48:57.560> us,<00:48:57.680> \"Okay,<00:48:57.920> yes,<00:48:58.200> we<00:48:58.320> are" + }, + { + "start": 2938.39, + "duration": 0.0, + "text": "context tells us, \"Okay, yes, we are" + }, + { + "start": 2938.4, + "duration": 0.0, + "text": "context tells us, \"Okay, yes, we are looking<00:48:58.640> at<00:48:58.720> this<00:48:58.920> with<00:48:59.120> this<00:48:59.280> parameter.\"" + }, + { + "start": 2941.31, + "duration": 0.0, + "text": "looking at this with this parameter.\"" + }, + { + "start": 2941.32, + "duration": 0.0, + "text": "looking at this with this parameter.\" Uh,<00:49:01.600> then<00:49:01.840> conditioning<00:49:02.440> on<00:49:02.560> the<00:49:02.640> data<00:49:02.960> does" + }, + { + "start": 2943.11, + "duration": 0.0, + "text": "Uh, then conditioning on the data does" + }, + { + "start": 2943.12, + "duration": 0.0, + "text": "Uh, then conditioning on the data does not<00:49:03.280> change<00:49:03.560> it.<00:49:03.680> So,<00:49:03.880> also<00:49:04.200> the<00:49:04.480> probability" + }, + { + "start": 2945.15, + "duration": 0.0, + "text": "not change it. So, also the probability" + }, + { + "start": 2945.16, + "duration": 0.0, + "text": "not change it. So, also the probability of" + }, + { + "start": 2946.03, + "duration": 0.0, + "text": "of" + }, + { + "start": 2946.04, + "duration": 0.0, + "text": "of either<00:49:06.320> one<00:49:06.520> of<00:49:06.640> them<00:49:06.880> being<00:49:07.160> true,<00:49:07.520> given<00:49:07.800> the" + }, + { + "start": 2947.87, + "duration": 0.0, + "text": "either one of them being true, given the" + }, + { + "start": 2947.88, + "duration": 0.0, + "text": "either one of them being true, given the data<00:49:08.120> and<00:49:08.240> the<00:49:08.320> context,<00:49:08.760> is<00:49:08.880> still<00:49:09.160> one.<00:49:09.440> That" + }, + { + "start": 2949.55, + "duration": 0.0, + "text": "data and the context, is still one. That" + }, + { + "start": 2949.56, + "duration": 0.0, + "text": "data and the context, is still one. That doesn't<00:49:09.800> change." + }, + { + "start": 2950.99, + "duration": 0.0, + "text": "doesn't change." + }, + { + "start": 2951.0, + "duration": 0.0, + "text": "doesn't change. But<00:49:11.120> now,<00:49:11.600> what" + }, + { + "start": 2952.71, + "duration": 0.0, + "text": "But now, what" + }, + { + "start": 2952.72, + "duration": 0.0, + "text": "But now, what if<00:49:12.920> the<00:49:13.040> model<00:49:13.360> has<00:49:13.560> also<00:49:13.760> another<00:49:14.000> parameter," + }, + { + "start": 2954.39, + "duration": 0.0, + "text": "if the model has also another parameter," + }, + { + "start": 2954.4, + "duration": 0.0, + "text": "if the model has also another parameter, omega," + }, + { + "start": 2955.43, + "duration": 0.0, + "text": "omega," + }, + { + "start": 2955.44, + "duration": 0.0, + "text": "omega, which<00:49:15.720> is<00:49:15.880> the<00:49:15.960> one<00:49:16.080> that<00:49:16.200> we<00:49:16.280> are<00:49:16.320> actually" + }, + { + "start": 2956.59, + "duration": 0.0, + "text": "which is the one that we are actually" + }, + { + "start": 2956.6, + "duration": 0.0, + "text": "which is the one that we are actually interested<00:49:17.040> in," + }, + { + "start": 2957.83, + "duration": 0.0, + "text": "interested in," + }, + { + "start": 2957.84, + "duration": 0.0, + "text": "interested in, uh," + }, + { + "start": 2958.47, + "duration": 0.0, + "text": "uh," + }, + { + "start": 2958.48, + "duration": 0.0, + "text": "uh, that<00:49:18.640> we<00:49:18.720> want<00:49:18.880> to<00:49:19.000> estimate,<00:49:19.400> that<00:49:19.480> we<00:49:19.520> want" + }, + { + "start": 2959.67, + "duration": 0.0, + "text": "that we want to estimate, that we want" + }, + { + "start": 2959.68, + "duration": 0.0, + "text": "that we want to estimate, that we want to<00:49:19.760> learn<00:49:19.920> from<00:49:20.040> the<00:49:20.120> data," + }, + { + "start": 2961.07, + "duration": 0.0, + "text": "to learn from the data," + }, + { + "start": 2961.08, + "duration": 0.0, + "text": "to learn from the data, um,<00:49:21.760> meaning<00:49:22.040> we<00:49:22.120> want<00:49:22.280> to<00:49:22.360> find<00:49:22.520> the" + }, + { + "start": 2962.55, + "duration": 0.0, + "text": "um, meaning we want to find the" + }, + { + "start": 2962.56, + "duration": 0.0, + "text": "um, meaning we want to find the probability<00:49:23.000> of<00:49:23.080> omega<00:49:23.440> given<00:49:23.680> the<00:49:23.800> data,<00:49:24.520> and" + }, + { + "start": 2964.67, + "duration": 0.0, + "text": "probability of omega given the data, and" + }, + { + "start": 2964.68, + "duration": 0.0, + "text": "probability of omega given the data, and not<00:49:25.000> probability" + }, + { + "start": 2966.03, + "duration": 0.0, + "text": "not probability" + }, + { + "start": 2966.04, + "duration": 0.0, + "text": "not probability probability<00:49:26.320> of<00:49:26.480> omega<00:49:26.800> and<00:49:27.080> A<00:49:27.640> given<00:49:27.880> the" + }, + { + "start": 2967.99, + "duration": 0.0, + "text": "probability of omega and A given the" + }, + { + "start": 2968.0, + "duration": 0.0, + "text": "probability of omega and A given the data,<00:49:28.320> where<00:49:28.520> here<00:49:29.120> comma<00:49:29.440> simply<00:49:29.680> means<00:49:30.080> it's" + }, + { + "start": 2970.19, + "duration": 0.0, + "text": "data, where here comma simply means it's" + }, + { + "start": 2970.2, + "duration": 0.0, + "text": "data, where here comma simply means it's just<00:49:30.400> an<00:49:30.520> and<00:49:30.760> operation,<00:49:31.240> right?<00:49:31.800> You<00:49:31.920> want" + }, + { + "start": 2972.07, + "duration": 0.0, + "text": "just an and operation, right? You want" + }, + { + "start": 2972.08, + "duration": 0.0, + "text": "just an and operation, right? You want to<00:49:32.160> know<00:49:32.320> both<00:49:32.600> simultaneously.<00:49:33.760> But<00:49:33.840> in<00:49:33.920> this" + }, + { + "start": 2974.07, + "duration": 0.0, + "text": "to know both simultaneously. But in this" + }, + { + "start": 2974.08, + "duration": 0.0, + "text": "to know both simultaneously. But in this case,<00:49:34.200> we<00:49:34.280> don't<00:49:34.480> want<00:49:34.640> that.<00:49:35.240> So,<00:49:35.360> how<00:49:35.480> do<00:49:35.600> we" + }, + { + "start": 2975.67, + "duration": 0.0, + "text": "case, we don't want that. So, how do we" + }, + { + "start": 2975.68, + "duration": 0.0, + "text": "case, we don't want that. So, how do we get<00:49:35.840> rid<00:49:36.040> of<00:49:36.160> this<00:49:36.400> what<00:49:36.560> is<00:49:36.680> called<00:49:37.040> a" + }, + { + "start": 2977.11, + "duration": 0.0, + "text": "get rid of this what is called a" + }, + { + "start": 2977.12, + "duration": 0.0, + "text": "get rid of this what is called a nuisance<00:49:37.560> parameter,<00:49:38.200> A?" + }, + { + "start": 2979.63, + "duration": 0.0, + "text": "nuisance parameter, A?" + }, + { + "start": 2979.64, + "duration": 0.0, + "text": "nuisance parameter, A? The<00:49:39.720> way<00:49:39.840> we<00:49:39.960> do<00:49:40.160> this<00:49:40.360> is<00:49:40.600> we<00:49:40.720> apply<00:49:41.200> the<00:49:41.360> sum" + }, + { + "start": 2981.55, + "duration": 0.0, + "text": "The way we do this is we apply the sum" + }, + { + "start": 2981.56, + "duration": 0.0, + "text": "The way we do this is we apply the sum and<00:49:41.680> product<00:49:42.000> rules." + }, + { + "start": 2982.95, + "duration": 0.0, + "text": "and product rules." + }, + { + "start": 2982.96, + "duration": 0.0, + "text": "and product rules. Um," + }, + { + "start": 2984.19, + "duration": 0.0, + "text": "Um," + }, + { + "start": 2984.2, + "duration": 0.0, + "text": "Um, we<00:49:44.520> start<00:49:44.840> from<00:49:45.760> uh,<00:49:46.000> this<00:49:46.480> statement<00:49:46.800> that<00:49:46.880> we" + }, + { + "start": 2986.95, + "duration": 0.0, + "text": "we start from uh, this statement that we" + }, + { + "start": 2986.96, + "duration": 0.0, + "text": "we start from uh, this statement that we just<00:49:47.120> saw<00:49:47.240> before,<00:49:47.600> that<00:49:47.760> is<00:49:47.880> still<00:49:48.080> equal<00:49:48.320> to" + }, + { + "start": 2988.47, + "duration": 0.0, + "text": "just saw before, that is still equal to" + }, + { + "start": 2988.48, + "duration": 0.0, + "text": "just saw before, that is still equal to one," + }, + { + "start": 2989.55, + "duration": 0.0, + "text": "one," + }, + { + "start": 2989.56, + "duration": 0.0, + "text": "one, and<00:49:49.800> we" + }, + { + "start": 2990.51, + "duration": 0.0, + "text": "and we" + }, + { + "start": 2990.52, + "duration": 0.0, + "text": "and we uh,<00:49:50.600> we<00:49:50.720> split<00:49:50.920> it<00:49:51.000> up.<00:49:51.640> Uh,<00:49:51.840> we<00:49:52.000> apply<00:49:52.240> the" + }, + { + "start": 2992.35, + "duration": 0.0, + "text": "uh, we split it up. Uh, we apply the" + }, + { + "start": 2992.36, + "duration": 0.0, + "text": "uh, we split it up. Uh, we apply the product<00:49:52.680> rule<00:49:52.960> and<00:49:53.720> basically<00:49:54.240> we<00:49:54.360> have<00:49:55.040> omega" + }, + { + "start": 2995.63, + "duration": 0.0, + "text": "product rule and basically we have omega" + }, + { + "start": 2995.64, + "duration": 0.0, + "text": "product rule and basically we have omega given<00:49:56.240> all<00:49:56.400> of<00:49:56.480> this<00:49:56.600> combination<00:49:57.160> of<00:49:57.240> the" + }, + { + "start": 2997.31, + "duration": 0.0, + "text": "given all of this combination of the" + }, + { + "start": 2997.32, + "duration": 0.0, + "text": "given all of this combination of the parameter<00:49:58.480> and<00:49:58.720> times<00:49:59.080> the<00:49:59.200> probability<00:50:00.040> the" + }, + { + "start": 3000.15, + "duration": 0.0, + "text": "parameter and times the probability the" + }, + { + "start": 3000.16, + "duration": 0.0, + "text": "parameter and times the probability the prior<00:50:00.360> probability<00:50:01.280> of<00:50:02.000> all<00:50:02.160> these<00:50:02.320> possible" + }, + { + "start": 3002.67, + "duration": 0.0, + "text": "prior probability of all these possible" + }, + { + "start": 3002.68, + "duration": 0.0, + "text": "prior probability of all these possible values<00:50:02.920> of<00:50:03.000> the<00:50:03.080> parameter.<00:50:03.960> And<00:50:04.080> since<00:50:04.320> this" + }, + { + "start": 3004.51, + "duration": 0.0, + "text": "values of the parameter. And since this" + }, + { + "start": 3004.52, + "duration": 0.0, + "text": "values of the parameter. And since this is<00:50:04.640> equal<00:50:04.880> to<00:50:05.000> one<00:50:05.360> in<00:50:05.400> the<00:50:05.480> second<00:50:05.800> part" + }, + { + "start": 3007.51, + "duration": 0.0, + "text": "is equal to one in the second part" + }, + { + "start": 3007.52, + "duration": 0.0, + "text": "is equal to one in the second part um" + }, + { + "start": 3009.03, + "duration": 0.0, + "text": "um" + }, + { + "start": 3009.04, + "duration": 0.0, + "text": "um and<00:50:09.440> this<00:50:09.560> parameter<00:50:09.960> does<00:50:10.120> not<00:50:10.280> give<00:50:10.480> us<00:50:10.720> any" + }, + { + "start": 3010.91, + "duration": 0.0, + "text": "and this parameter does not give us any" + }, + { + "start": 3010.92, + "duration": 0.0, + "text": "and this parameter does not give us any additional<00:50:11.280> information<00:50:11.840> on<00:50:11.920> the<00:50:12.000> second" + }, + { + "start": 3012.27, + "duration": 0.0, + "text": "additional information on the second" + }, + { + "start": 3012.28, + "duration": 0.0, + "text": "additional information on the second parameter," + }, + { + "start": 3013.95, + "duration": 0.0, + "text": "parameter," + }, + { + "start": 3013.96, + "duration": 0.0, + "text": "parameter, uh<00:50:14.080> this<00:50:14.320> essentially<00:50:15.360> all<00:50:15.480> of<00:50:15.560> this<00:50:15.840> is<00:50:16.000> equal" + }, + { + "start": 3016.27, + "duration": 0.0, + "text": "uh this essentially all of this is equal" + }, + { + "start": 3016.28, + "duration": 0.0, + "text": "uh this essentially all of this is equal to<00:50:16.600> just<00:50:16.800> the<00:50:16.880> probability<00:50:17.680> of" + }, + { + "start": 3018.75, + "duration": 0.0, + "text": "to just the probability of" + }, + { + "start": 3018.76, + "duration": 0.0, + "text": "to just the probability of the<00:50:18.880> parameter<00:50:19.240> that<00:50:19.360> we<00:50:19.480> want.<00:50:20.160> So,<00:50:20.520> this<00:50:20.760> is" + }, + { + "start": 3020.83, + "duration": 0.0, + "text": "the parameter that we want. So, this is" + }, + { + "start": 3020.84, + "duration": 0.0, + "text": "the parameter that we want. So, this is the<00:50:20.920> expression<00:50:21.720> that<00:50:21.920> we're<00:50:22.040> looking<00:50:22.280> for." + }, + { + "start": 3022.43, + "duration": 0.0, + "text": "the expression that we're looking for." + }, + { + "start": 3022.44, + "duration": 0.0, + "text": "the expression that we're looking for. We<00:50:22.560> have<00:50:22.680> found<00:50:23.000> it.<00:50:23.760> Um" + }, + { + "start": 3026.28, + "duration": 0.0, + "text": "And<00:50:26.360> this<00:50:26.480> is<00:50:26.560> again<00:50:26.760> a<00:50:26.800> consequence<00:50:27.360> of<00:50:27.480> the" + }, + { + "start": 3027.55, + "duration": 0.0, + "text": "And this is again a consequence of the" + }, + { + "start": 3027.56, + "duration": 0.0, + "text": "And this is again a consequence of the fact<00:50:27.880> that<00:50:28.280> this<00:50:28.440> kind<00:50:28.680> of<00:50:28.960> massive<00:50:29.800> compound" + }, + { + "start": 3030.27, + "duration": 0.0, + "text": "fact that this kind of massive compound" + }, + { + "start": 3030.28, + "duration": 0.0, + "text": "fact that this kind of massive compound statement<00:50:30.800> is<00:50:30.960> already<00:50:31.200> contained<00:50:31.560> in<00:50:31.640> the" + }, + { + "start": 3031.71, + "duration": 0.0, + "text": "statement is already contained in the" + }, + { + "start": 3031.72, + "duration": 0.0, + "text": "statement is already contained in the context.<00:50:32.560> And<00:50:32.680> so,<00:50:32.880> it<00:50:32.960> doesn't<00:50:33.240> bear<00:50:33.680> any" + }, + { + "start": 3033.95, + "duration": 0.0, + "text": "context. And so, it doesn't bear any" + }, + { + "start": 3033.96, + "duration": 0.0, + "text": "context. And so, it doesn't bear any additional<00:50:34.360> influence<00:50:35.280> on<00:50:35.400> the<00:50:35.480> knowledge<00:50:35.840> of" + }, + { + "start": 3036.07, + "duration": 0.0, + "text": "additional influence on the knowledge of" + }, + { + "start": 3036.08, + "duration": 0.0, + "text": "additional influence on the knowledge of omega<00:50:36.440> because<00:50:36.840> it<00:50:37.000> doesn't<00:50:37.280> add<00:50:37.520> anything" + }, + { + "start": 3037.83, + "duration": 0.0, + "text": "omega because it doesn't add anything" + }, + { + "start": 3037.84, + "duration": 0.0, + "text": "omega because it doesn't add anything that<00:50:37.960> is<00:50:38.080> not<00:50:38.280> already<00:50:38.520> contained<00:50:38.880> in<00:50:39.040> T." + }, + { + "start": 3042.56, + "duration": 0.0, + "text": "So,<00:50:42.840> how<00:50:43.080> how<00:50:43.320> do<00:50:43.400> we<00:50:43.480> make<00:50:43.720> use<00:50:43.880> of<00:50:43.960> this?<00:50:44.720> Um" + }, + { + "start": 3045.91, + "duration": 0.0, + "text": "So, how how do we make use of this? Um" + }, + { + "start": 3045.92, + "duration": 0.0, + "text": "So, how how do we make use of this? Um we<00:50:46.080> make<00:50:46.320> use<00:50:46.560> of<00:50:46.720> the<00:50:46.800> distributivity<00:50:47.760> of" + }, + { + "start": 3048.03, + "duration": 0.0, + "text": "we make use of the distributivity of" + }, + { + "start": 3048.04, + "duration": 0.0, + "text": "we make use of the distributivity of logical<00:50:48.400> operations.<00:50:48.840> So,<00:50:48.960> we<00:50:49.080> have<00:50:49.200> seen" + }, + { + "start": 3049.47, + "duration": 0.0, + "text": "logical operations. So, we have seen" + }, + { + "start": 3049.48, + "duration": 0.0, + "text": "logical operations. So, we have seen that<00:50:49.680> this<00:50:50.160> expression" + }, + { + "start": 3051.59, + "duration": 0.0, + "text": "that this expression" + }, + { + "start": 3051.6, + "duration": 0.0, + "text": "that this expression gives<00:50:51.880> us<00:50:51.960> the<00:50:52.080> probability<00:50:52.600> that<00:50:52.760> we<00:50:52.840> want." + }, + { + "start": 3053.39, + "duration": 0.0, + "text": "gives us the probability that we want." + }, + { + "start": 3053.4, + "duration": 0.0, + "text": "gives us the probability that we want. This<00:50:53.520> is<00:50:53.600> the<00:50:53.640> thing<00:50:53.800> that<00:50:53.920> we<00:50:54.000> saw<00:50:54.160> just" + }, + { + "start": 3054.35, + "duration": 0.0, + "text": "This is the thing that we saw just" + }, + { + "start": 3054.36, + "duration": 0.0, + "text": "This is the thing that we saw just above." + }, + { + "start": 3055.75, + "duration": 0.0, + "text": "above." + }, + { + "start": 3055.76, + "duration": 0.0, + "text": "above. And<00:50:56.160> now<00:50:56.440> we<00:50:56.560> just<00:50:56.720> simply<00:50:57.000> make<00:50:57.160> use<00:50:57.320> of<00:50:57.400> the" + }, + { + "start": 3057.47, + "duration": 0.0, + "text": "And now we just simply make use of the" + }, + { + "start": 3057.48, + "duration": 0.0, + "text": "And now we just simply make use of the distributivity<00:50:58.360> of<00:50:58.600> the<00:50:58.760> and<00:50:59.360> over<00:50:59.560> the<00:50:59.680> or" + }, + { + "start": 3059.87, + "duration": 0.0, + "text": "distributivity of the and over the or" + }, + { + "start": 3059.88, + "duration": 0.0, + "text": "distributivity of the and over the or operations<00:51:00.440> and<00:51:00.560> we<00:51:00.680> obtain<00:51:00.960> kind<00:51:01.160> of<00:51:01.760> this" + }, + { + "start": 3061.99, + "duration": 0.0, + "text": "operations and we obtain kind of this" + }, + { + "start": 3062.0, + "duration": 0.0, + "text": "operations and we obtain kind of this massive<00:51:02.360> compound<00:51:02.640> statement<00:51:03.080> where<00:51:03.240> we<00:51:03.320> have" + }, + { + "start": 3063.47, + "duration": 0.0, + "text": "massive compound statement where we have" + }, + { + "start": 3063.48, + "duration": 0.0, + "text": "massive compound statement where we have omega" + }, + { + "start": 3064.91, + "duration": 0.0, + "text": "omega" + }, + { + "start": 3064.92, + "duration": 0.0, + "text": "omega and<00:51:05.120> A1<00:51:05.640> or<00:51:05.880> omega<00:51:06.240> and<00:51:06.440> A2<00:51:07.080> and<00:51:07.320> so<00:51:07.520> on<00:51:07.760> or" + }, + { + "start": 3067.95, + "duration": 0.0, + "text": "and A1 or omega and A2 and so on or" + }, + { + "start": 3067.96, + "duration": 0.0, + "text": "and A1 or omega and A2 and so on or omega<00:51:08.320> and<00:51:08.600> AN." + }, + { + "start": 3069.75, + "duration": 0.0, + "text": "omega and AN." + }, + { + "start": 3069.76, + "duration": 0.0, + "text": "omega and AN. And<00:51:10.440> since<00:51:10.920> the<00:51:11.040> or<00:51:11.240> operation<00:51:11.800> is<00:51:12.000> always" + }, + { + "start": 3073.27, + "duration": 0.0, + "text": "And since the or operation is always" + }, + { + "start": 3073.28, + "duration": 0.0, + "text": "And since the or operation is always a<00:51:13.320> sum,<00:51:14.000> we<00:51:14.120> can<00:51:14.280> split<00:51:14.560> this<00:51:14.720> up<00:51:14.920> into" + }, + { + "start": 3076.31, + "duration": 0.0, + "text": "a sum, we can split this up into" + }, + { + "start": 3076.32, + "duration": 0.0, + "text": "a sum, we can split this up into different<00:51:16.520> probabilities<00:51:17.640> that<00:51:17.840> we<00:51:17.920> just" + }, + { + "start": 3078.11, + "duration": 0.0, + "text": "different probabilities that we just" + }, + { + "start": 3078.12, + "duration": 0.0, + "text": "different probabilities that we just simply<00:51:18.600> sum<00:51:18.920> together." + }, + { + "start": 3080.11, + "duration": 0.0, + "text": "simply sum together." + }, + { + "start": 3080.12, + "duration": 0.0, + "text": "simply sum together. Um<00:51:20.720> and<00:51:20.840> this<00:51:21.080> gives<00:51:21.280> us<00:51:21.440> a<00:51:21.480> general<00:51:21.840> recipe" + }, + { + "start": 3082.31, + "duration": 0.0, + "text": "Um and this gives us a general recipe" + }, + { + "start": 3082.32, + "duration": 0.0, + "text": "Um and this gives us a general recipe for<00:51:22.640> essentially<00:51:23.200> what<00:51:23.320> is<00:51:23.400> called<00:51:23.640> averaging" + }, + { + "start": 3084.15, + "duration": 0.0, + "text": "for essentially what is called averaging" + }, + { + "start": 3084.16, + "duration": 0.0, + "text": "for essentially what is called averaging out<00:51:24.520> the<00:51:24.600> nuisance<00:51:24.920> parameter.<00:51:25.600> So,<00:51:25.800> in<00:51:25.960> a" + }, + { + "start": 3085.99, + "duration": 0.0, + "text": "out the nuisance parameter. So, in a" + }, + { + "start": 3086.0, + "duration": 0.0, + "text": "out the nuisance parameter. So, in a continuous<00:51:26.600> case," + }, + { + "start": 3087.63, + "duration": 0.0, + "text": "continuous case," + }, + { + "start": 3087.64, + "duration": 0.0, + "text": "continuous case, we<00:51:27.800> obtain<00:51:28.200> in<00:51:28.320> this<00:51:28.520> beautiful<00:51:28.880> simple" + }, + { + "start": 3089.15, + "duration": 0.0, + "text": "we obtain in this beautiful simple" + }, + { + "start": 3089.16, + "duration": 0.0, + "text": "we obtain in this beautiful simple formula" + }, + { + "start": 3090.23, + "duration": 0.0, + "text": "formula" + }, + { + "start": 3090.24, + "duration": 0.0, + "text": "formula that<00:51:30.480> we<00:51:30.600> can<00:51:30.720> just<00:51:31.000> integrate<00:51:31.520> out<00:51:31.800> the" + }, + { + "start": 3091.83, + "duration": 0.0, + "text": "that we can just integrate out the" + }, + { + "start": 3091.84, + "duration": 0.0, + "text": "that we can just integrate out the nuisance<00:51:32.160> parameter.<00:51:32.480> We<00:51:32.560> just<00:51:32.720> need<00:51:32.840> to" + }, + { + "start": 3093.91, + "duration": 0.0, + "text": "nuisance parameter. We just need to" + }, + { + "start": 3093.92, + "duration": 0.0, + "text": "nuisance parameter. We just need to sum<00:51:34.320> all<00:51:34.480> the<00:51:34.560> probabilities" + }, + { + "start": 3096.23, + "duration": 0.0, + "text": "sum all the probabilities" + }, + { + "start": 3096.24, + "duration": 0.0, + "text": "sum all the probabilities of<00:51:36.440> the<00:51:36.520> kind<00:51:36.680> of<00:51:36.760> the<00:51:36.880> value<00:51:37.200> of<00:51:37.320> omega<00:51:37.560> that" + }, + { + "start": 3097.67, + "duration": 0.0, + "text": "of the kind of the value of omega that" + }, + { + "start": 3097.68, + "duration": 0.0, + "text": "of the kind of the value of omega that we're<00:51:37.760> looking<00:51:38.000> for<00:51:38.800> over<00:51:39.120> all<00:51:39.360> possible" + }, + { + "start": 3099.75, + "duration": 0.0, + "text": "we're looking for over all possible" + }, + { + "start": 3099.76, + "duration": 0.0, + "text": "we're looking for over all possible values<00:51:40.200> of<00:51:40.440> A." + }, + { + "start": 3101.75, + "duration": 0.0, + "text": "values of A." + }, + { + "start": 3101.76, + "duration": 0.0, + "text": "values of A. And<00:51:42.360> this<00:51:42.520> is<00:51:42.640> very<00:51:43.160> very<00:51:43.400> very<00:51:43.640> powerful" + }, + { + "start": 3104.03, + "duration": 0.0, + "text": "And this is very very very powerful" + }, + { + "start": 3104.04, + "duration": 0.0, + "text": "And this is very very very powerful because<00:51:44.440> in<00:51:44.560> a<00:51:44.640> high<00:51:45.680> when<00:51:45.920> a<00:51:45.960> problem<00:51:46.240> is<00:51:46.360> very" + }, + { + "start": 3106.67, + "duration": 0.0, + "text": "because in a high when a problem is very" + }, + { + "start": 3106.68, + "duration": 0.0, + "text": "because in a high when a problem is very high<00:51:46.800> dimensional," + }, + { + "start": 3108.35, + "duration": 0.0, + "text": "high dimensional," + }, + { + "start": 3108.36, + "duration": 0.0, + "text": "high dimensional, trying<00:51:48.720> to<00:51:48.840> estimate" + }, + { + "start": 3110.19, + "duration": 0.0, + "text": "trying to estimate" + }, + { + "start": 3110.2, + "duration": 0.0, + "text": "trying to estimate all<00:51:50.400> the<00:51:50.520> parameters<00:51:50.960> simultaneously<00:51:51.640> is" + }, + { + "start": 3111.83, + "duration": 0.0, + "text": "all the parameters simultaneously is" + }, + { + "start": 3111.84, + "duration": 0.0, + "text": "all the parameters simultaneously is often<00:51:52.560> impossible<00:51:53.240> or<00:51:53.480> it's<00:51:53.760> intractable" + }, + { + "start": 3114.51, + "duration": 0.0, + "text": "often impossible or it's intractable" + }, + { + "start": 3114.52, + "duration": 0.0, + "text": "often impossible or it's intractable with<00:51:55.160> with<00:51:55.280> the<00:51:55.360> computation<00:51:55.880> that<00:51:56.000> we<00:51:56.080> have" + }, + { + "start": 3116.27, + "duration": 0.0, + "text": "with with the computation that we have" + }, + { + "start": 3116.28, + "duration": 0.0, + "text": "with with the computation that we have available.<00:51:57.280> And<00:51:57.880> this<00:51:58.080> kind<00:51:58.240> of<00:51:58.440> averaging" + }, + { + "start": 3118.83, + "duration": 0.0, + "text": "available. And this kind of averaging" + }, + { + "start": 3118.84, + "duration": 0.0, + "text": "available. And this kind of averaging out<00:51:59.120> makes<00:51:59.480> many<00:52:00.240> such<00:52:00.480> problems" + }, + { + "start": 3121.11, + "duration": 0.0, + "text": "out makes many such problems" + }, + { + "start": 3121.12, + "duration": 0.0, + "text": "out makes many such problems approachable<00:52:01.680> so<00:52:01.800> that<00:52:01.920> we<00:52:02.000> can<00:52:02.280> actually<00:52:02.560> get" + }, + { + "start": 3122.95, + "duration": 0.0, + "text": "approachable so that we can actually get" + }, + { + "start": 3122.96, + "duration": 0.0, + "text": "approachable so that we can actually get an<00:52:03.080> interesting<00:52:03.520> answer<00:52:03.840> or<00:52:04.040> any<00:52:04.240> answer<00:52:04.560> at" + }, + { + "start": 3124.67, + "duration": 0.0, + "text": "an interesting answer or any answer at" + }, + { + "start": 3124.68, + "duration": 0.0, + "text": "an interesting answer or any answer at all<00:52:04.840> actually.<00:52:05.720> But<00:52:05.880> in<00:52:05.960> the<00:52:06.080> frequentist" + }, + { + "start": 3126.71, + "duration": 0.0, + "text": "all actually. But in the frequentist" + }, + { + "start": 3126.72, + "duration": 0.0, + "text": "all actually. But in the frequentist practice<00:52:07.080> this<00:52:07.240> is<00:52:07.360> impossible" + }, + { + "start": 3128.63, + "duration": 0.0, + "text": "practice this is impossible" + }, + { + "start": 3128.64, + "duration": 0.0, + "text": "practice this is impossible simply<00:52:08.960> because<00:52:09.320> they<00:52:09.400> don't<00:52:09.720> assign" + }, + { + "start": 3130.15, + "duration": 0.0, + "text": "simply because they don't assign" + }, + { + "start": 3130.16, + "duration": 0.0, + "text": "simply because they don't assign probabilities<00:52:10.760> to<00:52:10.880> parameters<00:52:11.440> and<00:52:11.520> so<00:52:11.720> all" + }, + { + "start": 3131.87, + "duration": 0.0, + "text": "probabilities to parameters and so all" + }, + { + "start": 3131.88, + "duration": 0.0, + "text": "probabilities to parameters and so all of<00:52:11.960> this<00:52:12.120> reasoning" + }, + { + "start": 3133.31, + "duration": 0.0, + "text": "of this reasoning" + }, + { + "start": 3133.32, + "duration": 0.0, + "text": "of this reasoning with<00:52:13.440> probability<00:52:13.920> of<00:52:14.120> A<00:52:14.240> the<00:52:14.720> of<00:52:14.920> omega<00:52:15.280> and<00:52:15.480> A" + }, + { + "start": 3135.63, + "duration": 0.0, + "text": "with probability of A the of omega and A" + }, + { + "start": 3135.64, + "duration": 0.0, + "text": "with probability of A the of omega and A does<00:52:16.400> not<00:52:16.560> make<00:52:16.720> any<00:52:16.840> sense." + }, + { + "start": 3139.48, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3141.27, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3141.28, + "duration": 0.0, + "text": "Um And<00:52:21.400> now<00:52:21.760> another,<00:52:22.400> which<00:52:22.560> is<00:52:22.640> probably<00:52:22.880> my" + }, + { + "start": 3143.03, + "duration": 0.0, + "text": "And now another, which is probably my" + }, + { + "start": 3143.04, + "duration": 0.0, + "text": "And now another, which is probably my favorite" + }, + { + "start": 3144.27, + "duration": 0.0, + "text": "favorite" + }, + { + "start": 3144.28, + "duration": 0.0, + "text": "favorite feature<00:52:24.720> of<00:52:24.840> the<00:52:25.200> of<00:52:25.320> the<00:52:25.440> Bayesian" + }, + { + "start": 3145.67, + "duration": 0.0, + "text": "feature of the of the Bayesian" + }, + { + "start": 3145.68, + "duration": 0.0, + "text": "feature of the of the Bayesian probability<00:52:26.080> calculus,<00:52:26.560> is<00:52:26.760> that<00:52:26.920> it<00:52:27.040> has<00:52:27.720> a" + }, + { + "start": 3147.79, + "duration": 0.0, + "text": "probability calculus, is that it has a" + }, + { + "start": 3147.8, + "duration": 0.0, + "text": "probability calculus, is that it has a built-in<00:52:28.680> Occam's<00:52:29.080> razor.<00:52:29.360> It<00:52:29.440> is<00:52:29.520> a" + }, + { + "start": 3149.59, + "duration": 0.0, + "text": "built-in Occam's razor. It is a" + }, + { + "start": 3149.6, + "duration": 0.0, + "text": "built-in Occam's razor. It is a quantification<00:52:30.360> of<00:52:30.520> the<00:52:30.640> notion<00:52:30.920> of<00:52:31.080> Occam's" + }, + { + "start": 3151.43, + "duration": 0.0, + "text": "quantification of the notion of Occam's" + }, + { + "start": 3151.44, + "duration": 0.0, + "text": "quantification of the notion of Occam's razor." + }, + { + "start": 3152.59, + "duration": 0.0, + "text": "razor." + }, + { + "start": 3152.6, + "duration": 0.0, + "text": "razor. Um" + }, + { + "start": 3153.75, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3153.76, + "duration": 0.0, + "text": "Um This<00:52:33.960> idea<00:52:34.520> is<00:52:34.800> usually<00:52:35.360> kind<00:52:35.640> of" + }, + { + "start": 3157.15, + "duration": 0.0, + "text": "This idea is usually kind of" + }, + { + "start": 3157.16, + "duration": 0.0, + "text": "This idea is usually kind of seen<00:52:37.720> as<00:52:38.080> a<00:52:38.760> qualitative<00:52:39.480> desiderata<00:52:40.320> of" + }, + { + "start": 3160.51, + "duration": 0.0, + "text": "seen as a qualitative desiderata of" + }, + { + "start": 3160.52, + "duration": 0.0, + "text": "seen as a qualitative desiderata of scientific<00:52:41.000> theories." + }, + { + "start": 3162.99, + "duration": 0.0, + "text": "scientific theories." + }, + { + "start": 3163.0, + "duration": 0.0, + "text": "scientific theories. The<00:52:43.200> the<00:52:43.280> exact<00:52:43.680> words<00:52:44.120> of<00:52:44.320> Occam<00:52:44.600> can<00:52:44.720> be" + }, + { + "start": 3164.79, + "duration": 0.0, + "text": "The the exact words of Occam can be" + }, + { + "start": 3164.8, + "duration": 0.0, + "text": "The the exact words of Occam can be translated<00:52:45.200> as<00:52:45.480> things<00:52:46.200> should<00:52:46.320> be<00:52:47.000> increased" + }, + { + "start": 3167.43, + "duration": 0.0, + "text": "translated as things should be increased" + }, + { + "start": 3167.44, + "duration": 0.0, + "text": "translated as things should be increased in<00:52:47.560> number<00:52:48.040> beyond<00:52:48.440> necessity<00:52:49.200> and<00:52:49.280> which<00:52:49.400> is" + }, + { + "start": 3169.47, + "duration": 0.0, + "text": "in number beyond necessity and which is" + }, + { + "start": 3169.48, + "duration": 0.0, + "text": "in number beyond necessity and which is usually<00:52:49.720> summarized<00:52:50.400> as<00:52:51.240> all<00:52:51.520> else<00:52:51.840> being" + }, + { + "start": 3172.07, + "duration": 0.0, + "text": "usually summarized as all else being" + }, + { + "start": 3172.08, + "duration": 0.0, + "text": "usually summarized as all else being equal<00:52:52.920> the<00:52:53.000> simplest<00:52:53.280> explanation<00:52:53.800> is" + }, + { + "start": 3173.91, + "duration": 0.0, + "text": "equal the simplest explanation is" + }, + { + "start": 3173.92, + "duration": 0.0, + "text": "equal the simplest explanation is usually<00:52:54.160> the<00:52:54.280> best<00:52:54.560> one." + }, + { + "start": 3175.31, + "duration": 0.0, + "text": "usually the best one." + }, + { + "start": 3175.32, + "duration": 0.0, + "text": "usually the best one. Um" + }, + { + "start": 3176.63, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3176.64, + "duration": 0.0, + "text": "Um Bayes'<00:52:56.880> theorem<00:52:57.200> has<00:52:57.440> a<00:52:57.640> mechanism<00:52:58.120> or<00:52:58.240> the" + }, + { + "start": 3178.67, + "duration": 0.0, + "text": "Bayes' theorem has a mechanism or the" + }, + { + "start": 3178.68, + "duration": 0.0, + "text": "Bayes' theorem has a mechanism or the Bayesian<00:52:59.080> calculus<00:52:59.480> in<00:52:59.560> general<00:52:59.840> has<00:53:00.000> a" + }, + { + "start": 3180.03, + "duration": 0.0, + "text": "Bayesian calculus in general has a" + }, + { + "start": 3180.04, + "duration": 0.0, + "text": "Bayesian calculus in general has a mechanism<00:53:00.480> that<00:53:00.600> does<00:53:00.920> exactly<00:53:01.240> this." + }, + { + "start": 3182.39, + "duration": 0.0, + "text": "mechanism that does exactly this." + }, + { + "start": 3182.4, + "duration": 0.0, + "text": "mechanism that does exactly this. Uh<00:53:02.800> meaning<00:53:03.680> more<00:53:03.920> complex<00:53:04.440> models" + }, + { + "start": 3185.83, + "duration": 0.0, + "text": "Uh meaning more complex models" + }, + { + "start": 3185.84, + "duration": 0.0, + "text": "Uh meaning more complex models um" + }, + { + "start": 3187.07, + "duration": 0.0, + "text": "um" + }, + { + "start": 3187.08, + "duration": 0.0, + "text": "um penalized<00:53:07.800> if<00:53:08.000> their<00:53:08.280> additional<00:53:08.800> predictive" + }, + { + "start": 3189.31, + "duration": 0.0, + "text": "penalized if their additional predictive" + }, + { + "start": 3189.32, + "duration": 0.0, + "text": "penalized if their additional predictive accuracy<00:53:10.360> doesn't<00:53:10.640> make<00:53:10.880> up<00:53:11.040> for<00:53:11.200> their" + }, + { + "start": 3191.35, + "duration": 0.0, + "text": "accuracy doesn't make up for their" + }, + { + "start": 3191.36, + "duration": 0.0, + "text": "accuracy doesn't make up for their predictive<00:53:12.000> flexibility." + }, + { + "start": 3193.51, + "duration": 0.0, + "text": "predictive flexibility." + }, + { + "start": 3193.52, + "duration": 0.0, + "text": "predictive flexibility. And<00:53:13.680> we<00:53:13.760> can<00:53:13.880> see<00:53:14.000> this<00:53:14.160> as<00:53:14.280> follows.<00:53:14.760> First," + }, + { + "start": 3195.75, + "duration": 0.0, + "text": "And we can see this as follows. First," + }, + { + "start": 3195.76, + "duration": 0.0, + "text": "And we can see this as follows. First, we<00:53:15.840> need<00:53:15.960> to<00:53:16.080> look<00:53:16.240> at<00:53:16.360> how<00:53:16.520> we<00:53:16.640> would<00:53:16.800> even" + }, + { + "start": 3196.99, + "duration": 0.0, + "text": "we need to look at how we would even" + }, + { + "start": 3197.0, + "duration": 0.0, + "text": "we need to look at how we would even compare<00:53:17.440> different<00:53:17.760> models." + }, + { + "start": 3198.99, + "duration": 0.0, + "text": "compare different models." + }, + { + "start": 3199.0, + "duration": 0.0, + "text": "compare different models. And<00:53:19.160> so<00:53:19.400> once<00:53:19.640> again<00:53:19.880> we<00:53:20.000> just<00:53:20.720> remember<00:53:21.120> that" + }, + { + "start": 3201.43, + "duration": 0.0, + "text": "And so once again we just remember that" + }, + { + "start": 3201.44, + "duration": 0.0, + "text": "And so once again we just remember that in<00:53:21.600> the<00:53:21.680> Bayesian<00:53:22.520> context<00:53:23.000> we<00:53:23.080> can<00:53:23.560> assign" + }, + { + "start": 3203.83, + "duration": 0.0, + "text": "in the Bayesian context we can assign" + }, + { + "start": 3203.84, + "duration": 0.0, + "text": "in the Bayesian context we can assign probabilities<00:53:24.400> to<00:53:24.640> any<00:53:24.840> proposition<00:53:25.800> and" + }, + { + "start": 3205.91, + "duration": 0.0, + "text": "probabilities to any proposition and" + }, + { + "start": 3205.92, + "duration": 0.0, + "text": "probabilities to any proposition and that<00:53:26.040> is<00:53:26.160> also<00:53:26.360> the<00:53:26.440> proposition<00:53:27.000> of<00:53:27.400> model<00:53:27.880> I" + }, + { + "start": 3208.31, + "duration": 0.0, + "text": "that is also the proposition of model I" + }, + { + "start": 3208.32, + "duration": 0.0, + "text": "that is also the proposition of model I of<00:53:28.840> a<00:53:28.880> choice<00:53:29.200> of<00:53:29.320> models<00:53:29.680> is<00:53:29.840> being<00:53:30.040> true." + }, + { + "start": 3211.07, + "duration": 0.0, + "text": "of a choice of models is being true." + }, + { + "start": 3211.08, + "duration": 0.0, + "text": "of a choice of models is being true. And<00:53:31.240> so<00:53:31.520> we<00:53:31.640> can<00:53:32.000> once<00:53:32.240> again<00:53:32.960> write<00:53:33.280> this<00:53:33.440> down" + }, + { + "start": 3213.75, + "duration": 0.0, + "text": "And so we can once again write this down" + }, + { + "start": 3213.76, + "duration": 0.0, + "text": "And so we can once again write this down in<00:53:33.960> the" + }, + { + "start": 3214.95, + "duration": 0.0, + "text": "in the" + }, + { + "start": 3214.96, + "duration": 0.0, + "text": "in the in<00:53:35.040> the<00:53:35.120> Bayes'<00:53:35.360> theorem<00:53:35.720> form." + }, + { + "start": 3217.55, + "duration": 0.0, + "text": "in the Bayes' theorem form." + }, + { + "start": 3217.56, + "duration": 0.0, + "text": "in the Bayes' theorem form. And<00:53:37.720> again<00:53:37.920> the<00:53:38.200> the<00:53:38.560> term<00:53:38.760> in<00:53:38.840> the" + }, + { + "start": 3218.87, + "duration": 0.0, + "text": "And again the the term in the" + }, + { + "start": 3218.88, + "duration": 0.0, + "text": "And again the the term in the denominator<00:53:39.520> is<00:53:39.640> just<00:53:39.960> a<00:53:40.000> The<00:53:40.440> constant" + }, + { + "start": 3221.31, + "duration": 0.0, + "text": "denominator is just a The constant" + }, + { + "start": 3221.32, + "duration": 0.0, + "text": "denominator is just a The constant obtained<00:53:41.720> by<00:53:41.840> summing<00:53:42.280> over<00:53:42.560> all" + }, + { + "start": 3222.83, + "duration": 0.0, + "text": "obtained by summing over all" + }, + { + "start": 3222.84, + "duration": 0.0, + "text": "obtained by summing over all possibilities<00:53:43.600> over<00:53:43.840> all<00:53:43.960> models." + }, + { + "start": 3227.04, + "duration": 0.0, + "text": "And<00:53:47.280> com-<00:53:47.600> model<00:53:47.840> comparison<00:53:48.560> is<00:53:49.160> um<00:53:49.680> very" + }, + { + "start": 3229.91, + "duration": 0.0, + "text": "And com- model comparison is um very" + }, + { + "start": 3229.92, + "duration": 0.0, + "text": "And com- model comparison is um very analogous<00:53:50.320> to<00:53:50.400> parameter<00:53:50.800> estimation.<00:53:51.880> Um" + }, + { + "start": 3232.47, + "duration": 0.0, + "text": "analogous to parameter estimation. Um" + }, + { + "start": 3232.48, + "duration": 0.0, + "text": "analogous to parameter estimation. Um the<00:53:53.080> posterior<00:53:53.680> probability<00:53:54.120> for<00:53:54.200> model<00:53:54.480> is" + }, + { + "start": 3234.59, + "duration": 0.0, + "text": "the posterior probability for model is" + }, + { + "start": 3234.6, + "duration": 0.0, + "text": "the posterior probability for model is proportional<00:53:55.240> to<00:53:55.400> its<00:53:55.640> prior<00:53:55.960> probability" + }, + { + "start": 3236.47, + "duration": 0.0, + "text": "proportional to its prior probability" + }, + { + "start": 3236.48, + "duration": 0.0, + "text": "proportional to its prior probability times<00:53:56.840> its<00:53:56.960> likelihood.<00:53:57.760> And<00:53:57.880> now<00:53:57.960> to<00:53:58.080> compare" + }, + { + "start": 3238.47, + "duration": 0.0, + "text": "times its likelihood. And now to compare" + }, + { + "start": 3238.48, + "duration": 0.0, + "text": "times its likelihood. And now to compare two<00:53:58.640> models,<00:53:59.040> we<00:53:59.160> can<00:53:59.360> compute<00:53:59.680> their<00:53:59.880> odds" + }, + { + "start": 3240.11, + "duration": 0.0, + "text": "two models, we can compute their odds" + }, + { + "start": 3240.12, + "duration": 0.0, + "text": "two models, we can compute their odds ratio.<00:54:00.600> We<00:54:00.720> compare<00:54:01.080> them<00:54:01.240> by<00:54:01.480> dividing<00:54:02.000> one" + }, + { + "start": 3242.11, + "duration": 0.0, + "text": "ratio. We compare them by dividing one" + }, + { + "start": 3242.12, + "duration": 0.0, + "text": "ratio. We compare them by dividing one by<00:54:02.240> the<00:54:02.360> other.<00:54:03.040> And<00:54:03.160> the<00:54:03.240> reason<00:54:03.760> we<00:54:04.360> only" + }, + { + "start": 3244.63, + "duration": 0.0, + "text": "by the other. And the reason we only" + }, + { + "start": 3244.64, + "duration": 0.0, + "text": "by the other. And the reason we only have<00:54:05.160> um" + }, + { + "start": 3247.15, + "duration": 0.0, + "text": "have um" + }, + { + "start": 3247.16, + "duration": 0.0, + "text": "have um Exactly.<00:54:07.520> Here<00:54:07.600> we<00:54:07.680> have<00:54:07.800> the<00:54:07.880> original" + }, + { + "start": 3248.19, + "duration": 0.0, + "text": "Exactly. Here we have the original" + }, + { + "start": 3248.2, + "duration": 0.0, + "text": "Exactly. Here we have the original ratio,<00:54:08.520> and<00:54:08.640> the<00:54:08.720> reason<00:54:09.080> why<00:54:09.640> we<00:54:09.760> can<00:54:10.040> write" + }, + { + "start": 3250.11, + "duration": 0.0, + "text": "ratio, and the reason why we can write" + }, + { + "start": 3250.12, + "duration": 0.0, + "text": "ratio, and the reason why we can write it<00:54:10.200> as<00:54:10.320> such" + }, + { + "start": 3251.39, + "duration": 0.0, + "text": "it as such" + }, + { + "start": 3251.4, + "duration": 0.0, + "text": "it as such with<00:54:11.840> sort<00:54:12.080> of<00:54:12.640> the<00:54:12.760> top<00:54:13.120> the<00:54:13.240> num-<00:54:13.600> the" + }, + { + "start": 3253.71, + "duration": 0.0, + "text": "with sort of the top the num- the" + }, + { + "start": 3253.72, + "duration": 0.0, + "text": "with sort of the top the num- the numerator<00:54:14.200> of<00:54:14.320> model<00:54:14.680> I<00:54:14.920> and<00:54:15.080> the<00:54:15.160> numerator" + }, + { + "start": 3255.63, + "duration": 0.0, + "text": "numerator of model I and the numerator" + }, + { + "start": 3255.64, + "duration": 0.0, + "text": "numerator of model I and the numerator of<00:54:15.720> model<00:54:16.000> J<00:54:16.360> is<00:54:16.520> because<00:54:17.440> um" + }, + { + "start": 3258.63, + "duration": 0.0, + "text": "of model J is because um" + }, + { + "start": 3258.64, + "duration": 0.0, + "text": "of model J is because um this<00:54:18.920> normalizing<00:54:19.440> constant<00:54:20.480> they<00:54:20.600> would" + }, + { + "start": 3260.75, + "duration": 0.0, + "text": "this normalizing constant they would" + }, + { + "start": 3260.76, + "duration": 0.0, + "text": "this normalizing constant they would have<00:54:20.880> in<00:54:20.960> common<00:54:21.320> because<00:54:21.600> we<00:54:21.720> are<00:54:21.800> studying" + }, + { + "start": 3262.19, + "duration": 0.0, + "text": "have in common because we are studying" + }, + { + "start": 3262.2, + "duration": 0.0, + "text": "have in common because we are studying them<00:54:22.360> within<00:54:22.560> the<00:54:22.640> same<00:54:22.960> context<00:54:23.560> and<00:54:24.000> with" + }, + { + "start": 3264.15, + "duration": 0.0, + "text": "them within the same context and with" + }, + { + "start": 3264.16, + "duration": 0.0, + "text": "them within the same context and with the<00:54:24.280> same<00:54:24.520> data.<00:54:25.240> This<00:54:25.440> is<00:54:25.920> uh<00:54:26.000> what<00:54:26.120> we're" + }, + { + "start": 3266.19, + "duration": 0.0, + "text": "the same data. This is uh what we're" + }, + { + "start": 3266.2, + "duration": 0.0, + "text": "the same data. This is uh what we're doing<00:54:26.480> here.<00:54:27.240> And<00:54:27.320> so<00:54:27.600> we<00:54:27.760> can<00:54:28.200> obtain<00:54:28.800> this" + }, + { + "start": 3269.39, + "duration": 0.0, + "text": "doing here. And so we can obtain this" + }, + { + "start": 3269.4, + "duration": 0.0, + "text": "doing here. And so we can obtain this kind<00:54:29.560> of<00:54:29.680> ratio,<00:54:30.040> which<00:54:30.160> is<00:54:30.280> very" + }, + { + "start": 3270.47, + "duration": 0.0, + "text": "kind of ratio, which is very" + }, + { + "start": 3270.48, + "duration": 0.0, + "text": "kind of ratio, which is very interesting.<00:54:31.440> And<00:54:31.680> since<00:54:31.960> usually<00:54:32.520> we<00:54:32.640> don't" + }, + { + "start": 3272.83, + "duration": 0.0, + "text": "interesting. And since usually we don't" + }, + { + "start": 3272.84, + "duration": 0.0, + "text": "interesting. And since usually we don't have<00:54:33.240> any<00:54:33.480> kind<00:54:33.680> of<00:54:33.840> reason<00:54:34.240> to<00:54:34.360> prefer<00:54:35.320> any" + }, + { + "start": 3275.55, + "duration": 0.0, + "text": "have any kind of reason to prefer any" + }, + { + "start": 3275.56, + "duration": 0.0, + "text": "have any kind of reason to prefer any model<00:54:35.880> over<00:54:36.040> the<00:54:36.160> other,<00:54:36.520> we<00:54:36.680> must<00:54:37.200> by" + }, + { + "start": 3277.31, + "duration": 0.0, + "text": "model over the other, we must by" + }, + { + "start": 3277.32, + "duration": 0.0, + "text": "model over the other, we must by consistency<00:54:38.000> assign<00:54:38.520> equal<00:54:38.840> prior" + }, + { + "start": 3279.07, + "duration": 0.0, + "text": "consistency assign equal prior" + }, + { + "start": 3279.08, + "duration": 0.0, + "text": "consistency assign equal prior probabilities<00:54:39.680> to<00:54:39.760> them.<00:54:40.000> Meaning<00:54:40.200> the" + }, + { + "start": 3280.23, + "duration": 0.0, + "text": "probabilities to them. Meaning the" + }, + { + "start": 3280.24, + "duration": 0.0, + "text": "probabilities to them. Meaning the probability<00:54:40.840> of<00:54:41.000> model<00:54:41.360> I<00:54:42.000> is<00:54:42.160> equal<00:54:42.320> to<00:54:42.400> the" + }, + { + "start": 3282.51, + "duration": 0.0, + "text": "probability of model I is equal to the" + }, + { + "start": 3282.52, + "duration": 0.0, + "text": "probability of model I is equal to the prior<00:54:42.880> probability<00:54:43.560> of<00:54:43.720> model<00:54:44.000> J." + }, + { + "start": 3285.23, + "duration": 0.0, + "text": "prior probability of model J." + }, + { + "start": 3285.24, + "duration": 0.0, + "text": "prior probability of model J. And<00:54:45.320> so<00:54:45.440> this<00:54:45.720> odds<00:54:46.040> ratio,<00:54:46.720> it's<00:54:46.880> called,<00:54:47.800> um" + }, + { + "start": 3288.19, + "duration": 0.0, + "text": "And so this odds ratio, it's called, um" + }, + { + "start": 3288.2, + "duration": 0.0, + "text": "And so this odds ratio, it's called, um becomes<00:54:48.680> the<00:54:48.760> ratio<00:54:49.240> of<00:54:49.400> the<00:54:49.960> global" + }, + { + "start": 3290.31, + "duration": 0.0, + "text": "becomes the ratio of the global" + }, + { + "start": 3290.32, + "duration": 0.0, + "text": "becomes the ratio of the global likelihoods<00:54:50.880> of<00:54:50.960> the<00:54:51.040> model." + }, + { + "start": 3293.07, + "duration": 0.0, + "text": "likelihoods of the model." + }, + { + "start": 3293.08, + "duration": 0.0, + "text": "likelihoods of the model. Uh<00:54:53.240> like<00:54:53.480> so.<00:54:54.000> And<00:54:54.200> this<00:54:54.440> little<00:54:54.600> expression" + }, + { + "start": 3295.03, + "duration": 0.0, + "text": "Uh like so. And this little expression" + }, + { + "start": 3295.04, + "duration": 0.0, + "text": "Uh like so. And this little expression is<00:54:55.160> usually<00:54:55.400> called<00:54:55.600> the<00:54:55.680> Bayes<00:54:55.960> factor," + }, + { + "start": 3296.31, + "duration": 0.0, + "text": "is usually called the Bayes factor," + }, + { + "start": 3296.32, + "duration": 0.0, + "text": "is usually called the Bayes factor, which<00:54:56.520> is<00:54:56.680> why<00:54:56.880> I<00:54:56.960> denote<00:54:57.240> it<00:54:57.360> as<00:54:57.720> B<00:54:58.200> IJ." + }, + { + "start": 3301.24, + "duration": 0.0, + "text": "And<00:55:01.840> um" + }, + { + "start": 3302.55, + "duration": 0.0, + "text": "And um" + }, + { + "start": 3302.56, + "duration": 0.0, + "text": "And um these<00:55:02.680> likelihoods<00:55:03.040> are<00:55:03.120> calculated<00:55:03.640> from" + }, + { + "start": 3303.75, + "duration": 0.0, + "text": "these likelihoods are calculated from" + }, + { + "start": 3303.76, + "duration": 0.0, + "text": "these likelihoods are calculated from the<00:55:03.840> product<00:55:04.200> rules<00:55:04.560> as<00:55:04.680> seen<00:55:04.880> before." + }, + { + "start": 3305.99, + "duration": 0.0, + "text": "the product rules as seen before." + }, + { + "start": 3306.0, + "duration": 0.0, + "text": "the product rules as seen before. Um" + }, + { + "start": 3307.15, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3307.16, + "duration": 0.0, + "text": "Um if<00:55:07.800> say<00:55:08.080> for<00:55:08.240> example<00:55:09.240> uh<00:55:09.680> model<00:55:10.280> I<00:55:10.680> has<00:55:10.920> only" + }, + { + "start": 3311.11, + "duration": 0.0, + "text": "if say for example uh model I has only" + }, + { + "start": 3311.12, + "duration": 0.0, + "text": "if say for example uh model I has only one<00:55:11.320> parameter," + }, + { + "start": 3312.83, + "duration": 0.0, + "text": "one parameter," + }, + { + "start": 3312.84, + "duration": 0.0, + "text": "one parameter, theta<00:55:13.080> one,<00:55:13.840> then<00:55:13.920> its<00:55:14.080> likelihood<00:55:14.640> will<00:55:15.280> be" + }, + { + "start": 3315.71, + "duration": 0.0, + "text": "theta one, then its likelihood will be" + }, + { + "start": 3315.72, + "duration": 0.0, + "text": "theta one, then its likelihood will be just<00:55:15.920> this,<00:55:16.280> right?<00:55:16.720> You<00:55:17.280> um" + }, + { + "start": 3319.92, + "duration": 0.0, + "text": "you<00:55:20.000> just" + }, + { + "start": 3321.91, + "duration": 0.0, + "text": "you just" + }, + { + "start": 3321.92, + "duration": 0.0, + "text": "you just then<00:55:22.200> aver-<00:55:22.520> average<00:55:22.920> out<00:55:23.640> the<00:55:24.320> the<00:55:25.000> the" + }, + { + "start": 3325.19, + "duration": 0.0, + "text": "then aver- average out the the the" + }, + { + "start": 3325.2, + "duration": 0.0, + "text": "then aver- average out the the the parameter,<00:55:25.800> and<00:55:25.920> you<00:55:25.960> just<00:55:26.200> obtain<00:55:26.600> the" + }, + { + "start": 3327.19, + "duration": 0.0, + "text": "parameter, and you just obtain the" + }, + { + "start": 3327.2, + "duration": 0.0, + "text": "parameter, and you just obtain the complete" + }, + { + "start": 3328.43, + "duration": 0.0, + "text": "complete" + }, + { + "start": 3328.44, + "duration": 0.0, + "text": "complete for<00:55:28.520> the<00:55:28.640> likelihood<00:55:29.120> of<00:55:29.200> the<00:55:29.320> model.<00:55:30.160> This<00:55:30.240> is" + }, + { + "start": 3330.35, + "duration": 0.0, + "text": "for the likelihood of the model. This is" + }, + { + "start": 3330.36, + "duration": 0.0, + "text": "for the likelihood of the model. This is essentially<00:55:30.760> the<00:55:30.840> observation<00:55:31.320> that<00:55:31.440> we<00:55:31.560> have" + }, + { + "start": 3332.15, + "duration": 0.0, + "text": "essentially the observation that we have" + }, + { + "start": 3332.16, + "duration": 0.0, + "text": "essentially the observation that we have seen<00:55:32.440> before.<00:55:33.200> However,<00:55:34.000> if<00:55:34.200> the<00:55:34.320> alternative" + }, + { + "start": 3334.87, + "duration": 0.0, + "text": "seen before. However, if the alternative" + }, + { + "start": 3334.88, + "duration": 0.0, + "text": "seen before. However, if the alternative model<00:55:35.400> J<00:55:35.800> has<00:55:36.040> three<00:55:36.240> parameters,<00:55:37.400> then<00:55:37.600> its" + }, + { + "start": 3337.71, + "duration": 0.0, + "text": "model J has three parameters, then its" + }, + { + "start": 3337.72, + "duration": 0.0, + "text": "model J has three parameters, then its likelihood<00:55:38.200> is<00:55:38.920> this<00:55:39.200> massive<00:55:39.520> expression." + }, + { + "start": 3340.71, + "duration": 0.0, + "text": "likelihood is this massive expression." + }, + { + "start": 3340.72, + "duration": 0.0, + "text": "likelihood is this massive expression. Um" + }, + { + "start": 3342.75, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3342.76, + "duration": 0.0, + "text": "Um and<00:55:42.840> because<00:55:43.160> of<00:55:43.280> both<00:55:43.960> prior<00:55:44.200> probabilities," + }, + { + "start": 3344.83, + "duration": 0.0, + "text": "and because of both prior probabilities," + }, + { + "start": 3344.84, + "duration": 0.0, + "text": "and because of both prior probabilities, meaning<00:55:45.320> this<00:55:45.560> probability<00:55:46.480> here,<00:55:47.400> uh<00:55:47.680> of" + }, + { + "start": 3348.03, + "duration": 0.0, + "text": "meaning this probability here, uh of" + }, + { + "start": 3348.04, + "duration": 0.0, + "text": "meaning this probability here, uh of parameter<00:55:48.560> one<00:55:49.120> from<00:55:49.320> the<00:55:49.400> first<00:55:49.680> model<00:55:50.080> and" + }, + { + "start": 3350.39, + "duration": 0.0, + "text": "parameter one from the first model and" + }, + { + "start": 3350.4, + "duration": 0.0, + "text": "parameter one from the first model and of<00:55:50.560> the" + }, + { + "start": 3351.43, + "duration": 0.0, + "text": "of the" + }, + { + "start": 3351.44, + "duration": 0.0, + "text": "of the three<00:55:51.520> parameters<00:55:52.000> of<00:55:52.080> the<00:55:52.160> second<00:55:52.440> model," + }, + { + "start": 3353.03, + "duration": 0.0, + "text": "three parameters of the second model," + }, + { + "start": 3353.04, + "duration": 0.0, + "text": "three parameters of the second model, they<00:55:53.200> both<00:55:53.680> must<00:55:54.000> individually<00:55:54.720> add<00:55:54.960> up<00:55:55.120> to" + }, + { + "start": 3355.23, + "duration": 0.0, + "text": "they both must individually add up to" + }, + { + "start": 3355.24, + "duration": 0.0, + "text": "they both must individually add up to one<00:55:55.440> because<00:55:55.680> they're<00:55:55.760> probabilities." + }, + { + "start": 3357.07, + "duration": 0.0, + "text": "one because they're probabilities." + }, + { + "start": 3357.08, + "duration": 0.0, + "text": "one because they're probabilities. Uh<00:55:57.200> it<00:55:57.400> means<00:55:57.640> that<00:55:57.760> in<00:55:57.880> the<00:55:57.960> second<00:55:58.320> case,<00:55:58.720> the" + }, + { + "start": 3358.83, + "duration": 0.0, + "text": "Uh it means that in the second case, the" + }, + { + "start": 3358.84, + "duration": 0.0, + "text": "Uh it means that in the second case, the prior<00:55:59.160> probability,<00:56:00.280> this<00:56:00.440> one<00:56:00.560> right<00:56:00.720> here," + }, + { + "start": 3361.67, + "duration": 0.0, + "text": "prior probability, this one right here," + }, + { + "start": 3361.68, + "duration": 0.0, + "text": "prior probability, this one right here, is<00:56:02.120> much<00:56:02.360> more<00:56:02.560> {quote}<00:56:03.000> diluted,<00:56:03.720> meaning<00:56:04.160> it" + }, + { + "start": 3364.35, + "duration": 0.0, + "text": "is much more {quote} diluted, meaning it" + }, + { + "start": 3364.36, + "duration": 0.0, + "text": "is much more {quote} diluted, meaning it is<00:56:05.080> um" + }, + { + "start": 3366.43, + "duration": 0.0, + "text": "is um" + }, + { + "start": 3366.44, + "duration": 0.0, + "text": "is um distributed<00:56:07.240> among<00:56:08.280> more<00:56:08.520> parameters<00:56:09.080> and" + }, + { + "start": 3369.19, + "duration": 0.0, + "text": "distributed among more parameters and" + }, + { + "start": 3369.2, + "duration": 0.0, + "text": "distributed among more parameters and more<00:56:09.360> values,<00:56:10.360> meaning<00:56:10.920> each<00:56:11.200> value<00:56:11.800> for<00:56:12.000> each" + }, + { + "start": 3372.11, + "duration": 0.0, + "text": "more values, meaning each value for each" + }, + { + "start": 3372.12, + "duration": 0.0, + "text": "more values, meaning each value for each parameter<00:56:12.640> is<00:56:12.840> much<00:56:13.240> much<00:56:13.480> much<00:56:13.720> less<00:56:14.080> likely" + }, + { + "start": 3374.55, + "duration": 0.0, + "text": "parameter is much much much less likely" + }, + { + "start": 3374.56, + "duration": 0.0, + "text": "parameter is much much much less likely to<00:56:14.680> begin<00:56:15.000> with." + }, + { + "start": 3377.96, + "duration": 0.0, + "text": "Um<00:56:18.680> and<00:56:18.880> if<00:56:19.120> this<00:56:19.320> means<00:56:19.480> that<00:56:19.640> if<00:56:19.800> we<00:56:19.960> add<00:56:20.120> a" + }, + { + "start": 3380.15, + "duration": 0.0, + "text": "Um and if this means that if we add a" + }, + { + "start": 3380.16, + "duration": 0.0, + "text": "Um and if this means that if we add a parameter<00:56:20.680> that<00:56:20.840> doesn't<00:56:21.160> drastically" + }, + { + "start": 3381.75, + "duration": 0.0, + "text": "parameter that doesn't drastically" + }, + { + "start": 3381.76, + "duration": 0.0, + "text": "parameter that doesn't drastically improve<00:56:22.080> the<00:56:22.200> likelihood,<00:56:23.320> severe<00:56:23.680> drop<00:56:23.960> in" + }, + { + "start": 3384.03, + "duration": 0.0, + "text": "improve the likelihood, severe drop in" + }, + { + "start": 3384.04, + "duration": 0.0, + "text": "improve the likelihood, severe drop in prior<00:56:24.320> density<00:56:25.560> caused<00:56:25.920> by<00:56:26.000> that<00:56:26.160> new" + }, + { + "start": 3386.27, + "duration": 0.0, + "text": "prior density caused by that new" + }, + { + "start": 3386.28, + "duration": 0.0, + "text": "prior density caused by that new dimension<00:56:26.760> will<00:56:26.920> drag<00:56:27.320> the<00:56:27.440> entire<00:56:27.840> integral" + }, + { + "start": 3388.27, + "duration": 0.0, + "text": "dimension will drag the entire integral" + }, + { + "start": 3388.28, + "duration": 0.0, + "text": "dimension will drag the entire integral down." + }, + { + "start": 3389.03, + "duration": 0.0, + "text": "down." + }, + { + "start": 3389.04, + "duration": 0.0, + "text": "down. Basically,<00:56:29.480> the<00:56:29.600> model<00:56:30.040> is" + }, + { + "start": 3390.99, + "duration": 0.0, + "text": "Basically, the model is" + }, + { + "start": 3391.0, + "duration": 0.0, + "text": "Basically, the model is the" + }, + { + "start": 3392.07, + "duration": 0.0, + "text": "the" + }, + { + "start": 3392.08, + "duration": 0.0, + "text": "the the<00:56:32.160> probability<00:56:32.640> calculus<00:56:33.160> is" + }, + { + "start": 3393.35, + "duration": 0.0, + "text": "the probability calculus is" + }, + { + "start": 3393.36, + "duration": 0.0, + "text": "the probability calculus is mathematically<00:56:34.080> penalizing<00:56:35.440> uh<00:56:35.680> models<00:56:36.360> for" + }, + { + "start": 3396.71, + "duration": 0.0, + "text": "mathematically penalizing uh models for" + }, + { + "start": 3396.72, + "duration": 0.0, + "text": "mathematically penalizing uh models for exploring<00:56:37.280> useless<00:56:37.640> dimensions." + }, + { + "start": 3398.79, + "duration": 0.0, + "text": "exploring useless dimensions." + }, + { + "start": 3398.8, + "duration": 0.0, + "text": "exploring useless dimensions. Uh<00:56:39.080> and<00:56:39.240> once<00:56:39.440> again,<00:56:39.720> the<00:56:39.840> frequentist" + }, + { + "start": 3400.31, + "duration": 0.0, + "text": "Uh and once again, the frequentist" + }, + { + "start": 3400.32, + "duration": 0.0, + "text": "Uh and once again, the frequentist methodology<00:56:40.880> cannot<00:56:41.480> do<00:56:41.720> any<00:56:41.960> of<00:56:42.040> this" + }, + { + "start": 3402.27, + "duration": 0.0, + "text": "methodology cannot do any of this" + }, + { + "start": 3402.28, + "duration": 0.0, + "text": "methodology cannot do any of this because<00:56:42.680> they<00:56:42.760> don't<00:56:42.960> assign<00:56:43.200> probabilities" + }, + { + "start": 3403.67, + "duration": 0.0, + "text": "because they don't assign probabilities" + }, + { + "start": 3403.68, + "duration": 0.0, + "text": "because they don't assign probabilities to<00:56:43.800> parameters" + }, + { + "start": 3404.95, + "duration": 0.0, + "text": "to parameters" + }, + { + "start": 3404.96, + "duration": 0.0, + "text": "to parameters and<00:56:45.680> uh<00:56:45.760> also<00:56:46.480> therefore<00:56:46.720> can't<00:56:46.920> use<00:56:47.080> this.<00:56:47.880> Uh" + }, + { + "start": 3407.95, + "duration": 0.0, + "text": "and uh also therefore can't use this. Uh" + }, + { + "start": 3407.96, + "duration": 0.0, + "text": "and uh also therefore can't use this. Uh and<00:56:48.040> they<00:56:48.120> also<00:56:48.320> don't<00:56:48.520> use<00:56:48.680> priors,<00:56:49.080> of" + }, + { + "start": 3409.19, + "duration": 0.0, + "text": "and they also don't use priors, of" + }, + { + "start": 3409.2, + "duration": 0.0, + "text": "and they also don't use priors, of course.<00:56:50.080> Um<00:56:50.640> instead,<00:56:51.320> in<00:56:51.480> order<00:56:51.680> to<00:56:52.520> um" + }, + { + "start": 3412.67, + "duration": 0.0, + "text": "course. Um instead, in order to um" + }, + { + "start": 3412.68, + "duration": 0.0, + "text": "course. Um instead, in order to um somehow" + }, + { + "start": 3413.83, + "duration": 0.0, + "text": "somehow" + }, + { + "start": 3413.84, + "duration": 0.0, + "text": "somehow avoid<00:56:54.320> the<00:56:54.440> overfitting<00:56:55.000> of<00:56:55.120> their<00:56:55.240> models" + }, + { + "start": 3415.59, + "duration": 0.0, + "text": "avoid the overfitting of their models" + }, + { + "start": 3415.6, + "duration": 0.0, + "text": "avoid the overfitting of their models when<00:56:55.720> they<00:56:55.800> do<00:56:55.960> maximum<00:56:56.400> likelihood" + }, + { + "start": 3416.87, + "duration": 0.0, + "text": "when they do maximum likelihood" + }, + { + "start": 3416.88, + "duration": 0.0, + "text": "when they do maximum likelihood estimation,<00:56:58.000> uh<00:56:58.320> they<00:56:58.480> have<00:56:58.640> to<00:56:58.720> invent<00:56:59.080> all" + }, + { + "start": 3419.19, + "duration": 0.0, + "text": "estimation, uh they have to invent all" + }, + { + "start": 3419.2, + "duration": 0.0, + "text": "estimation, uh they have to invent all sorts<00:56:59.440> of<00:56:59.680> ad<00:56:59.840> hoc<00:57:00.360> rules<00:57:00.840> that<00:57:01.200> don't<00:57:01.560> follow" + }, + { + "start": 3421.83, + "duration": 0.0, + "text": "sorts of ad hoc rules that don't follow" + }, + { + "start": 3421.84, + "duration": 0.0, + "text": "sorts of ad hoc rules that don't follow from<00:57:01.960> the<00:57:02.040> rules<00:57:02.200> of<00:57:02.320> probability<00:57:02.920> and" + }, + { + "start": 3423.07, + "duration": 0.0, + "text": "from the rules of probability and" + }, + { + "start": 3423.08, + "duration": 0.0, + "text": "from the rules of probability and therefore<00:57:03.440> are<00:57:03.560> suboptimal<00:57:04.160> or<00:57:04.240> even<00:57:04.440> wrong." + }, + { + "start": 3425.43, + "duration": 0.0, + "text": "therefore are suboptimal or even wrong." + }, + { + "start": 3425.44, + "duration": 0.0, + "text": "therefore are suboptimal or even wrong. And<00:57:05.600> some<00:57:05.760> of<00:57:05.880> these<00:57:06.120> ad<00:57:06.280> hoc<00:57:06.400> fixes<00:57:06.760> are<00:57:06.880> very" + }, + { + "start": 3427.11, + "duration": 0.0, + "text": "And some of these ad hoc fixes are very" + }, + { + "start": 3427.12, + "duration": 0.0, + "text": "And some of these ad hoc fixes are very famous.<00:57:07.560> You're<00:57:07.840> all<00:57:07.960> familiar<00:57:08.280> with<00:57:08.400> them." + }, + { + "start": 3428.55, + "duration": 0.0, + "text": "famous. You're all familiar with them." + }, + { + "start": 3428.56, + "duration": 0.0, + "text": "famous. You're all familiar with them. We<00:57:08.640> have<00:57:08.880> validation<00:57:09.520> data<00:57:09.800> sets," + }, + { + "start": 3430.39, + "duration": 0.0, + "text": "We have validation data sets," + }, + { + "start": 3430.4, + "duration": 0.0, + "text": "We have validation data sets, cross-validation," + }, + { + "start": 3431.91, + "duration": 0.0, + "text": "cross-validation," + }, + { + "start": 3431.92, + "duration": 0.0, + "text": "cross-validation, regularization,<00:57:13.320> uh<00:57:13.440> some<00:57:13.560> kind<00:57:13.680> of" + }, + { + "start": 3433.75, + "duration": 0.0, + "text": "regularization, uh some kind of" + }, + { + "start": 3433.76, + "duration": 0.0, + "text": "regularization, uh some kind of information<00:57:14.200> criterion." + }, + { + "start": 3436.63, + "duration": 0.0, + "text": "information criterion." + }, + { + "start": 3436.64, + "duration": 0.0, + "text": "information criterion. And<00:57:16.680> especially<00:57:17.080> in<00:57:17.240> the<00:57:17.440> neural<00:57:17.680> network<00:57:18.480> uh" + }, + { + "start": 3438.59, + "duration": 0.0, + "text": "And especially in the neural network uh" + }, + { + "start": 3438.6, + "duration": 0.0, + "text": "And especially in the neural network uh side,<00:57:18.840> we<00:57:18.920> have<00:57:19.120> early<00:57:19.320> stopping<00:57:19.800> and<00:57:19.920> dropout" + }, + { + "start": 3440.39, + "duration": 0.0, + "text": "side, we have early stopping and dropout" + }, + { + "start": 3440.4, + "duration": 0.0, + "text": "side, we have early stopping and dropout and<00:57:20.520> all<00:57:20.640> of<00:57:20.720> these<00:57:20.920> other<00:57:21.280> strange<00:57:21.680> and" + }, + { + "start": 3441.79, + "duration": 0.0, + "text": "and all of these other strange and" + }, + { + "start": 3441.8, + "duration": 0.0, + "text": "and all of these other strange and arbitrary<00:57:22.200> rules<00:57:22.480> that<00:57:22.760> somehow<00:57:23.280> try<00:57:23.560> to" + }, + { + "start": 3443.83, + "duration": 0.0, + "text": "arbitrary rules that somehow try to" + }, + { + "start": 3443.84, + "duration": 0.0, + "text": "arbitrary rules that somehow try to avoid" + }, + { + "start": 3445.07, + "duration": 0.0, + "text": "avoid" + }, + { + "start": 3445.08, + "duration": 0.0, + "text": "avoid uh<00:57:25.240> you<00:57:25.360> know,<00:57:25.560> just<00:57:25.840> the<00:57:25.920> model<00:57:26.200> finding" + }, + { + "start": 3447.19, + "duration": 0.0, + "text": "uh you know, just the model finding" + }, + { + "start": 3447.2, + "duration": 0.0, + "text": "uh you know, just the model finding the<00:57:27.360> greediest<00:57:27.960> way<00:57:28.120> to<00:57:28.240> fit<00:57:28.520> all<00:57:28.680> the<00:57:28.760> points" + }, + { + "start": 3449.03, + "duration": 0.0, + "text": "the greediest way to fit all the points" + }, + { + "start": 3449.04, + "duration": 0.0, + "text": "the greediest way to fit all the points in<00:57:29.120> the<00:57:29.160> data<00:57:29.400> perfectly." + }, + { + "start": 3451.83, + "duration": 0.0, + "text": "in the data perfectly." + }, + { + "start": 3451.84, + "duration": 0.0, + "text": "in the data perfectly. And<00:57:32.560> uh<00:57:32.640> so<00:57:32.960> we<00:57:33.160> come<00:57:33.600> to<00:57:33.920> basically<00:57:34.280> the" + }, + { + "start": 3454.35, + "duration": 0.0, + "text": "And uh so we come to basically the" + }, + { + "start": 3454.36, + "duration": 0.0, + "text": "And uh so we come to basically the conclusion<00:57:34.800> of<00:57:34.880> the<00:57:34.920> presentation,<00:57:35.520> which<00:57:35.720> is" + }, + { + "start": 3457.07, + "duration": 0.0, + "text": "conclusion of the presentation, which is" + }, + { + "start": 3457.08, + "duration": 0.0, + "text": "conclusion of the presentation, which is the<00:57:37.160> Bayesian<00:57:37.760> probability<00:57:38.240> calculus" + }, + { + "start": 3458.67, + "duration": 0.0, + "text": "the Bayesian probability calculus" + }, + { + "start": 3458.68, + "duration": 0.0, + "text": "the Bayesian probability calculus teaches<00:57:39.000> us<00:57:39.160> that<00:57:39.280> the<00:57:39.400> best<00:57:39.680> practice<00:57:40.200> or" + }, + { + "start": 3460.91, + "duration": 0.0, + "text": "teaches us that the best practice or" + }, + { + "start": 3460.92, + "duration": 0.0, + "text": "teaches us that the best practice or good<00:57:41.120> practice<00:57:41.560> is<00:57:41.760> simply<00:57:42.680> that" + }, + { + "start": 3463.59, + "duration": 0.0, + "text": "good practice is simply that" + }, + { + "start": 3463.6, + "duration": 0.0, + "text": "good practice is simply that since<00:57:43.880> the<00:57:44.240> rules<00:57:44.560> of<00:57:45.080> manipulating" + }, + { + "start": 3465.59, + "duration": 0.0, + "text": "since the rules of manipulating" + }, + { + "start": 3465.6, + "duration": 0.0, + "text": "since the rules of manipulating probability<00:57:46.560> as<00:57:46.720> a<00:57:46.760> degree<00:57:47.080> of<00:57:47.200> implication" + }, + { + "start": 3468.79, + "duration": 0.0, + "text": "probability as a degree of implication" + }, + { + "start": 3468.8, + "duration": 0.0, + "text": "probability as a degree of implication are<00:57:49.000> unique,<00:57:49.960> it<00:57:50.120> means<00:57:50.400> that<00:57:50.600> any<00:57:50.800> deviation" + }, + { + "start": 3471.31, + "duration": 0.0, + "text": "are unique, it means that any deviation" + }, + { + "start": 3471.32, + "duration": 0.0, + "text": "are unique, it means that any deviation from<00:57:51.520> them<00:57:51.800> is<00:57:52.040> bound<00:57:52.320> to<00:57:52.440> lead<00:57:52.640> to<00:57:52.760> some" + }, + { + "start": 3473.07, + "duration": 0.0, + "text": "from them is bound to lead to some" + }, + { + "start": 3473.08, + "duration": 0.0, + "text": "from them is bound to lead to some suboptimal<00:57:54.400> paradoxical<00:57:55.120> or<00:57:55.240> even<00:57:55.840> really" + }, + { + "start": 3476.15, + "duration": 0.0, + "text": "suboptimal paradoxical or even really" + }, + { + "start": 3476.16, + "duration": 0.0, + "text": "suboptimal paradoxical or even really really<00:57:56.400> incorrect<00:57:56.840> results." + }, + { + "start": 3477.83, + "duration": 0.0, + "text": "really incorrect results." + }, + { + "start": 3477.84, + "duration": 0.0, + "text": "really incorrect results. And<00:57:58.200> that<00:57:58.320> is<00:57:58.520> why<00:57:59.320> it's<00:57:59.560> very<00:57:59.800> important<00:58:00.200> in" + }, + { + "start": 3480.27, + "duration": 0.0, + "text": "And that is why it's very important in" + }, + { + "start": 3480.28, + "duration": 0.0, + "text": "And that is why it's very important in the<00:58:00.360> literature<00:58:00.600> there's<00:58:00.720> great<00:58:01.000> focus<00:58:01.360> on" + }, + { + "start": 3481.47, + "duration": 0.0, + "text": "the literature there's great focus on" + }, + { + "start": 3481.48, + "duration": 0.0, + "text": "the literature there's great focus on somehow<00:58:01.920> not<00:58:02.240> deviating<00:58:02.640> from<00:58:02.760> these<00:58:02.960> rules." + }, + { + "start": 3483.27, + "duration": 0.0, + "text": "somehow not deviating from these rules." + }, + { + "start": 3483.28, + "duration": 0.0, + "text": "somehow not deviating from these rules. And<00:58:03.800> in<00:58:03.920> fact,<00:58:04.240> there's<00:58:04.440> so<00:58:04.680> often<00:58:04.960> when" + }, + { + "start": 3485.07, + "duration": 0.0, + "text": "And in fact, there's so often when" + }, + { + "start": 3485.08, + "duration": 0.0, + "text": "And in fact, there's so often when someone<00:58:05.840> tries<00:58:06.280> to<00:58:06.440> invent<00:58:06.800> a<00:58:06.840> new<00:58:07.080> trick<00:58:08.080> that" + }, + { + "start": 3488.35, + "duration": 0.0, + "text": "someone tries to invent a new trick that" + }, + { + "start": 3488.36, + "duration": 0.0, + "text": "someone tries to invent a new trick that somehow<00:58:08.640> doesn't<00:58:08.960> follow<00:58:09.160> from<00:58:09.280> the<00:58:09.360> rules," + }, + { + "start": 3489.59, + "duration": 0.0, + "text": "somehow doesn't follow from the rules," + }, + { + "start": 3489.6, + "duration": 0.0, + "text": "somehow doesn't follow from the rules, and<00:58:09.680> it<00:58:09.760> doesn't<00:58:10.040> take<00:58:10.240> long<00:58:10.480> for<00:58:10.600> someone<00:58:10.840> to" + }, + { + "start": 3490.95, + "duration": 0.0, + "text": "and it doesn't take long for someone to" + }, + { + "start": 3490.96, + "duration": 0.0, + "text": "and it doesn't take long for someone to find,<00:58:11.240> \"Okay,<00:58:11.480> this<00:58:11.640> actually<00:58:11.880> leads<00:58:12.000> to<00:58:12.120> some" + }, + { + "start": 3492.27, + "duration": 0.0, + "text": "find, \"Okay, this actually leads to some" + }, + { + "start": 3492.28, + "duration": 0.0, + "text": "find, \"Okay, this actually leads to some paradoxes<00:58:12.760> down<00:58:12.920> the<00:58:13.000> road.\"" + }, + { + "start": 3494.11, + "duration": 0.0, + "text": "paradoxes down the road.\"" + }, + { + "start": 3494.12, + "duration": 0.0, + "text": "paradoxes down the road.\" Um" + }, + { + "start": 3495.55, + "duration": 0.0, + "text": "Um" + }, + { + "start": 3495.56, + "duration": 0.0, + "text": "Um And<00:58:15.680> additionally,<00:58:16.480> one<00:58:16.720> needs<00:58:16.880> to<00:58:16.960> be<00:58:17.120> very" + }, + { + "start": 3497.39, + "duration": 0.0, + "text": "And additionally, one needs to be very" + }, + { + "start": 3497.4, + "duration": 0.0, + "text": "And additionally, one needs to be very very<00:58:17.640> careful<00:58:17.920> with<00:58:18.080> handling<00:58:18.400> infinite" + }, + { + "start": 3498.79, + "duration": 0.0, + "text": "very careful with handling infinite" + }, + { + "start": 3498.8, + "duration": 0.0, + "text": "very careful with handling infinite sets.<00:58:19.200> This<00:58:19.400> is<00:58:19.560> what<00:58:19.720> I<00:58:19.760> mentioned<00:58:20.120> at<00:58:20.200> the" + }, + { + "start": 3500.27, + "duration": 0.0, + "text": "sets. This is what I mentioned at the" + }, + { + "start": 3500.28, + "duration": 0.0, + "text": "sets. This is what I mentioned at the beginning." + }, + { + "start": 3501.31, + "duration": 0.0, + "text": "beginning." + }, + { + "start": 3501.32, + "duration": 0.0, + "text": "beginning. And<00:58:21.640> carelessness<00:58:22.120> in<00:58:22.200> this<00:58:22.360> context<00:58:22.840> can" + }, + { + "start": 3502.95, + "duration": 0.0, + "text": "And carelessness in this context can" + }, + { + "start": 3502.96, + "duration": 0.0, + "text": "And carelessness in this context can lead<00:58:23.080> to<00:58:23.200> paradoxes<00:58:23.680> that<00:58:23.800> has<00:58:24.200> that<00:58:24.320> have" + }, + { + "start": 3504.59, + "duration": 0.0, + "text": "lead to paradoxes that has that have" + }, + { + "start": 3504.6, + "duration": 0.0, + "text": "lead to paradoxes that has that have nothing<00:58:25.080> to<00:58:25.240> do" + }, + { + "start": 3506.11, + "duration": 0.0, + "text": "nothing to do" + }, + { + "start": 3506.12, + "duration": 0.0, + "text": "nothing to do with<00:58:26.200> the<00:58:26.280> theory<00:58:26.600> of<00:58:26.720> probability,<00:58:27.280> but" + }, + { + "start": 3507.43, + "duration": 0.0, + "text": "with the theory of probability, but" + }, + { + "start": 3507.44, + "duration": 0.0, + "text": "with the theory of probability, but instead<00:58:27.800> follow<00:58:28.120> from<00:58:28.360> the<00:58:28.680> unsound" + }, + { + "start": 3509.07, + "duration": 0.0, + "text": "instead follow from the unsound" + }, + { + "start": 3509.08, + "duration": 0.0, + "text": "instead follow from the unsound mathematical<00:58:29.520> practices.<00:58:30.600> Some<00:58:30.760> of<00:58:30.800> the" + }, + { + "start": 3510.87, + "duration": 0.0, + "text": "mathematical practices. Some of the" + }, + { + "start": 3510.88, + "duration": 0.0, + "text": "mathematical practices. Some of the famous<00:58:31.200> paradoxes<00:58:31.680> are<00:58:31.840> the" + }, + { + "start": 3512.63, + "duration": 0.0, + "text": "famous paradoxes are the" + }, + { + "start": 3512.64, + "duration": 0.0, + "text": "famous paradoxes are the Bertrand's<00:58:33.040> paradox" + }, + { + "start": 3514.35, + "duration": 0.0, + "text": "Bertrand's paradox" + }, + { + "start": 3514.36, + "duration": 0.0, + "text": "Bertrand's paradox and<00:58:34.560> the<00:58:34.680> Borel-Cantelli<00:58:35.760> Borel-Cantelli" + }, + { + "start": 3516.43, + "duration": 0.0, + "text": "and the Borel-Cantelli Borel-Cantelli" + }, + { + "start": 3516.44, + "duration": 0.0, + "text": "and the Borel-Cantelli Borel-Cantelli paradox.<00:58:37.560> And<00:58:37.840> you<00:58:37.920> can<00:58:38.080> avoid<00:58:38.520> them<00:58:38.760> by" + }, + { + "start": 3518.91, + "duration": 0.0, + "text": "paradox. And you can avoid them by" + }, + { + "start": 3518.92, + "duration": 0.0, + "text": "paradox. And you can avoid them by simply<00:58:39.320> using<00:58:39.640> finite<00:58:40.120> sets<00:58:40.880> or<00:58:41.080> infinite" + }, + { + "start": 3521.47, + "duration": 0.0, + "text": "simply using finite sets or infinite" + }, + { + "start": 3521.48, + "duration": 0.0, + "text": "simply using finite sets or infinite sets<00:58:41.640> that<00:58:41.760> are<00:58:41.840> produced<00:58:42.360> as<00:58:42.640> the" + }, + { + "start": 3522.75, + "duration": 0.0, + "text": "sets that are produced as the" + }, + { + "start": 3522.76, + "duration": 0.0, + "text": "sets that are produced as the well-behaved<00:58:43.240> limit<00:58:43.600> of<00:58:43.800> a<00:58:43.840> sequence<00:58:44.440> of" + }, + { + "start": 3524.59, + "duration": 0.0, + "text": "well-behaved limit of a sequence of" + }, + { + "start": 3524.6, + "duration": 0.0, + "text": "well-behaved limit of a sequence of finite<00:58:44.880> sets.<00:58:45.680> So," + }, + { + "start": 3526.75, + "duration": 0.0, + "text": "finite sets. So," + }, + { + "start": 3526.76, + "duration": 0.0, + "text": "finite sets. So, if<00:58:46.960> you<00:58:47.120> have<00:58:47.320> once<00:58:47.480> again<00:58:47.640> this<00:58:47.800> question<00:58:48.200> of" + }, + { + "start": 3528.35, + "duration": 0.0, + "text": "if you have once again this question of" + }, + { + "start": 3528.36, + "duration": 0.0, + "text": "if you have once again this question of like,<00:58:48.600> \"Okay,<00:58:48.840> you<00:58:48.960> have<00:58:49.160> somehow<00:58:49.800> the<00:58:49.920> real" + }, + { + "start": 3530.15, + "duration": 0.0, + "text": "like, \"Okay, you have somehow the real" + }, + { + "start": 3530.16, + "duration": 0.0, + "text": "like, \"Okay, you have somehow the real numbers<00:58:51.000> in<00:58:51.120> some<00:58:51.280> kind<00:58:51.480> of<00:58:51.560> range,<00:58:51.840> what<00:58:52.000> is" + }, + { + "start": 3532.11, + "duration": 0.0, + "text": "numbers in some kind of range, what is" + }, + { + "start": 3532.12, + "duration": 0.0, + "text": "numbers in some kind of range, what is the<00:58:52.200> probability<00:58:52.720> that<00:58:52.920> between<00:58:53.760> that<00:58:54.000> you" + }, + { + "start": 3534.23, + "duration": 0.0, + "text": "the probability that between that you" + }, + { + "start": 3534.24, + "duration": 0.0, + "text": "the probability that between that you cannot<00:58:54.560> pick<00:58:54.760> out<00:58:54.960> a<00:58:55.000> rational<00:58:55.400> number<00:58:55.680> from" + }, + { + "start": 3535.79, + "duration": 0.0, + "text": "cannot pick out a rational number from" + }, + { + "start": 3535.8, + "duration": 0.0, + "text": "cannot pick out a rational number from this<00:58:56.080> from<00:58:56.280> the<00:58:56.360> space?\"" + }, + { + "start": 3537.31, + "duration": 0.0, + "text": "this from the space?\"" + }, + { + "start": 3537.32, + "duration": 0.0, + "text": "this from the space?\" The<00:58:57.560> the<00:58:57.680> frequentists<00:58:58.200> or<00:58:58.280> measure" + }, + { + "start": 3538.59, + "duration": 0.0, + "text": "The the frequentists or measure" + }, + { + "start": 3538.6, + "duration": 0.0, + "text": "The the frequentists or measure theoretic<00:58:59.400> probability<00:58:59.880> calculus<00:59:00.200> will<00:59:00.320> tell" + }, + { + "start": 3540.47, + "duration": 0.0, + "text": "theoretic probability calculus will tell" + }, + { + "start": 3540.48, + "duration": 0.0, + "text": "theoretic probability calculus will tell you<00:59:00.560> the<00:59:00.640> probability<00:59:01.120> is<00:59:01.240> zero." + }, + { + "start": 3542.15, + "duration": 0.0, + "text": "you the probability is zero." + }, + { + "start": 3542.16, + "duration": 0.0, + "text": "you the probability is zero. But<00:59:02.240> that<00:59:02.440> is<00:59:02.640> of<00:59:02.760> course<00:59:02.960> absurd<00:59:03.360> because<00:59:03.680> it" + }, + { + "start": 3543.79, + "duration": 0.0, + "text": "But that is of course absurd because it" + }, + { + "start": 3543.8, + "duration": 0.0, + "text": "But that is of course absurd because it is<00:59:03.960> possible<00:59:04.440> to<00:59:04.560> draw<00:59:05.280> a<00:59:05.360> rational<00:59:05.720> number." + }, + { + "start": 3546.11, + "duration": 0.0, + "text": "is possible to draw a rational number." + }, + { + "start": 3546.12, + "duration": 0.0, + "text": "is possible to draw a rational number. And<00:59:06.240> so,<00:59:06.920> this<00:59:07.080> is<00:59:07.680> an<00:59:07.800> example<00:59:08.400> of<00:59:08.680> a<00:59:09.200> of<00:59:09.360> a" + }, + { + "start": 3549.39, + "duration": 0.0, + "text": "And so, this is an example of a of a" + }, + { + "start": 3549.4, + "duration": 0.0, + "text": "And so, this is an example of a of a result<00:59:09.840> that" + }, + { + "start": 3550.59, + "duration": 0.0, + "text": "result that" + }, + { + "start": 3550.6, + "duration": 0.0, + "text": "result that leads<00:59:10.880> to<00:59:10.960> paradoxes,<00:59:12.000> and<00:59:12.120> it<00:59:12.200> is<00:59:12.320> actually" + }, + { + "start": 3552.67, + "duration": 0.0, + "text": "leads to paradoxes, and it is actually" + }, + { + "start": 3552.68, + "duration": 0.0, + "text": "leads to paradoxes, and it is actually just<00:59:12.840> a<00:59:12.880> consequence<00:59:13.480> of<00:59:13.600> an<00:59:13.760> ill-posed" + }, + { + "start": 3554.23, + "duration": 0.0, + "text": "just a consequence of an ill-posed" + }, + { + "start": 3554.24, + "duration": 0.0, + "text": "just a consequence of an ill-posed question<00:59:14.680> and<00:59:14.840> has<00:59:15.000> nothing<00:59:15.320> to<00:59:15.440> do" + }, + { + "start": 3556.39, + "duration": 0.0, + "text": "question and has nothing to do" + }, + { + "start": 3556.4, + "duration": 0.0, + "text": "question and has nothing to do with<00:59:17.120> the<00:59:17.200> probability<00:59:17.640> calculus<00:59:18.080> per<00:59:18.280> se." + }, + { + "start": 3559.55, + "duration": 0.0, + "text": "with the probability calculus per se." + }, + { + "start": 3559.56, + "duration": 0.0, + "text": "with the probability calculus per se. Um<00:59:19.960> yes." + }, + { + "start": 3561.19, + "duration": 0.0, + "text": "Um yes." + }, + { + "start": 3561.2, + "duration": 0.0, + "text": "Um yes. And<00:59:21.520> there<00:59:21.640> we<00:59:21.720> go.<00:59:22.120> That's<00:59:22.320> it." + }, + { + "start": 3563.79, + "duration": 0.0, + "text": "And there we go. That's it." + }, + { + "start": 3563.8, + "duration": 0.0, + "text": "And there we go. That's it. If<00:59:23.840> you<00:59:23.920> have<00:59:24.000> any<00:59:24.120> questions,<00:59:24.600> feel<00:59:24.760> free<00:59:24.880> to" + }, + { + "start": 3564.99, + "duration": 0.0, + "text": "If you have any questions, feel free to" + }, + { + "start": 3565.0, + "duration": 0.0, + "text": "If you have any questions, feel free to to" + }, + { + "start": 3567.16, + "duration": 0.0, + "text": "I<00:59:27.240> hope<00:59:27.360> it<00:59:27.480> was<00:59:27.600> interesting." + }, + { + "start": 3571.32, + "duration": 0.0, + "text": "Oh,<00:59:31.440> you<00:59:31.520> couldn't<00:59:31.760> see<00:59:31.840> the<00:59:31.960> slides?" + } + ], + "plain": "So,<00:00:01.120> we're<00:00:01.200> going<00:00:01.320> to<00:00:01.400> talk<00:00:01.600> about\nSo, we're going to talk about\nSo, we're going to talk about probability<00:00:02.560> today<00:00:03.160> and<00:00:03.520> we're<00:00:03.600> going<00:00:03.800> to\nprobability today and we're going to\nprobability today and we're going to give<00:00:04.360> a<00:00:04.440> very<00:00:04.584> [snorts]\ngive a very [snorts]\ngive a very [snorts] um\num\num overlooked<00:00:07.040> and<00:00:07.160> underdeveloped<00:00:07.680> approach\noverlooked and underdeveloped approach\noverlooked and underdeveloped approach that<00:00:08.960> sees<00:00:09.280> probability<00:00:10.200> theory<00:00:10.680> as<00:00:10.920> an\nthat sees probability theory as an\nthat sees probability theory as an extension<00:00:11.400> of<00:00:11.520> logic.\nextension of logic.\nextension of logic. Famously,<00:00:13.280> one<00:00:13.440> of<00:00:13.520> the<00:00:14.000> first\nFamously, one of the first\nFamously, one of the first scientists<00:00:15.920> and<00:00:16.040> mathematicians<00:00:16.600> to<00:00:16.920> develop\nscientists and mathematicians to develop\nscientists and mathematicians to develop this<00:00:17.440> idea<00:00:17.720> was<00:00:17.880> Laplace,<00:00:18.440> who<00:00:18.680> in<00:00:18.840> 1819<00:00:19.360> said,\nthis idea was Laplace, who in 1819 said,\nthis idea was Laplace, who in 1819 said, \"Probability<00:00:20.120> theory<00:00:20.520> is<00:00:20.680> nothing<00:00:20.960> but\n\"Probability theory is nothing but\n\"Probability theory is nothing but common<00:00:21.440> sense<00:00:22.080> reduced<00:00:22.520> to<00:00:22.600> calculation.\"\ncommon sense reduced to calculation.\"\ncommon sense reduced to calculation.\" And<00:00:24.920> we<00:00:25.000> will<00:00:25.120> see<00:00:25.240> today<00:00:25.520> what<00:00:25.640> that<00:00:25.800> means\nAnd we will see today what that means\nAnd we will see today what that means exactly.\nexactly.\nexactly. So,<00:00:27.760> first<00:00:27.920> we're<00:00:28.000> going<00:00:28.120> to<00:00:28.200> look<00:00:28.360> at<00:00:28.600> the\nSo, first we're going to look at the\nSo, first we're going to look at the different<00:00:28.880> definitions<00:00:29.280> of<00:00:29.320> probability.\ndifferent definitions of probability.\ndifferent definitions of probability. We're<00:00:30.120> going<00:00:30.240> to<00:00:30.320> talk<00:00:30.520> about<00:00:30.720> some<00:00:30.880> classical\nWe're going to talk about some classical\nWe're going to talk about some classical logic,<00:00:32.160> then<00:00:32.599> some<00:00:32.800> lattice<00:00:33.120> theory<00:00:33.640> because\nlogic, then some lattice theory because\nlogic, then some lattice theory because this<00:00:34.120> is<00:00:34.240> how<00:00:34.480> we're<00:00:34.600> going<00:00:34.800> to\nthis is how we're going to\nthis is how we're going to derive<00:00:36.000> our<00:00:36.120> foundations.\nderive our foundations.\nderive our foundations. We're<00:00:37.680> going<00:00:37.800> to<00:00:38.280> derive<00:00:38.640> the<00:00:38.720> famous<00:00:39.080> sum\nWe're going to derive the famous sum\nWe're going to derive the famous sum rule<00:00:39.760> and<00:00:39.920> the<00:00:40.000> product<00:00:40.360> rules<00:00:40.640> of\nrule and the product rules of\nrule and the product rules of probability<00:00:41.440> that<00:00:41.600> you<00:00:41.720> all<00:00:41.880> know.<00:00:42.720> Uh<00:00:43.200> we're\nprobability that you all know. Uh we're\nprobability that you all know. Uh we're going<00:00:43.440> to<00:00:43.520> talk<00:00:43.680> about<00:00:44.720> how<00:00:44.840> this<00:00:45.040> leads<00:00:45.280> to\ngoing to talk about how this leads to\ngoing to talk about how this leads to Bayesian<00:00:45.720> inference<00:00:46.360> with<00:00:46.840> Bayes'<00:00:47.160> rule<00:00:47.880> and\nBayesian inference with Bayes' rule and\nBayesian inference with Bayes' rule and then<00:00:48.360> some<00:00:48.680> unique<00:00:48.960> powers<00:00:49.680> of<00:00:50.080> Bayesian\nthen some unique powers of Bayesian\nthen some unique powers of Bayesian inference.\ninference.\ninference. All<00:00:52.760> right.<00:00:53.120> So,\nAll right. So,\nAll right. So, nowadays<00:00:55.480> there<00:00:55.720> is<00:00:56.680> two<00:00:57.280> big<00:00:58.120> definitions<00:00:58.760> of\nnowadays there is two big definitions of\nnowadays there is two big definitions of probability<00:00:59.680> that<00:00:59.880> kind<00:01:00.040> of<00:01:00.360> contend<00:01:00.880> for<00:01:01.880> the\nprobability that kind of contend for the\nprobability that kind of contend for the spot<00:01:02.320> of<00:01:02.400> being<00:01:02.560> correct.\nspot of being correct.\nspot of being correct. And<00:01:04.239> that<00:01:04.400> is<00:01:04.600> the<00:01:05.040> frequentist\nAnd that is the frequentist\nAnd that is the frequentist uh\nuh\nuh interpretation,<00:01:07.880> which<00:01:08.080> sees<00:01:08.320> probability\ninterpretation, which sees probability\ninterpretation, which sees probability as<00:01:09.080> sort<00:01:09.280> of<00:01:09.680> the<00:01:09.800> limit<00:01:10.120> of<00:01:10.240> the<00:01:10.320> frequency<00:01:11.520> uh\nas sort of the limit of the frequency uh\nas sort of the limit of the frequency uh of<00:01:11.920> an<00:01:12.000> event<00:01:12.280> happening,<00:01:13.200> and<00:01:13.640> the\nof an event happening, and the\nof an event happening, and the plausibility<00:01:14.400> approach,<00:01:14.840> which<00:01:14.960> is<00:01:15.040> the\nplausibility approach, which is the\nplausibility approach, which is the Bayesian<00:01:15.400> approach,<00:01:15.840> which<00:01:16.080> sees\nBayesian approach, which sees\nBayesian approach, which sees probability<00:01:17.360> simply<00:01:18.440> as<00:01:18.640> a<00:01:18.680> quantification\nprobability simply as a quantification\nprobability simply as a quantification of<00:01:20.720> how<00:01:21.400> plausible<00:01:22.600> an<00:01:22.720> event<00:01:23.200> or<00:01:23.320> a\nof how plausible an event or a\nof how plausible an event or a proposition<00:01:23.920> is<00:01:24.480> given<00:01:25.200> our<00:01:25.480> state<00:01:25.680> of\nproposition is given our state of\nproposition is given our state of knowledge<00:01:26.160> or<00:01:26.480> our<00:01:26.560> state<00:01:26.720> of<00:01:26.840> ignorance,\nknowledge or our state of ignorance,\nknowledge or our state of ignorance, depending<00:01:27.680> on<00:01:27.840> how<00:01:28.040> you<00:01:28.400> look<00:01:28.560> at<00:01:28.640> it.\ndepending on how you look at it.\ndepending on how you look at it. So,<00:01:30.000> for<00:01:30.200> example,<00:01:31.160> imagine<00:01:32.080> um<00:01:32.640> that<00:01:32.800> we're\nSo, for example, imagine um that we're\nSo, for example, imagine um that we're doing<00:01:33.160> the<00:01:33.240> very<00:01:33.600> simple<00:01:33.840> experiment<00:01:34.200> of\ndoing the very simple experiment of\ndoing the very simple experiment of tossing<00:01:34.600> a<00:01:34.640> coin.\ntossing a coin.\ntossing a coin. And<00:01:36.040> imagine<00:01:36.320> this<00:01:36.480> is<00:01:36.640> just<00:01:36.840> a<00:01:36.880> regular<00:01:37.200> coin,\nAnd imagine this is just a regular coin,\nAnd imagine this is just a regular coin, it's<00:01:37.720> a<00:01:37.760> fair<00:01:38.040> coin,<00:01:38.360> you<00:01:38.440> know,<00:01:38.560> nothing\nit's a fair coin, you know, nothing\nit's a fair coin, you know, nothing weird<00:01:39.120> is<00:01:39.200> going<00:01:39.520> on.<00:01:40.160> Why<00:01:40.320> do<00:01:40.440> we<00:01:40.560> say<00:01:40.760> the\nweird is going on. Why do we say the\nweird is going on. Why do we say the probability<00:01:41.440> is<00:01:41.600> 50%<00:01:42.440> uh<00:01:42.640> 50%?<00:01:43.720> The\nprobability is 50% uh 50%? The\nprobability is 50% uh 50%? The frequentists<00:01:44.320> would<00:01:44.480> say<00:01:44.720> that<00:01:44.960> because<00:01:45.760> if\nfrequentists would say that because if\nfrequentists would say that because if you<00:01:46.840> keep<00:01:47.400> flipping<00:01:47.720> the<00:01:47.800> coins,<00:01:48.800> the<00:01:49.080> ratio\nyou keep flipping the coins, the ratio\nyou keep flipping the coins, the ratio of<00:01:49.680> the<00:01:49.800> two<00:01:50.000> outcomes<00:01:50.840> will<00:01:51.120> eventually<00:01:52.120> uh\nof the two outcomes will eventually uh\nof the two outcomes will eventually uh approach<00:01:52.920> one,<00:01:53.280> meaning<00:01:53.960> uh\napproach one, meaning uh\napproach one, meaning uh that<00:01:54.840> the<00:01:54.920> probability<00:01:55.400> that<00:01:55.600> either<00:01:55.760> one<00:01:55.960> the\nthat the probability that either one the\nthat the probability that either one the the<00:01:56.200> fraction<00:01:56.640> of<00:01:56.720> either<00:01:56.920> of<00:01:57.040> one<00:01:57.200> happens\nthe fraction of either of one happens\nthe fraction of either of one happens approaches<00:01:58.520> one<00:01:58.680> half.\napproaches one half.\napproaches one half. Whereas<00:02:00.160> the<00:02:00.280> Bayesian<00:02:00.760> would<00:02:00.920> say<00:02:01.280> that<00:02:02.120> we\nWhereas the Bayesian would say that we\nWhereas the Bayesian would say that we say<00:02:02.680> the<00:02:02.760> probability<00:02:03.400> is<00:02:03.560> one<00:02:03.720> half<00:02:04.080> because\nsay the probability is one half because\nsay the probability is one half because we<00:02:05.760> don't<00:02:05.960> have<00:02:06.160> any<00:02:06.360> reason<00:02:06.680> for<00:02:06.840> prefer<00:02:07.200> any\nwe don't have any reason for prefer any\nwe don't have any reason for prefer any of<00:02:07.520> the<00:02:07.640> two<00:02:07.760> sides<00:02:08.320> given<00:02:08.600> our<00:02:09.000> ignorance<00:02:09.479> of\nof the two sides given our ignorance of\nof the two sides given our ignorance of the\nthe\nthe properties<00:02:10.800> of<00:02:10.960> the<00:02:11.800> the<00:02:11.880> coin.\nproperties of the the coin.\nproperties of the the coin. And<00:02:13.200> therefore<00:02:13.640> we<00:02:13.800> must<00:02:14.400> by<00:02:14.560> consistency\nAnd therefore we must by consistency\nAnd therefore we must by consistency assign<00:02:15.520> an<00:02:15.640> equal<00:02:15.880> probability<00:02:16.360> weight<00:02:16.680> to\nassign an equal probability weight to\nassign an equal probability weight to both<00:02:17.320> and<00:02:17.440> hence<00:02:18.360> 50%.\nNow<00:02:21.760> there<00:02:22.040> are<00:02:22.120> some<00:02:22.920> important<00:02:23.320> problems\nNow there are some important problems\nNow there are some important problems with<00:02:23.800> the<00:02:23.920> frequentist<00:02:24.320> definition.<00:02:25.400> The\nwith the frequentist definition. The\nwith the frequentist definition. The most<00:02:25.640> important<00:02:26.000> one<00:02:26.160> is<00:02:26.320> it<00:02:26.440> cannot<00:02:26.720> assign\nmost important one is it cannot assign\nmost important one is it cannot assign probabilities<00:02:27.600> to<00:02:27.680> single<00:02:27.959> events.<00:02:28.320> So<00:02:28.720> in\nprobabilities to single events. So in\nprobabilities to single events. So in the<00:02:28.920> frequentist<00:02:29.480> framework<00:02:29.920> which<00:02:30.120> is<00:02:30.640> the\nthe frequentist framework which is the\nthe frequentist framework which is the orthodox<00:02:31.160> framework,<00:02:31.480> the<00:02:31.560> one<00:02:31.720> that<00:02:31.840> you\northodox framework, the one that you\northodox framework, the one that you were<00:02:32.000> taught<00:02:32.959> at<00:02:33.200> lectures<00:02:33.720> with<00:02:33.880> the<00:02:34.840> T<00:02:35.040> tests\nwere taught at lectures with the T tests\nwere taught at lectures with the T tests and<00:02:35.760> and<00:02:35.920> P<00:02:36.080> values<00:02:36.720> and<00:02:36.840> confidence\nand and P values and confidence\nand and P values and confidence intervals,<00:02:38.200> it<00:02:38.880> cannot<00:02:39.080> assign\nintervals, it cannot assign\nintervals, it cannot assign probabilities<00:02:39.680> to<00:02:39.760> single<00:02:40.000> events.<00:02:40.240> So<00:02:40.320> for\nprobabilities to single events. So for\nprobabilities to single events. So for example<00:02:40.760> the<00:02:40.800> statement<00:02:41.160> the<00:02:41.200> probability\nexample the statement the probability\nexample the statement the probability tomorrow<00:02:42.160> it<00:02:42.280> is<00:02:42.440> sunny\ntomorrow it is sunny\ntomorrow it is sunny doesn't<00:02:43.640> really<00:02:43.800> have<00:02:43.920> a<00:02:43.959> meaning<00:02:44.480> in<00:02:44.640> this\ndoesn't really have a meaning in this\ndoesn't really have a meaning in this interpretation.<00:02:45.400> The<00:02:45.480> probability<00:02:45.920> that\ninterpretation. The probability that\ninterpretation. The probability that a<00:02:46.800> certain<00:02:47.040> person<00:02:47.440> wins<00:02:47.680> an<00:02:47.760> election<00:02:48.160> or\na certain person wins an election or\na certain person wins an election or that<00:02:48.360> certain<00:02:48.800> person<00:02:49.120> wins<00:02:49.360> a<00:02:49.440> race<00:02:49.760> doesn't\nthat certain person wins a race doesn't\nthat certain person wins a race doesn't have<00:02:50.120> a<00:02:50.160> meaning<00:02:50.520> because<00:02:51.080> these<00:02:51.280> events<00:02:51.600> are\nhave a meaning because these events are\nhave a meaning because these events are unique.\nunique.\nunique. Also<00:02:53.760> the<00:02:54.080> the<00:02:54.600> notion\nAlso the the notion\nAlso the the notion the<00:02:56.200> validity<00:02:56.560> of<00:02:56.640> this<00:02:56.760> notion<00:02:57.040> of\nthe validity of this notion of\nthe validity of this notion of probability\nprobability\nprobability intuitively<00:02:58.959> relies<00:02:59.480> on<00:02:59.600> the<00:02:59.680> law<00:02:59.920> of<00:03:00.040> large\nintuitively relies on the law of large\nintuitively relies on the law of large numbers<00:03:00.959> which<00:03:01.320> in<00:03:01.440> turn<00:03:01.640> depends<00:03:02.239> of\nnumbers which in turn depends of\nnumbers which in turn depends of previous<00:03:03.080> definition<00:03:03.440> of<00:03:03.480> probability<00:03:03.959> which\nprevious definition of probability which\nprevious definition of probability which creates<00:03:04.600> some<00:03:05.040> circularity<00:03:05.600> issues.\ncreates some circularity issues.\ncreates some circularity issues. Also<00:03:07.680> this<00:03:07.880> is<00:03:08.120> probably<00:03:08.400> the<00:03:08.519> biggest\nAlso this is probably the biggest\nAlso this is probably the biggest the<00:03:09.680> biggest<00:03:10.000> hint<00:03:10.239> that<00:03:10.360> there's<00:03:10.560> something\nthe biggest hint that there's something\nthe biggest hint that there's something off.\noff.\noff. It<00:03:12.400> relies<00:03:12.840> on<00:03:13.040> some\nIt relies on some\nIt relies on some some<00:03:13.959> kind<00:03:14.080> of<00:03:14.200> notion<00:03:14.480> of<00:03:14.600> randomness<00:03:15.239> which\nsome kind of notion of randomness which\nsome kind of notion of randomness which is<00:03:15.760> very<00:03:16.120> hard<00:03:16.440> if<00:03:16.760> not<00:03:17.000> impossible<00:03:17.440> to\nis very hard if not impossible to\nis very hard if not impossible to define.<00:03:18.000> In<00:03:18.080> fact<00:03:18.280> many<00:03:18.480> textbooks<00:03:18.840> say<00:03:19.480> say\ndefine. In fact many textbooks say say\ndefine. In fact many textbooks say say it<00:03:19.800> out<00:03:20.000> loud<00:03:20.239> that<00:03:20.320> there<00:03:20.440> is<00:03:20.519> not<00:03:20.720> really\nit out loud that there is not really\nit out loud that there is not really good<00:03:21.080> definition<00:03:21.440> for<00:03:21.560> this.<00:03:22.080> Also<00:03:22.280> because\ngood definition for this. Also because\ngood definition for this. Also because simultaneously<00:03:23.280> it<00:03:23.400> requires<00:03:23.880> that<00:03:24.040> the\nsimultaneously it requires that the\nsimultaneously it requires that the different<00:03:24.800> events<00:03:25.160> and<00:03:25.239> different\ndifferent events and different\ndifferent events and different repetitions<00:03:26.280> have<00:03:26.519> to<00:03:26.600> be<00:03:26.720> also<00:03:26.920> be<00:03:27.200> equal<00:03:27.760> to\nrepetitions have to be also be equal to\nrepetitions have to be also be equal to the<00:03:27.959> others<00:03:28.320> in<00:03:28.480> some<00:03:28.720> way<00:03:28.880> that<00:03:29.000> is<00:03:29.320> well\nthe others in some way that is well\nthe others in some way that is well equally<00:03:30.120> hard<00:03:30.360> to<00:03:30.440> define.\nequally hard to define.\nequally hard to define. Um\nUm\nUm Also<00:03:33.760> it<00:03:33.880> is<00:03:34.640> you<00:03:34.800> could<00:03:34.920> say\nAlso it is you could say\nAlso it is you could say epistemologically<00:03:35.880> clunky<00:03:36.200> because<00:03:36.640> it<00:03:36.840> ever\nepistemologically clunky because it ever\nepistemologically clunky because it ever only<00:03:37.440> deals<00:03:37.800> with<00:03:37.959> the<00:03:38.080> probability<00:03:38.680> of<00:03:38.840> the\nonly deals with the probability of the\nonly deals with the probability of the data<00:03:39.640> given<00:03:39.880> the<00:03:39.959> hypothesis.<00:03:41.160> This<00:03:41.400> is<00:03:41.680> what\ndata given the hypothesis. This is what\ndata given the hypothesis. This is what you<00:03:42.000> do<00:03:42.360> when<00:03:42.560> you<00:03:42.680> do<00:03:42.880> what's<00:03:43.080> called<00:03:43.440> maximum\nyou do when you do what's called maximum\nyou do when you do what's called maximum likelihood.<00:03:44.840> You<00:03:45.000> just<00:03:45.200> see<00:03:46.000> what<00:03:46.880> values<00:03:47.239> of\nlikelihood. You just see what values of\nlikelihood. You just see what values of the<00:03:47.400> parameters<00:03:47.880> of<00:03:47.959> your<00:03:48.080> model<00:03:48.320> just\nthe parameters of your model just\nthe parameters of your model just maximizes<00:03:49.760> the<00:03:49.840> probability<00:03:50.320> of<00:03:50.400> the<00:03:50.519> data\nmaximizes the probability of the data\nmaximizes the probability of the data given<00:03:51.080> your<00:03:51.280> model.\ngiven your model.\ngiven your model. However,<00:03:52.960> it<00:03:53.080> doesn't<00:03:53.840> uh<00:03:53.920> give<00:03:54.200> you<00:03:54.640> a<00:03:54.720> way<00:03:54.920> to\nHowever, it doesn't uh give you a way to\nHowever, it doesn't uh give you a way to assign<00:03:55.720> probabilities.\nassign probabilities.\nassign probabilities. Sorry,<00:03:58.400> it<00:03:58.480> doesn't<00:03:58.680> have<00:03:58.760> a<00:03:58.800> way<00:03:58.880> to<00:03:59.000> assign\nSorry, it doesn't have a way to assign\nSorry, it doesn't have a way to assign probabilities<00:04:00.120> to<00:04:00.360> a<00:04:00.400> hypothesis<00:04:01.440> because\nprobabilities to a hypothesis because\nprobabilities to a hypothesis because um<00:04:03.120> hypothesis,<00:04:03.640> once<00:04:03.800> again,<00:04:04.280> I<00:04:04.960> are<00:04:05.120> only<00:04:06.080> or\num hypothesis, once again, I are only or\num hypothesis, once again, I are only or the<00:04:07.040> truth<00:04:07.600> is<00:04:07.920> sort<00:04:08.080> of<00:04:08.200> a<00:04:08.280> fixed<00:04:08.640> thing.<00:04:08.880> It's\nthe truth is sort of a fixed thing. It's\nthe truth is sort of a fixed thing. It's a<00:04:09.040> constant<00:04:09.600> and<00:04:09.680> therefore<00:04:10.400> this\na constant and therefore this\na constant and therefore this methodology<00:04:10.959> does<00:04:11.080> not<00:04:11.240> allow<00:04:12.000> you<00:04:12.080> to<00:04:12.200> give\nmethodology does not allow you to give\nmethodology does not allow you to give it<00:04:12.440> a<00:04:12.480> probability.\nit a probability.\nit a probability. Also,<00:04:14.360> it<00:04:14.560> has<00:04:14.800> the<00:04:15.320> sort<00:04:15.520> of<00:04:16.239> methodological\nAlso, it has the sort of methodological\nAlso, it has the sort of methodological uh<00:04:17.840> handicap<00:04:18.320> that<00:04:18.560> it<00:04:19.000> ignores<00:04:19.600> any<00:04:20.359> prior\nuh handicap that it ignores any prior\nuh handicap that it ignores any prior information<00:04:21.079> or<00:04:21.160> prior<00:04:21.400> knowledge<00:04:21.760> from\ninformation or prior knowledge from\ninformation or prior knowledge from either<00:04:22.160> previous<00:04:22.440> estimation<00:04:23.080> or\neither previous estimation or\neither previous estimation or uh<00:04:24.640> prior<00:04:24.840> results<00:04:25.240> in<00:04:25.360> a<00:04:25.400> completely\nuh prior results in a completely\nuh prior results in a completely different<00:04:26.640> experiment.<00:04:27.800> And<00:04:28.000> it<00:04:28.120> often\ndifferent experiment. And it often\ndifferent experiment. And it often provides<00:04:28.800> only<00:04:28.960> point<00:04:29.280> estimates,<00:04:29.800> which\nprovides only point estimates, which\nprovides only point estimates, which makes<00:04:30.640> it<00:04:31.160> very<00:04:31.360> hard<00:04:31.520> if<00:04:31.600> not<00:04:31.720> impossible<00:04:32.160> to\nmakes it very hard if not impossible to\nmakes it very hard if not impossible to assess<00:04:32.680> the<00:04:32.800> uncertainty<00:04:33.520> of<00:04:34.120> the<00:04:34.240> result.\nassess the uncertainty of the result.\nassess the uncertainty of the result. Uh<00:04:36.440> in<00:04:36.560> an<00:04:36.680> in<00:04:36.800> an<00:04:36.880> attempt<00:04:37.120> to<00:04:37.200> circumvent\nUh in an in an attempt to circumvent\nUh in an in an attempt to circumvent these<00:04:37.800> issues,<00:04:38.320> the<00:04:38.400> methodology<00:04:38.919> has<00:04:39.560> uh\nthese issues, the methodology has uh\nthese issues, the methodology has uh made<00:04:40.520> it<00:04:40.600> so<00:04:40.680> that<00:04:40.800> scientists<00:04:41.320> now<00:04:41.600> commonly\nmade it so that scientists now commonly\nmade it so that scientists now commonly reason<00:04:42.480> about<00:04:43.400> uh\nreason about uh\nreason about uh what<00:04:44.200> the<00:04:44.280> world<00:04:44.600> would<00:04:44.720> be<00:04:44.960> in<00:04:45.160> different\nwhat the world would be in different\nwhat the world would be in different imaginary<00:04:46.200> universes.<00:04:46.800> So,<00:04:46.919> for<00:04:47.040> example,<00:04:47.440> if\nimaginary universes. So, for example, if\nimaginary universes. So, for example, if you<00:04:47.640> wanted<00:04:47.880> to<00:04:47.960> estimate<00:04:48.520> the<00:04:48.640> mass<00:04:48.960> of<00:04:49.080> a\nyou wanted to estimate the mass of a\nyou wanted to estimate the mass of a planet,<00:04:50.160> uh<00:04:50.680> this<00:04:50.840> methodology<00:04:51.320> in<00:04:51.440> per<00:04:51.640> se\nplanet, uh this methodology in per se\nplanet, uh this methodology in per se would<00:04:52.040> not<00:04:52.200> allow<00:04:52.520> you<00:04:52.680> to<00:04:52.880> assign<00:04:53.160> a\nwould not allow you to assign a\nwould not allow you to assign a probability<00:04:53.840> to<00:04:53.960> this<00:04:54.160> value<00:04:54.480> because<00:04:54.720> it's<00:04:54.840> a\nprobability to this value because it's a\nprobability to this value because it's a constant.<00:04:55.440> There<00:04:55.560> is<00:04:55.680> only<00:04:55.840> one<00:04:56.000> true<00:04:56.200> value.\nconstant. There is only one true value.\nconstant. There is only one true value. And<00:04:57.200> so,<00:04:57.440> it<00:04:57.560> forces<00:04:57.840> you<00:04:57.919> to<00:04:58.040> imagine,<00:04:58.520> oh,\nAnd so, it forces you to imagine, oh,\nAnd so, it forces you to imagine, oh, what<00:04:58.960> if<00:04:59.160> there<00:04:59.280> were<00:04:59.480> different<00:04:59.840> worlds<00:05:00.400> in\nwhat if there were different worlds in\nwhat if there were different worlds in which<00:05:01.160> different<00:05:01.440> version<00:05:01.760> of<00:05:01.840> this<00:05:02.000> planet\nwhich different version of this planet\nwhich different version of this planet uh<00:05:02.760> existed?<00:05:03.760> And<00:05:03.960> so,<00:05:04.200> you<00:05:04.320> reason<00:05:04.600> about\nuh existed? And so, you reason about\nuh existed? And so, you reason about this.<00:05:05.640> And<00:05:06.480> uh<00:05:06.640> I'm<00:05:06.840> I'm<00:05:06.919> sure<00:05:07.120> you<00:05:07.240> are<00:05:07.400> all\nthis. And uh I'm I'm sure you are all\nthis. And uh I'm I'm sure you are all familiar<00:05:08.120> with<00:05:08.680> uh<00:05:08.720> what<00:05:08.840> a<00:05:08.919> p-value<00:05:09.520> is.<00:05:09.960> Uh\nfamiliar with uh what a p-value is. Uh\nfamiliar with uh what a p-value is. Uh And<00:05:10.880> there's<00:05:11.120> a<00:05:11.200> this<00:05:11.360> famous<00:05:11.680> critique<00:05:12.040> by\nAnd there's a this famous critique by\nAnd there's a this famous critique by Sir<00:05:12.400> Havel<00:05:12.919> Sir<00:05:13.160> Harold<00:05:13.480> Jeffreys<00:05:14.560> which<00:05:14.720> said\nSir Havel Sir Harold Jeffreys which said\nSir Havel Sir Harold Jeffreys which said that<00:05:15.200> what<00:05:15.400> p<00:05:15.640> implies<00:05:16.160> is<00:05:16.280> that<00:05:16.480> the\nthat what p implies is that the\nthat what p implies is that the hypothesis<00:05:17.200> that<00:05:17.440> may<00:05:17.600> be<00:05:17.720> true<00:05:18.520> may<00:05:18.600> be\nhypothesis that may be true may be\nhypothesis that may be true may be rejected<00:05:19.160> because<00:05:19.400> it<00:05:19.480> does<00:05:19.640> not<00:05:20.000> predict<00:05:20.320> it\nrejected because it does not predict it\nrejected because it does not predict it observable<00:05:20.880> results<00:05:21.240> that<00:05:21.400> have<00:05:21.560> not\nobservable results that have not\nobservable results that have not occurred.\noccurred.\noccurred. So,<00:05:23.400> uh<00:05:23.840> just<00:05:24.000> as<00:05:24.080> a<00:05:24.120> reminder,<00:05:24.480> the<00:05:24.600> p-value\nSo, uh just as a reminder, the p-value\nSo, uh just as a reminder, the p-value is<00:05:25.560> the<00:05:25.680> probability<00:05:26.360> that<00:05:26.560> you<00:05:26.640> get<00:05:26.800> a\nis the probability that you get a\nis the probability that you get a statistic<00:05:27.520> that<00:05:27.800> is<00:05:28.560> equal<00:05:28.760> to<00:05:28.840> the<00:05:28.960> one<00:05:29.080> that\nstatistic that is equal to the one that\nstatistic that is equal to the one that you<00:05:29.280> got<00:05:29.800> from<00:05:29.960> the<00:05:30.040> data<00:05:30.400> or<00:05:30.680> more<00:05:30.960> extreme,\nyou got from the data or more extreme,\nyou got from the data or more extreme, so<00:05:31.680> further<00:05:31.960> up<00:05:32.080> the<00:05:32.200> tails.<00:05:33.080> And<00:05:33.280> while<00:05:33.600> this\nso further up the tails. And while this\nso further up the tails. And while this intuitively<00:05:34.400> seems<00:05:34.640> like<00:05:34.760> a<00:05:34.800> good<00:05:35.040> reason<00:05:35.400> to\nintuitively seems like a good reason to\nintuitively seems like a good reason to reject<00:05:35.880> the<00:05:35.960> null<00:05:36.160> hypothesis,\nreject the null hypothesis,\nreject the null hypothesis, uh<00:05:37.720> it<00:05:37.840> also<00:05:38.080> means<00:05:38.320> that<00:05:38.440> the<00:05:38.520> null\nuh it also means that the null\nuh it also means that the null hypothesis<00:05:39.320> has<00:05:39.760> failed<00:05:40.320> to<00:05:40.440> produce<00:05:41.000> data\nhypothesis has failed to produce data\nhypothesis has failed to produce data that<00:05:41.760> is<00:05:41.960> further<00:05:42.240> up<00:05:42.360> the<00:05:42.440> tail.<00:05:42.760> So,<00:05:43.640> uh<00:05:43.840> it's\nthat is further up the tail. So, uh it's\nthat is further up the tail. So, uh it's actually<00:05:44.480> not<00:05:44.880> entirely<00:05:45.240> clear<00:05:45.600> why<00:05:46.160> this\nactually not entirely clear why this\nactually not entirely clear why this should<00:05:46.600> be<00:05:46.720> a<00:05:46.760> good<00:05:47.000> criterion<00:05:47.480> to<00:05:47.600> reject<00:05:47.880> it.\nshould be a good criterion to reject it.\nshould be a good criterion to reject it. Uh<00:05:49.280> and<00:05:49.400> also<00:05:49.640> it's<00:05:49.800> equally\nUh and also it's equally\nUh and also it's equally it's<00:05:51.120> equally<00:05:51.360> problematic<00:05:52.000> to<00:05:52.160> say<00:05:52.560> why<00:05:53.400> we\nit's equally problematic to say why we\nit's equally problematic to say why we can<00:05:53.800> interpret<00:05:54.160> the<00:05:54.240> confidence<00:05:54.720> interval<00:05:55.200> at\ncan interpret the confidence interval at\ncan interpret the confidence interval at all<00:05:55.480> because<00:05:55.760> a<00:05:55.800> confidence<00:05:56.280> interval\nall because a confidence interval\nall because a confidence interval doesn't<00:05:56.960> tell<00:05:57.200> us\ndoesn't tell us\ndoesn't tell us the<00:05:58.120> probability<00:05:58.920> of<00:05:59.560> the<00:06:00.120> null<00:06:00.720> being<00:06:00.960> true\nthe probability of the null being true\nthe probability of the null being true or<00:06:01.320> the<00:06:01.400> alternative<00:06:01.760> hypothesis<00:06:02.200> being\nor the alternative hypothesis being\nor the alternative hypothesis being true.<00:06:02.720> It<00:06:02.840> only<00:06:03.040> tells<00:06:03.320> us<00:06:03.440> that<00:06:03.680> if<00:06:04.560> we<00:06:04.760> repeat\ntrue. It only tells us that if we repeat\ntrue. It only tells us that if we repeat this<00:06:05.280> experiment<00:06:05.720> infinitely<00:06:06.080> many<00:06:06.240> times,\nthis experiment infinitely many times,\nthis experiment infinitely many times, then<00:06:07.040> say<00:06:07.240> 95%<00:06:07.880> of<00:06:07.960> the<00:06:08.040> time<00:06:08.360> or<00:06:08.480> whatever\nthen say 95% of the time or whatever\nthen say 95% of the time or whatever interval<00:06:09.560> we<00:06:09.640> choose,<00:06:10.440> the<00:06:10.520> statistics<00:06:10.960> is\ninterval we choose, the statistics is\ninterval we choose, the statistics is going<00:06:11.240> to<00:06:11.320> be<00:06:11.400> the<00:06:11.520> statistic<00:06:11.880> that<00:06:12.000> we<00:06:12.080> look\ngoing to be the statistic that we look\ngoing to be the statistic that we look is<00:06:12.320> going<00:06:12.440> to<00:06:12.520> be<00:06:12.600> in<00:06:12.760> there.\nSo,<00:06:15.520> again,<00:06:15.720> we<00:06:15.800> have<00:06:15.880> this<00:06:16.040> inversion\nSo, again, we have this inversion\nSo, again, we have this inversion problem<00:06:16.760> from<00:06:17.120> we<00:06:17.240> have<00:06:17.400> probability<00:06:17.840> of<00:06:17.919> the\nproblem from we have probability of the\nproblem from we have probability of the data<00:06:18.440> given<00:06:18.680> a<00:06:18.720> hypothesis,<00:06:19.280> but<00:06:19.400> not<00:06:19.520> the\ndata given a hypothesis, but not the\ndata given a hypothesis, but not the probability<00:06:19.960> of<00:06:20.040> the<00:06:20.120> hypothesis<00:06:20.840> given<00:06:21.120> the\nprobability of the hypothesis given the\nprobability of the hypothesis given the data.\ndata.\ndata. Now,<00:06:23.240> we're<00:06:23.360> going<00:06:23.480> to<00:06:23.520> talk<00:06:23.720> about<00:06:24.160> the\nNow, we're going to talk about the\nNow, we're going to talk about the alternative,<00:06:25.280> um\nalternative, um\nalternative, um which<00:06:26.560> has<00:06:26.760> been<00:06:26.880> famously<00:06:27.280> advocated<00:06:28.200> in<00:06:28.480> the\nwhich has been famously advocated in the\nwhich has been famously advocated in the last<00:06:28.760> century<00:06:29.080> by<00:06:29.919> Edwin<00:06:30.160> Thompson<00:06:30.440> James,\nlast century by Edwin Thompson James,\nlast century by Edwin Thompson James, legendary<00:06:31.400> physicist<00:06:31.960> and<00:06:32.160> statistician.\nlegendary physicist and statistician.\nlegendary physicist and statistician. And<00:06:33.960> to<00:06:34.040> do<00:06:34.160> that,<00:06:34.480> we're<00:06:34.600> going<00:06:34.720> to<00:06:34.800> do<00:06:35.000> first\nAnd to do that, we're going to do first\nAnd to do that, we're going to do first of<00:06:35.320> all<00:06:35.440> a<00:06:35.480> thought<00:06:35.560> experiment,<00:06:36.040> and<00:06:36.240> I\nof all a thought experiment, and I\nof all a thought experiment, and I really<00:06:36.919> want<00:06:37.080> you<00:06:37.160> to<00:06:37.840> think<00:06:38.040> along<00:06:38.240> with\nreally want you to think along with\nreally want you to think along with this.<00:06:38.919> Suppose<00:06:39.280> that<00:06:39.440> you<00:06:39.600> are<00:06:39.760> a<00:06:39.800> policeman.\nthis. Suppose that you are a policeman.\nthis. Suppose that you are a policeman. Uh<00:06:40.919> you<00:06:41.240> patrol<00:06:41.600> the<00:06:41.680> streets<00:06:41.960> at<00:06:42.080> night,<00:06:42.560> and\nUh you patrol the streets at night, and\nUh you patrol the streets at night, and suddenly<00:06:43.240> you<00:06:43.360> hear<00:06:43.520> an<00:06:43.640> alarm.<00:06:44.520> Uh<00:06:44.600> there's<00:06:44.760> a\nsuddenly you hear an alarm. Uh there's a\nsuddenly you hear an alarm. Uh there's a jewelry<00:06:45.160> store,<00:06:45.600> and<00:06:45.840> a<00:06:46.080> a<00:06:46.120> man<00:06:46.440> with<00:06:46.560> a<00:06:46.600> mask\njewelry store, and a a man with a mask\njewelry store, and a a man with a mask comes<00:06:47.320> out<00:06:47.560> with<00:06:47.720> a<00:06:48.520> with<00:06:48.640> a<00:06:48.720> bag<00:06:48.960> on<00:06:49.040> his\ncomes out with a with a bag on his\ncomes out with a with a bag on his shoulder.<00:06:49.880> You<00:06:50.040> arrest<00:06:50.400> him,<00:06:50.520> and<00:06:50.600> it<00:06:50.640> turns\nshoulder. You arrest him, and it turns\nshoulder. You arrest him, and it turns out<00:06:51.000> the<00:06:51.080> bag<00:06:51.480> is<00:06:51.680> full<00:06:51.880> of<00:06:52.120> jewels.<00:06:53.040> Now,<00:06:53.200> I\nout the bag is full of jewels. Now, I\nout the bag is full of jewels. Now, I want<00:06:53.440> you<00:06:53.520> to<00:06:53.640> tell<00:06:53.840> me,<00:06:54.000> do<00:06:54.160> you<00:06:54.240> think<00:06:54.520> this\nwant you to tell me, do you think this\nwant you to tell me, do you think this is<00:06:54.800> a<00:06:54.840> crime?<00:06:55.320> Did<00:06:55.480> you<00:06:55.600> witness<00:06:55.840> a<00:06:55.880> crime?\nis a crime? Did you witness a crime?\nis a crime? Did you witness a crime? What<00:06:57.680> would<00:06:57.800> you<00:06:57.880> say?<00:06:58.320> Your<00:06:58.440> gut<00:06:58.680> instinct.\n>> Of<00:07:06.800> course,<00:07:07.000> it's<00:07:07.120> a<00:07:07.160> crime.\n>> Is<00:07:10.200> it<00:07:10.280> a<00:07:10.320> crime<00:07:10.600> or<00:07:10.680> is<00:07:10.800> this<00:07:11.000> man<00:07:11.240> honest?\nIs<00:07:32.520> anybody<00:07:32.800> talking<00:07:33.160> or<00:07:33.280> am<00:07:33.400> I<00:07:33.440> not<00:07:33.600> hearing\nIs anybody talking or am I not hearing\nIs anybody talking or am I not hearing you?\nyou?\nyou? Okay.\n>> That's<00:07:58.760> all<00:07:58.800> right.\n>> That's all right.\n>> That's all right. >> Don't<00:07:59.920> worry.\n>> Don't worry.\n>> Don't worry. >> Okay,<00:08:01.040> I<00:08:01.080> think<00:08:01.280> we<00:08:01.360> can<00:08:01.480> all<00:08:01.600> agree<00:08:01.880> this<00:08:02.200> this\n>> Okay, I think we can all agree this this\n>> Okay, I think we can all agree this this looks<00:08:02.600> like<00:08:02.760> a<00:08:02.800> crime,<00:08:03.160> right?<00:08:03.760> And<00:08:03.840> now<00:08:04.240> my\nlooks like a crime, right? And now my\nlooks like a crime, right? And now my second<00:08:04.760> question,<00:08:05.200> and<00:08:05.280> the<00:08:05.360> more<00:08:05.520> important\nsecond question, and the more important\nsecond question, and the more important question<00:08:06.200> actually,<00:08:06.480> is\nquestion actually, is\nquestion actually, is did<00:08:07.640> you<00:08:07.960> come<00:08:08.200> to<00:08:08.280> this<00:08:08.400> conclusion<00:08:08.840> through\ndid you come to this conclusion through\ndid you come to this conclusion through deduction<00:08:10.160> or<00:08:10.320> through<00:08:10.600> induction?\nNo,<00:08:53.560> yes.<00:08:54.480> Uh<00:08:55.040> it<00:08:55.160> is<00:08:55.360> induction,<00:08:56.000> and<00:08:56.160> you<00:08:56.280> can\nNo, yes. Uh it is induction, and you can\nNo, yes. Uh it is induction, and you can easily<00:08:57.640> see<00:08:57.839> that<00:08:58.080> because<00:08:58.360> you<00:08:58.440> can<00:08:58.640> imagine\neasily see that because you can imagine\neasily see that because you can imagine you<00:08:59.920> can<00:09:00.080> come<00:09:00.280> up<00:09:00.480> with<00:09:00.920> reasons<00:09:01.400> why<00:09:01.480> this\nyou can come up with reasons why this\nyou can come up with reasons why this might<00:09:01.880> have<00:09:02.040> happened<00:09:02.560> that<00:09:02.720> are<00:09:02.800> completely\nmight have happened that are completely\nmight have happened that are completely innocent.<00:09:03.839> I<00:09:03.880> mean,<00:09:04.600> if<00:09:04.720> you<00:09:04.800> want<00:09:04.960> to<00:09:05.520> say<00:09:06.400> uh\ninnocent. I mean, if you want to say uh\ninnocent. I mean, if you want to say uh a<00:09:06.520> particularly<00:09:06.920> ridiculous<00:09:07.360> one,<00:09:07.520> imagine\na particularly ridiculous one, imagine\na particularly ridiculous one, imagine that<00:09:08.040> the\nthat the\nthat the the<00:09:09.120> the<00:09:09.360> the<00:09:09.480> man<00:09:09.760> in<00:09:09.839> the<00:09:09.920> mask<00:09:10.200> is<00:09:10.280> actually\nthe the the man in the mask is actually\nthe the the man in the mask is actually the<00:09:10.640> store<00:09:11.000> owner,<00:09:11.760> and<00:09:12.120> there<00:09:12.240> was<00:09:12.400> a<00:09:12.440> fire\nthe store owner, and there was a fire\nthe store owner, and there was a fire alarm,<00:09:13.120> and<00:09:13.240> he<00:09:13.360> just<00:09:14.040> uh\nalarm, and he just uh\nalarm, and he just uh wanted<00:09:15.000> to<00:09:15.080> save<00:09:15.280> his<00:09:15.440> jewels,<00:09:15.880> and<00:09:16.680> he\nwanted to save his jewels, and he\nwanted to save his jewels, and he somehow<00:09:17.000> forgot<00:09:17.280> his<00:09:17.400> keys,<00:09:17.680> so<00:09:17.760> he<00:09:17.880> saw<00:09:18.040> no\nsomehow forgot his keys, so he saw no\nsomehow forgot his keys, so he saw no other<00:09:18.440> way<00:09:18.760> than<00:09:19.080> to<00:09:19.520> uh<00:09:19.560> break<00:09:19.800> in,<00:09:20.000> and<00:09:20.600> he\nother way than to uh break in, and he\nother way than to uh break in, and he also<00:09:20.839> had<00:09:20.960> the<00:09:21.040> mask<00:09:21.360> on<00:09:21.480> hand<00:09:21.720> to<00:09:21.800> keep\nalso had the mask on hand to keep\nalso had the mask on hand to keep himself<00:09:22.240> warm.<00:09:22.440> You<00:09:22.520> know,<00:09:22.800> you<00:09:22.920> could<00:09:23.080> have\nhimself warm. You know, you could have\nhimself warm. You know, you could have imagined<00:09:23.440> some<00:09:23.560> ridiculous<00:09:24.040> scenario.<00:09:24.440> This\nimagined some ridiculous scenario. This\nimagined some ridiculous scenario. This is<00:09:24.839> possible.<00:09:25.720> So,<00:09:25.800> just<00:09:25.960> by<00:09:26.040> the<00:09:26.120> very<00:09:26.320> fact\nis possible. So, just by the very fact\nis possible. So, just by the very fact that<00:09:26.680> an<00:09:26.760> alternative<00:09:27.200> is<00:09:27.440> possible<00:09:27.880> tells\nthat an alternative is possible tells\nthat an alternative is possible tells you<00:09:28.160> that<00:09:28.280> this<00:09:28.440> cannot<00:09:28.760> be<00:09:28.839> deduction,\nyou that this cannot be deduction,\nyou that this cannot be deduction, because<00:09:30.120> otherwise<00:09:31.040> uh<00:09:31.120> the<00:09:31.200> result<00:09:31.480> would\nbecause otherwise uh the result would\nbecause otherwise uh the result would have<00:09:31.680> been<00:09:31.839> uniquely<00:09:32.200> determined.\nhave been uniquely determined.\nhave been uniquely determined. And<00:09:34.160> uh\nAnd uh\nAnd uh unfortunately,<00:09:35.160> induction<00:09:35.600> gets<00:09:35.800> a<00:09:35.839> bit<00:09:35.960> of<00:09:36.040> a\nunfortunately, induction gets a bit of a\nunfortunately, induction gets a bit of a bad<00:09:36.600> reputation,<00:09:37.400> mainly<00:09:37.800> because<00:09:38.080> of<00:09:38.160> some\nbad reputation, mainly because of some\nbad reputation, mainly because of some philosophers<00:09:39.040> that<00:09:39.200> were<00:09:39.320> not<00:09:39.840> scientists,\nphilosophers that were not scientists,\nphilosophers that were not scientists, but<00:09:40.560> that's<00:09:40.720> another<00:09:40.960> point.<00:09:41.840> Um,\nbut that's another point. Um,\nbut that's another point. Um, but<00:09:43.080> we're<00:09:43.200> going<00:09:43.320> to<00:09:43.360> show<00:09:43.600> that<00:09:44.200> what\nbut we're going to show that what\nbut we're going to show that what science<00:09:44.800> does<00:09:45.280> for<00:09:45.400> the<00:09:45.480> most<00:09:45.760> part<00:09:45.960> is\nscience does for the most part is\nscience does for the most part is actually<00:09:46.440> just<00:09:46.640> induction<00:09:47.160> and<00:09:47.480> deduction<00:09:47.960> is\nactually just induction and deduction is\nactually just induction and deduction is mostly<00:09:48.600> the\nmostly the\nmostly the realm<00:09:50.000> of<00:09:50.160> what<00:09:50.400> mathematics<00:09:50.920> does.\nrealm of what mathematics does.\nrealm of what mathematics does. And\nAnd\nAnd to<00:09:54.000> understand<00:09:54.360> the<00:09:54.440> difference<00:09:54.720> further,\nto understand the difference further,\nto understand the difference further, I'm<00:09:55.360> sure<00:09:55.560> you're<00:09:55.680> all<00:09:55.800> aware<00:09:56.120> of<00:09:56.240> the<00:09:56.320> kind<00:09:56.640> of\nI'm sure you're all aware of the kind of\nI'm sure you're all aware of the kind of reasoning<00:09:57.760> that<00:09:58.080> Sherlock<00:09:58.400> Holmes<00:09:58.720> does.\nreasoning that Sherlock Holmes does.\nreasoning that Sherlock Holmes does. Now,<00:09:59.800> if<00:09:59.920> I<00:09:59.960> remember<00:10:00.240> correctly,<00:10:00.600> he<00:10:00.720> usually\nNow, if I remember correctly, he usually\nNow, if I remember correctly, he usually says<00:10:01.400> this<00:10:01.560> is<00:10:01.680> deduction,<00:10:02.160> but<00:10:02.360> also<00:10:02.720> all<00:10:02.880> of\nsays this is deduction, but also all of\nsays this is deduction, but also all of these<00:10:03.120> kind<00:10:03.360> of<00:10:03.480> like<00:10:03.680> little<00:10:03.880> tricks<00:10:04.200> and\nthese kind of like little tricks and\nthese kind of like little tricks and connections<00:10:04.680> that<00:10:04.800> he<00:10:05.000> makes<00:10:05.560> actually<00:10:05.840> all\nconnections that he makes actually all\nconnections that he makes actually all inductive.\ninductive.\ninductive. And<00:10:07.520> so<00:10:07.880> you<00:10:08.000> can<00:10:08.240> you<00:10:08.360> can<00:10:08.480> tell<00:10:08.680> that\nAnd so you can you can tell that\nAnd so you can you can tell that even<00:10:09.680> though<00:10:09.920> induction<00:10:10.520> is<00:10:10.720> has<00:10:10.880> this\neven though induction is has this\neven though induction is has this reputation<00:10:11.400> of<00:10:11.520> being<00:10:11.880> very<00:10:12.120> weak<00:10:12.360> reasoning,\nreputation of being very weak reasoning,\nreputation of being very weak reasoning, we<00:10:12.920> can<00:10:13.080> be<00:10:13.240> very,<00:10:13.640> very<00:10:13.880> confident<00:10:14.400> about<00:10:14.800> the\nwe can be very, very confident about the\nwe can be very, very confident about the conclusions<00:10:15.320> that<00:10:15.440> we<00:10:15.560> take<00:10:15.720> from<00:10:15.920> it.\nconclusions that we take from it.\nconclusions that we take from it. Now,<00:10:17.960> here<00:10:18.400> are<00:10:18.600> the<00:10:19.200> most<00:10:19.480> fundamental<00:10:19.840> rules\nNow, here are the most fundamental rules\nNow, here are the most fundamental rules of<00:10:20.120> probability.\nof probability.\nof probability. Uh,\nUh,\nUh, in<00:10:22.440> general<00:10:22.800> in<00:10:22.920> the<00:10:23.000> Bayesian<00:10:23.320> methodology,\nin general in the Bayesian methodology,\nin general in the Bayesian methodology, probability<00:10:24.480> is<00:10:24.640> seen<00:10:24.880> as<00:10:25.040> a<00:10:25.160> quantification\nprobability is seen as a quantification\nprobability is seen as a quantification of<00:10:26.000> the<00:10:26.120> plausibility<00:10:26.840> of<00:10:27.000> a<00:10:27.080> proposition<00:10:27.600> of\nof the plausibility of a proposition of\nof the plausibility of a proposition of a<00:10:27.760> statement<00:10:28.680> where<00:10:28.840> a<00:10:28.880> value<00:10:29.200> of<00:10:29.360> zero\na statement where a value of zero\na statement where a value of zero corresponds<00:10:30.360> to<00:10:30.440> impossibility<00:10:31.200> and<00:10:31.320> a<00:10:31.360> value\ncorresponds to impossibility and a value\ncorresponds to impossibility and a value of<00:10:31.760> one<00:10:32.160> corresponds<00:10:32.560> to<00:10:32.640> certainty.\nof one corresponds to certainty.\nof one corresponds to certainty. Now,<00:10:33.960> if<00:10:34.160> your<00:10:34.280> mind<00:10:34.520> immediately<00:10:34.960> jumps<00:10:35.360> to\nNow, if your mind immediately jumps to\nNow, if your mind immediately jumps to the\nthe\nthe uh,<00:10:36.880> case<00:10:37.320> of<00:10:37.520> sampling<00:10:37.960> a<00:10:38.000> rational<00:10:38.400> number\nuh, case of sampling a rational number\nuh, case of sampling a rational number from<00:10:39.000> a<00:10:39.360> set<00:10:39.600> of<00:10:39.720> irrationals,<00:10:40.400> don't<00:10:40.560> worry,\nfrom a set of irrationals, don't worry,\nfrom a set of irrationals, don't worry, we'll<00:10:40.880> talk<00:10:41.080> about<00:10:41.280> that<00:10:41.400> later<00:10:41.680> because<00:10:41.920> this\nwe'll talk about that later because this\nwe'll talk about that later because this seems<00:10:42.200> to<00:10:42.280> contradict<00:10:42.720> this.\nseems to contradict this.\nseems to contradict this. Uh,<00:10:44.120> but<00:10:44.320> let's<00:10:44.440> start<00:10:44.640> with<00:10:44.720> the<00:10:44.800> basics.\nUh, but let's start with the basics.\nUh, but let's start with the basics. Now,<00:10:45.400> we<00:10:45.480> have<00:10:45.600> the<00:10:45.680> product<00:10:46.000> rule\nNow, we have the product rule\nNow, we have the product rule uh,<00:10:47.360> that<00:10:47.480> allows<00:10:47.839> us<00:10:48.560> uh,<00:10:48.880> to<00:10:49.000> find<00:10:49.560> the\nuh, that allows us uh, to find the\nuh, that allows us uh, to find the probability<00:10:50.320> of<00:10:51.040> uh,<00:10:51.360> compound<00:10:51.760> statement<00:10:52.120> of\nprobability of uh, compound statement of\nprobability of uh, compound statement of A<00:10:52.400> and<00:10:52.520> B<00:10:53.280> given<00:10:53.600> some<00:10:53.720> context<00:10:54.160> C.<00:10:54.880> And<00:10:55.000> then\nA and B given some context C. And then\nA and B given some context C. And then we<00:10:55.240> have<00:10:55.440> the<00:10:55.920> sum<00:10:56.240> rule<00:10:56.480> which<00:10:56.640> allows<00:10:57.000> us<00:10:57.160> to\nwe have the sum rule which allows us to\nwe have the sum rule which allows us to uh,<00:10:58.640> combine<00:10:58.920> the<00:10:59.000> probabilities<00:10:59.560> of<00:10:59.760> two\nuh, combine the probabilities of two\nuh, combine the probabilities of two complements.\ncomplements.\ncomplements. And<00:11:02.360> you<00:11:02.440> notice<00:11:02.760> that\nAnd you notice that\nAnd you notice that uh,<00:11:03.960> we<00:11:04.080> always<00:11:04.440> have<00:11:04.800> this<00:11:05.600> uh,<00:11:06.080> given<00:11:06.400> C\nuh, we always have this uh, given C\nuh, we always have this uh, given C at<00:11:07.560> the<00:11:07.680> end<00:11:08.360> on<00:11:08.440> the<00:11:08.560> right<00:11:08.839> side<00:11:09.600> of<00:11:09.720> the\nat the end on the right side of the\nat the end on the right side of the expression<00:11:10.320> and<00:11:10.480> this<00:11:11.160> uh,<00:11:11.280> means<00:11:11.600> that\nexpression and this uh, means that\nexpression and this uh, means that in<00:11:12.880> the\nin the\nin the in<00:11:14.040> the<00:11:14.120> Bayesian<00:11:14.360> methodology,<00:11:14.960> we<00:11:15.080> don't\nin the Bayesian methodology, we don't\nin the Bayesian methodology, we don't think<00:11:15.960> of<00:11:16.120> probabilities<00:11:16.600> as<00:11:16.720> somehow\nthink of probabilities as somehow\nthink of probabilities as somehow existing<00:11:17.400> in<00:11:17.520> a<00:11:17.560> vacuum<00:11:18.160> or<00:11:18.280> existing<00:11:18.640> as<00:11:18.720> a\nexisting in a vacuum or existing as a\nexisting in a vacuum or existing as a property<00:11:19.360> of<00:11:19.480> the<00:11:19.560> experiment<00:11:20.080> or<00:11:20.160> the<00:11:20.280> object\nproperty of the experiment or the object\nproperty of the experiment or the object that<00:11:20.760> you're<00:11:20.839> studying,<00:11:21.520> but<00:11:21.640> they<00:11:21.720> always\nthat you're studying, but they always\nthat you're studying, but they always depend<00:11:22.600> on<00:11:23.240> what<00:11:23.520> knowledge<00:11:24.200> you<00:11:24.680> have<00:11:25.000> at\ndepend on what knowledge you have at\ndepend on what knowledge you have at this<00:11:25.200> point<00:11:25.520> and<00:11:25.640> this<00:11:25.760> is<00:11:26.480> uh,\nthis point and this is uh,\nthis point and this is uh, why<00:11:27.240> we<00:11:27.360> always<00:11:27.640> write<00:11:27.839> this<00:11:27.960> at<00:11:28.080> the<00:11:28.200> end.\nwhy we always write this at the end.\nwhy we always write this at the end. Usually<00:11:28.839> then<00:11:29.080> it<00:11:29.200> is<00:11:29.400> omitted<00:11:30.000> when<00:11:30.120> it's\nUsually then it is omitted when it's\nUsually then it is omitted when it's clear<00:11:30.520> that<00:11:30.640> it's<00:11:30.800> always<00:11:31.040> going<00:11:31.160> to<00:11:31.240> be<00:11:31.320> the\nclear that it's always going to be the\nclear that it's always going to be the same<00:11:31.720> letter<00:11:31.960> in<00:11:32.040> your<00:11:32.160> calculations,<00:11:32.760> but<00:11:32.960> it\nsame letter in your calculations, but it\nsame letter in your calculations, but it is<00:11:33.200> always<00:11:33.520> there<00:11:33.680> implicitly.\nis always there implicitly.\nis always there implicitly. Um<00:11:35.600> now<00:11:35.720> we're<00:11:35.840> going<00:11:35.960> to<00:11:36.040> just<00:11:36.240> quickly<00:11:36.440> talk\nUm now we're going to just quickly talk\nUm now we're going to just quickly talk about<00:11:36.840> some<00:11:36.960> classical<00:11:37.400> logic.<00:11:37.960> Uh<00:11:38.240> I'm<00:11:38.360> sure\nabout some classical logic. Uh I'm sure\nabout some classical logic. Uh I'm sure you're<00:11:38.800> all<00:11:38.960> familiar<00:11:39.280> with<00:11:39.400> this,<00:11:39.520> so<00:11:39.640> we're\nyou're all familiar with this, so we're\nyou're all familiar with this, so we're going<00:11:39.840> to<00:11:39.920> be<00:11:40.320> quick<00:11:40.560> here.\ngoing to be quick here.\ngoing to be quick here. Um\nUm\nUm we're<00:11:43.200> going<00:11:43.320> to<00:11:43.400> see<00:11:43.480> the<00:11:43.600> probability<00:11:44.160> is<00:11:44.600> an\nwe're going to see the probability is an\nwe're going to see the probability is an extension<00:11:45.080> of<00:11:45.200> logic<00:11:45.720> where<00:11:46.200> classical<00:11:46.680> logic\nextension of logic where classical logic\nextension of logic where classical logic uh<00:11:47.840> is<00:11:48.000> just<00:11:48.360> a<00:11:48.400> special<00:11:48.680> case.\nuh is just a special case.\nuh is just a special case. And<00:11:50.200> the<00:11:50.400> most<00:11:50.960> straightforward<00:11:51.640> way<00:11:51.840> to\nAnd the most straightforward way to\nAnd the most straightforward way to approach<00:11:52.360> classical<00:11:52.720> logic<00:11:53.000> is<00:11:53.120> with<00:11:53.360> truth\napproach classical logic is with truth\napproach classical logic is with truth tables.\ntables.\ntables. Um\nUm\nUm and<00:11:56.800> with<00:11:56.920> the<00:11:57.000> operations<00:11:57.960> and<00:11:58.520> or<00:11:58.920> and<00:11:59.080> not.\nand with the operations and or and not.\nand with the operations and or and not. These<00:12:00.200> are<00:12:00.320> an<00:12:00.440> adequate<00:12:00.800> set,<00:12:01.120> meaning<00:12:01.560> that\nThese are an adequate set, meaning that\nThese are an adequate set, meaning that they<00:12:03.080> uh<00:12:03.240> allow<00:12:03.520> us<00:12:03.680> to<00:12:03.760> express<00:12:04.480> any<00:12:04.720> truth\nthey uh allow us to express any truth\nthey uh allow us to express any truth value<00:12:05.280> depending<00:12:05.640> on<00:12:05.760> any<00:12:06.000> input.\nvalue depending on any input.\nvalue depending on any input. Uh<00:12:07.240> there<00:12:07.400> are<00:12:07.480> actually<00:12:07.800> smaller<00:12:08.160> adequate\nUh there are actually smaller adequate\nUh there are actually smaller adequate sets,<00:12:08.760> the<00:12:08.840> most<00:12:09.120> famous<00:12:09.440> one<00:12:09.640> being<00:12:10.240> NAND,\nsets, the most famous one being NAND,\nsets, the most famous one being NAND, which<00:12:11.200> is<00:12:11.280> used<00:12:11.560> extensively<00:12:12.000> in<00:12:12.120> computer\nwhich is used extensively in computer\nwhich is used extensively in computer chip<00:12:12.600> design,<00:12:12.960> and<00:12:13.080> there's<00:12:13.200> also<00:12:13.400> XOR.\nchip design, and there's also XOR.\nchip design, and there's also XOR. Um<00:12:15.640> but<00:12:15.920> and<00:12:16.080> and<00:12:16.200> or<00:12:16.480> are<00:12:16.560> just<00:12:16.720> very\nUm but and and or are just very\nUm but and and or are just very convenient,<00:12:17.960> and<00:12:18.280> so<00:12:19.160> uh<00:12:19.400> we<00:12:19.560> will<00:12:19.960> use<00:12:20.080> them.\nconvenient, and so uh we will use them.\nconvenient, and so uh we will use them. Now<00:12:21.760> I<00:12:21.800> just<00:12:22.040> also<00:12:22.280> want<00:12:22.440> to<00:12:22.480> draw<00:12:22.640> your\nNow I just also want to draw your\nNow I just also want to draw your attention<00:12:23.040> to<00:12:23.120> these<00:12:23.600> to<00:12:23.680> this<00:12:23.840> notation.<00:12:24.760> And\nattention to these to this notation. And\nattention to these to this notation. And and<00:12:25.040> or<00:12:25.840> uh<00:12:25.880> the<00:12:26.080> and<00:12:26.480> is<00:12:26.640> somehow<00:12:27.040> described\nand or uh the and is somehow described\nand or uh the and is somehow described as<00:12:27.560> this<00:12:27.720> like<00:12:28.600> uh<00:12:28.720> hat.\nas this like uh hat.\nas this like uh hat. And<00:12:30.160> the<00:12:30.240> or<00:12:30.600> is<00:12:30.760> kind<00:12:30.960> of<00:12:31.040> this<00:12:31.440> valley.<00:12:32.400> While\nAnd the or is kind of this valley. While\nAnd the or is kind of this valley. While the<00:12:32.760> not<00:12:33.000> is<00:12:33.120> just<00:12:33.400> a<00:12:33.880> line<00:12:34.400> over<00:12:34.640> the<00:12:35.200> actual\nthe not is just a line over the actual\nthe not is just a line over the actual proposition.\nHere<00:12:39.440> is<00:12:39.560> a<00:12:39.680> some<00:12:40.280> uh<00:12:40.360> famous<00:12:40.680> truth<00:12:40.880> tables\nHere is a some uh famous truth tables\nHere is a some uh famous truth tables and<00:12:41.440> what<00:12:41.600> they<00:12:41.680> might<00:12:41.880> look<00:12:42.040> like.<00:12:42.280> These<00:12:42.440> are\nand what they might look like. These are\nand what they might look like. These are actually<00:12:42.720> kind<00:12:42.880> of<00:12:42.960> important<00:12:43.960> uh<00:12:44.520> for<00:12:44.640> the\nactually kind of important uh for the\nactually kind of important uh for the rest<00:12:44.920> of<00:12:45.000> the<00:12:45.040> presentation,<00:12:45.680> but<00:12:45.960> I'm<00:12:46.080> not\nrest of the presentation, but I'm not\nrest of the presentation, but I'm not going<00:12:46.680> to<00:12:46.760> go<00:12:46.839> into<00:12:47.040> them.<00:12:47.360> If<00:12:47.960> you<00:12:48.120> ever<00:12:48.360> wish\ngoing to go into them. If you ever wish\ngoing to go into them. If you ever wish to<00:12:48.920> have<00:12:49.040> some<00:12:49.160> explanation<00:12:49.640> for<00:12:49.760> why\nto have some explanation for why\nto have some explanation for why something<00:12:50.160> happens,<00:12:50.880> I<00:12:50.960> can<00:12:51.080> come<00:12:51.240> back<00:12:51.440> here\nsomething happens, I can come back here\nsomething happens, I can come back here and<00:12:51.960> show<00:12:52.160> you.\nand show you.\nand show you. So<00:12:53.800> how<00:12:53.960> can<00:12:54.120> we<00:12:54.200> check<00:12:54.520> if<00:12:54.800> some<00:12:55.000> kind<00:12:55.240> of\nSo how can we check if some kind of\nSo how can we check if some kind of complex<00:12:56.080> compound<00:12:56.520> statement<00:12:57.360> uh<00:12:57.520> can<00:12:57.760> be\ncomplex compound statement uh can be\ncomplex compound statement uh can be true?\ntrue?\ntrue? We<00:12:59.000> can<00:12:59.200> do<00:12:59.440> what<00:12:59.600> is<00:12:59.720> called<00:13:00.200> reduction<00:13:00.680> to\nWe can do what is called reduction to\nWe can do what is called reduction to the<00:13:01.000> disjunctive<00:13:01.880> normal<00:13:02.120> form,<00:13:02.480> meaning<00:13:03.360> we\nthe disjunctive normal form, meaning we\nthe disjunctive normal form, meaning we uh\nuh\nuh split<00:13:05.600> the<00:13:05.680> statement<00:13:06.040> up<00:13:06.240> into<00:13:06.600> all<00:13:07.000> of<00:13:07.200> its\nsplit the statement up into all of its\nsplit the statement up into all of its uh<00:13:08.000> minus<00:13:08.800> uh<00:13:08.880> it's<00:13:09.080> all<00:13:09.200> of<00:13:09.280> its<00:13:09.440> smallest\nuh minus uh it's all of its smallest\nuh minus uh it's all of its smallest form,<00:13:10.120> so<00:13:10.200> to<00:13:10.280> speak,<00:13:11.160> by<00:13:11.640> applying<00:13:12.040> the<00:13:12.160> join\nform, so to speak, by applying the join\nform, so to speak, by applying the join operation,<00:13:13.160> meaning<00:13:13.400> the<00:13:13.520> and<00:13:13.760> operation.\noperation, meaning the and operation.\noperation, meaning the and operation. And<00:13:15.200> then<00:13:15.520> we<00:13:15.640> can<00:13:15.880> we<00:13:16.000> could<00:13:16.160> combine<00:13:16.520> those\nAnd then we can we could combine those\nAnd then we can we could combine those again<00:13:17.040> with<00:13:17.240> a<00:13:17.520> with<00:13:17.760> increasing<00:13:18.200> amount<00:13:18.400> of\nagain with a with increasing amount of\nagain with a with increasing amount of or<00:13:18.760> operations<00:13:19.280> to<00:13:19.360> come<00:13:19.560> back<00:13:19.760> to<00:13:19.839> the\nor operations to come back to the\nor operations to come back to the original<00:13:20.240> statement.<00:13:21.120> And<00:13:21.320> if<00:13:21.560> any<00:13:21.760> of<00:13:21.920> those\noriginal statement. And if any of those\noriginal statement. And if any of those atoms<00:13:23.400> that<00:13:23.520> cannot<00:13:23.800> be<00:13:23.960> further<00:13:24.520> combined\natoms that cannot be further combined\natoms that cannot be further combined with<00:13:25.440> with<00:13:25.640> meet<00:13:25.839> statements,<00:13:26.760> um<00:13:27.200> then<00:13:27.320> the\nwith with meet statements, um then the\nwith with meet statements, um then the whole<00:13:27.760> if<00:13:27.960> any<00:13:28.120> of<00:13:28.280> those<00:13:28.560> can<00:13:28.800> be<00:13:28.920> true,<00:13:29.160> then\nwhole if any of those can be true, then\nwhole if any of those can be true, then the<00:13:29.400> whole<00:13:29.560> statement<00:13:29.880> can<00:13:30.160> also<00:13:30.400> be<00:13:30.520> true.\nthe whole statement can also be true.\nthe whole statement can also be true. Now<00:13:31.240> this<00:13:31.400> doesn't<00:13:31.720> tell<00:13:31.880> us<00:13:32.000> whether<00:13:32.200> it<00:13:32.280> is\nNow this doesn't tell us whether it is\nNow this doesn't tell us whether it is true<00:13:32.480> or<00:13:32.560> not,<00:13:32.680> but<00:13:32.840> this<00:13:32.960> is<00:13:33.040> how<00:13:33.160> we<00:13:33.320> find<00:13:33.640> out\ntrue or not, but this is how we find out\ntrue or not, but this is how we find out if<00:13:33.880> it<00:13:34.080> could<00:13:34.320> possibly<00:13:34.720> be<00:13:34.800> true.<00:13:35.040> Because<00:13:35.400> if\nif it could possibly be true. Because if\nif it could possibly be true. Because if all<00:13:36.120> the<00:13:36.240> atoms\nall the atoms\nall the atoms have<00:13:37.840> contradictions<00:13:38.320> in<00:13:38.400> them,<00:13:38.600> then<00:13:38.760> the\nhave contradictions in them, then the\nhave contradictions in them, then the compound<00:13:39.200> statement<00:13:39.480> obviously<00:13:39.720> cannot<00:13:39.960> be\ncompound statement obviously cannot be\ncompound statement obviously cannot be true<00:13:40.240> either.\ntrue either.\ntrue either. For<00:13:41.960> example,<00:13:42.280> the<00:13:42.400> compound<00:13:42.840> statement<00:13:43.480> A<00:13:43.640> or\nFor example, the compound statement A or\nFor example, the compound statement A or B<00:13:44.000> can<00:13:44.160> be<00:13:44.240> divided<00:13:44.560> into<00:13:44.880> A\nB can be divided into A\nB can be divided into A individually<00:13:46.600> and<00:13:46.720> B<00:13:46.840> individually,<00:13:47.560> but\nindividually and B individually, but\nindividually and B individually, but then<00:13:47.800> also<00:13:47.960> further<00:13:48.280> into<00:13:48.920> A<00:13:49.040> and<00:13:49.200> not<00:13:49.440> B,<00:13:49.920> A\nthen also further into A and not B, A\nthen also further into A and not B, A and<00:13:50.200> B,<00:13:50.600> not<00:13:50.920> A<00:13:51.000> and<00:13:51.120> B,<00:13:51.400> and<00:13:51.560> not<00:13:51.840> A<00:13:51.880> and<00:13:52.000> not<00:13:52.160> B.\nand B, not A and B, and not A and not B.\nand B, not A and B, and not A and not B. And<00:13:52.560> these<00:13:52.800> are<00:13:52.920> then<00:13:53.080> the<00:13:53.200> atoms<00:13:53.560> because<00:13:53.840> you\nAnd these are then the atoms because you\nAnd these are then the atoms because you just<00:13:54.160> this<00:13:54.320> doesn't<00:13:54.560> make<00:13:54.680> sense<00:13:54.880> to<00:13:55.000> combine\njust this doesn't make sense to combine\njust this doesn't make sense to combine them<00:13:55.440> further\nthem further\nthem further with\nwith\nwith with<00:13:58.040> meet<00:13:58.200> operations.\nwith meet operations.\nwith meet operations. Now<00:14:00.880> from\nNow from\nNow from from<00:14:02.320> this<00:14:02.480> operation,<00:14:03.680> we<00:14:03.800> can<00:14:03.960> actually\nfrom this operation, we can actually\nfrom this operation, we can actually obtain<00:14:04.840> some<00:14:05.040> kind<00:14:05.280> of<00:14:05.720> order\nobtain some kind of order\nobtain some kind of order by<00:14:07.400> implication.<00:14:08.480> So\nby implication. So\nby implication. So we<00:14:10.280> know<00:14:10.560> that<00:14:10.960> these<00:14:11.280> are<00:14:11.360> the<00:14:11.440> most\nwe know that these are the most\nwe know that these are the most fundamental<00:14:12.160> building<00:14:12.400> blocks<00:14:13.360> here<00:14:13.560> at<00:14:13.640> the\nfundamental building blocks here at the\nfundamental building blocks here at the bottom,<00:14:14.520> and<00:14:14.760> these<00:14:14.920> statements<00:14:15.280> above<00:14:16.160> are\nbottom, and these statements above are\nbottom, and these statements above are just<00:14:16.560> the<00:14:16.640> combinations<00:14:17.360> with<00:14:17.520> the<00:14:18.000> with<00:14:18.080> the\njust the combinations with the with the\njust the combinations with the with the join<00:14:18.440> operations.<00:14:18.960> And<00:14:19.120> this<00:14:19.280> means<00:14:19.520> that\njoin operations. And this means that\njoin operations. And this means that this<00:14:21.600> element<00:14:22.000> here<00:14:22.320> implies<00:14:22.680> this<00:14:22.800> one\nthis element here implies this one\nthis element here implies this one above.\nUm\nUm\nUm and<00:14:27.800> here<00:14:28.080> in<00:14:28.200> this<00:14:28.400> representation<00:14:29.000> actually\nand here in this representation actually\nand here in this representation actually not<00:14:29.880> A<00:14:30.040> and<00:14:30.200> not<00:14:30.400> B<00:14:30.560> is<00:14:30.720> ignored<00:14:31.200> because<00:14:31.680> our\nnot A and not B is ignored because our\nnot A and not B is ignored because our top<00:14:32.160> element<00:14:32.520> here<00:14:32.800> is<00:14:33.120> A<00:14:33.240> or<00:14:33.360> B.\ntop element here is A or B.\ntop element here is A or B. And<00:14:34.800> if<00:14:34.960> we<00:14:35.080> assume<00:14:35.360> that<00:14:35.480> this<00:14:35.640> is<00:14:35.760> true,<00:14:36.040> then\nAnd if we assume that this is true, then\nAnd if we assume that this is true, then at<00:14:36.320> least<00:14:36.560> one<00:14:37.240> of<00:14:37.440> A<00:14:38.240> and<00:14:38.360> B<00:14:38.520> must<00:14:38.839> be<00:14:38.920> true.\nat least one of A and B must be true.\nat least one of A and B must be true. And<00:14:39.360> so<00:14:39.480> the<00:14:39.600> statement<00:14:39.920> is<00:14:40.040> necessarily\nAnd so the statement is necessarily\nAnd so the statement is necessarily false.<00:14:40.800> So<00:14:41.480> A<00:14:42.000> and<00:14:42.160> not<00:14:42.400> B<00:14:42.560> is<00:14:42.680> sort<00:14:42.800> of<00:14:43.000> the\nfalse. So A and not B is sort of the\nfalse. So A and not B is sort of the complement<00:14:43.680> to<00:14:43.800> to<00:14:43.880> the<00:14:44.000> top<00:14:44.240> statement.\ncomplement to to the top statement.\ncomplement to to the top statement. Therefore,<00:14:44.880> it<00:14:44.960> doesn't<00:14:45.200> feature<00:14:45.560> in<00:14:45.640> this.\nTherefore, it doesn't feature in this.\nTherefore, it doesn't feature in this. And<00:14:47.400> um\nAnd um\nAnd um exactly.<00:14:49.680> We<00:14:49.800> assume<00:14:50.280> that<00:14:50.560> this<00:14:50.960> top\nexactly. We assume that this top\nexactly. We assume that this top statement<00:14:51.480> is<00:14:51.600> true.\nstatement is true.\nstatement is true. And<00:14:52.880> from<00:14:53.040> this\nAnd from this\nAnd from this we<00:14:54.120> write<00:14:54.280> it<00:14:54.360> with<00:14:54.520> this<00:14:54.720> arrow,\nwe write it with this arrow,\nwe write it with this arrow, skinny<00:14:56.280> T<00:14:56.480> symbol.\nAnd<00:14:59.200> from<00:14:59.320> this<00:14:59.480> we<00:14:59.600> can<00:14:59.800> essentially<00:15:00.200> obtain\nAnd from this we can essentially obtain\nAnd from this we can essentially obtain a<00:15:00.600> lattice<00:15:01.160> that<00:15:01.760> has<00:15:02.000> elements<00:15:02.560> that<00:15:02.800> are\na lattice that has elements that are\na lattice that has elements that are ordered<00:15:03.440> by<00:15:03.960> implication.\nordered by implication.\nordered by implication. They<00:15:05.920> also<00:15:06.120> have<00:15:06.360> this<00:15:07.160> optional<00:15:07.960> bottom\nThey also have this optional bottom\nThey also have this optional bottom element<00:15:08.600> that<00:15:08.720> is<00:15:08.880> just<00:15:09.200> the<00:15:09.720> absurdity,<00:15:10.280> the\nelement that is just the absurdity, the\nelement that is just the absurdity, the contradiction,<00:15:10.960> this<00:15:11.160> makes<00:15:11.400> sense.<00:15:12.200> Um<00:15:13.240> and\ncontradiction, this makes sense. Um and\ncontradiction, this makes sense. Um and to<00:15:14.240> understand<00:15:14.800> what's<00:15:14.960> going<00:15:15.200> on<00:15:15.400> here,<00:15:15.600> it's\nto understand what's going on here, it's\nto understand what's going on here, it's very<00:15:16.000> easy<00:15:16.240> to<00:15:16.360> kind<00:15:16.560> of<00:15:16.640> see<00:15:16.800> an<00:15:17.160> isomorphism\nvery easy to kind of see an isomorphism\nvery easy to kind of see an isomorphism to\nto\nto a<00:15:18.960> lattice<00:15:19.640> of<00:15:20.000> sets<00:15:20.280> that<00:15:20.480> are<00:15:20.560> ordered<00:15:20.880> by\na lattice of sets that are ordered by\na lattice of sets that are ordered by inclusion,<00:15:21.560> by<00:15:21.680> set<00:15:21.920> inclusion,\ninclusion, by set inclusion,\ninclusion, by set inclusion, where<00:15:23.720> um<00:15:23.840> um\nwhere um um\nwhere um um you<00:15:25.440> just<00:15:25.600> see<00:15:26.520> uh<00:15:26.600> you<00:15:26.680> just<00:15:26.920> basically<00:15:27.200> make\nyou just see uh you just basically make\nyou just see uh you just basically make an<00:15:27.440> equivalent<00:15:27.960> of<00:15:28.320> the<00:15:28.560> the<00:15:28.720> labels<00:15:29.120> of<00:15:29.240> which\nan equivalent of the the labels of which\nan equivalent of the the labels of which statements<00:15:29.760> can<00:15:29.880> be<00:15:29.960> true,<00:15:30.560> and<00:15:30.680> you<00:15:30.760> can<00:15:30.880> see\nstatements can be true, and you can see\nstatements can be true, and you can see that<00:15:31.360> E<00:15:31.680> F<00:15:32.000> contains<00:15:32.480> E<00:15:32.680> and<00:15:32.840> F,<00:15:33.000> of<00:15:33.120> course,\nthat E F contains E and F, of course,\nthat E F contains E and F, of course, and<00:15:33.600> this<00:15:33.720> is<00:15:34.120> this<00:15:34.400> is<00:15:34.680> more<00:15:34.840> or<00:15:34.920> less<00:15:35.040> the\nand this is this is more or less the\nand this is this is more or less the fact<00:15:35.320> that<00:15:35.480> E<00:15:35.680> implies<00:15:36.200> E<00:15:36.320> or<00:15:36.480> F.\nAnd<00:15:39.120> it<00:15:39.200> is<00:15:39.280> with<00:15:39.440> these<00:15:39.640> lattice<00:15:39.880> structures\nAnd it is with these lattice structures\nAnd it is with these lattice structures that<00:15:40.480> we're<00:15:40.600> going<00:15:40.800> to\nthat we're going to\nthat we're going to uh\nuh\nuh find<00:15:42.840> the<00:15:43.360> the<00:15:43.440> fundamental<00:15:43.840> rules<00:15:44.000> of\nfind the the fundamental rules of\nfind the the fundamental rules of probability.<00:15:44.960> If<00:15:45.160> any<00:15:45.360> of<00:15:45.720> you<00:15:46.000> are<00:15:46.120> familiar\nprobability. If any of you are familiar\nprobability. If any of you are familiar with<00:15:46.760> the<00:15:46.880> work<00:15:47.240> of<00:15:47.840> uh<00:15:48.080> of<00:15:48.240> Cox,\nwith the work of uh of Cox,\nwith the work of uh of Cox, he<00:15:49.800> derived<00:15:50.400> the<00:15:50.760> the<00:15:50.880> rules<00:15:51.080> of<00:15:51.160> probability\nhe derived the the rules of probability\nhe derived the the rules of probability by<00:15:52.240> assuming<00:15:52.960> also\nby assuming also\nby assuming also uh<00:15:54.720> continuity<00:15:55.560> of<00:15:55.920> the<00:15:56.040> probability<00:15:56.600> and\nuh continuity of the probability and\nuh continuity of the probability and differentiability,<00:15:57.680> and<00:15:57.880> the<00:15:58.000> beauty<00:15:58.360> of\ndifferentiability, and the beauty of\ndifferentiability, and the beauty of this<00:15:58.680> approach<00:15:59.120> is<00:15:59.320> that<00:15:59.640> it<00:15:59.760> does<00:15:59.960> not<00:16:00.200> assume\nthis approach is that it does not assume\nthis approach is that it does not assume these<00:16:00.680> things.\nthese things.\nthese things. Uh<00:16:01.800> it's<00:16:02.120> much<00:16:02.320> more<00:16:02.440> simpler.\nUh it's much more simpler.\nUh it's much more simpler. That's<00:16:03.800> much<00:16:03.920> simpler,<00:16:04.160> sorry.\nThat's much simpler, sorry.\nThat's much simpler, sorry. So,<00:16:05.760> what<00:16:05.920> is<00:16:06.000> a<00:16:06.040> lattice?<00:16:06.920> Um<00:16:07.240> I'm<00:16:07.360> assuming\nSo, what is a lattice? Um I'm assuming\nSo, what is a lattice? Um I'm assuming most<00:16:08.280> of<00:16:08.400> you<00:16:08.480> have<00:16:08.600> probably<00:16:09.320> seen<00:16:09.600> this\nmost of you have probably seen this\nmost of you have probably seen this before,<00:16:10.280> so<00:16:10.640> I'm<00:16:10.800> going<00:16:10.960> to<00:16:11.040> be<00:16:11.640> rather<00:16:11.800> quick\nbefore, so I'm going to be rather quick\nbefore, so I'm going to be rather quick here<00:16:12.120> as<00:16:12.240> well.\nhere as well.\nhere as well. Um<00:16:14.160> a<00:16:14.200> lattice<00:16:14.560> is<00:16:14.760> a<00:16:14.800> kind<00:16:15.080> of<00:16:15.240> partially\nUm a lattice is a kind of partially\nUm a lattice is a kind of partially ordered<00:16:15.840> set,<00:16:16.080> or<00:16:16.160> they<00:16:16.240> call<00:16:16.400> it<00:16:16.440> a<00:16:16.480> poset.\nordered set, or they call it a poset.\nordered set, or they call it a poset. What<00:16:17.280> is<00:16:17.360> a<00:16:17.400> poset?<00:16:17.840> A<00:16:17.920> poset<00:16:18.400> is<00:16:18.640> a<00:16:18.840> a<00:16:18.880> set<00:16:19.720> with\nWhat is a poset? A poset is a a set with\nWhat is a poset? A poset is a a set with a<00:16:19.880> binary<00:16:20.200> ordering<00:16:20.520> relation<00:16:21.400> with<00:16:21.680> the\na binary ordering relation with the\na binary ordering relation with the property<00:16:22.760> of<00:16:22.920> reflexivity,<00:16:23.680> antisymmetry,\nproperty of reflexivity, antisymmetry,\nproperty of reflexivity, antisymmetry, and<00:16:24.560> transitivity.\nand transitivity.\nand transitivity. And<00:16:27.600> we<00:16:27.720> can<00:16:27.839> also<00:16:28.080> say<00:16:28.400> that<00:16:28.839> if<00:16:29.080> A<00:16:29.200> is<00:16:29.320> not<00:16:29.560> B\nAnd we can also say that if A is not B\nAnd we can also say that if A is not B and<00:16:30.640> B<00:16:31.200> is<00:16:31.960> uh<00:16:32.080> above<00:16:32.520> A,<00:16:32.760> then<00:16:33.400> B<00:16:33.560> is<00:16:33.680> strictly\nand B is uh above A, then B is strictly\nand B is uh above A, then B is strictly above<00:16:34.320> A.\nabove A.\nabove A. >> [snorts]\n>> [snorts]\n>> [snorts] >> Um\n>> Um\n>> Um And<00:16:37.360> we<00:16:37.440> can<00:16:37.640> also<00:16:38.000> use<00:16:38.760> uh<00:16:39.000> this<00:16:39.160> notation\nAnd we can also use uh this notation\nAnd we can also use uh this notation to<00:16:40.400> show<00:16:40.800> that<00:16:41.240> B<00:16:41.560> is<00:16:41.800> sort<00:16:41.960> of<00:16:42.280> the<00:16:42.440> immediate\nto show that B is sort of the immediate\nto show that B is sort of the immediate superior<00:16:43.600> to<00:16:43.760> A.\nTo<00:16:49.000> understand<00:16:49.240> what<00:16:49.360> a<00:16:49.400> lattice<00:16:49.680> is,<00:16:49.839> you\nTo understand what a lattice is, you\nTo understand what a lattice is, you need<00:16:50.160> to<00:16:50.200> understand<00:16:50.600> two<00:16:50.720> more<00:16:50.839> concepts.\nneed to understand two more concepts.\nneed to understand two more concepts. Um<00:16:52.360> imagine<00:16:53.240> uh<00:16:53.600> a<00:16:53.680> subset<00:16:54.839> X<00:16:55.320> of<00:16:55.440> a<00:16:55.720> of<00:16:55.880> a<00:16:55.920> poset\nUm imagine uh a subset X of a of a poset\nUm imagine uh a subset X of a of a poset P.\nP.\nP. We<00:16:57.360> can<00:16:57.880> talk<00:16:58.080> about<00:16:58.320> an<00:16:58.400> element<00:16:59.040> A<00:16:59.360> in<00:16:59.520> P<00:16:59.760> that\nWe can talk about an element A in P that\nWe can talk about an element A in P that contains<00:17:00.760> every<00:17:01.000> element<00:17:02.080> uh<00:17:02.320> of<00:17:02.520> X,<00:17:03.280> meaning\ncontains every element uh of X, meaning\ncontains every element uh of X, meaning it<00:17:03.680> is<00:17:03.800> superior<00:17:04.240> to<00:17:04.400> all<00:17:04.600> of<00:17:04.720> them<00:17:04.959> in<00:17:05.079> the\nit is superior to all of them in the\nit is superior to all of them in the in<00:17:06.400> the<00:17:06.480> hierarchy.<00:17:07.400> Then<00:17:07.680> A<00:17:07.839> is<00:17:08.000> called<00:17:08.400> an\nin the hierarchy. Then A is called an\nin the hierarchy. Then A is called an upper<00:17:08.800> bound<00:17:09.199> of<00:17:09.319> the<00:17:09.400> subset<00:17:09.839> X.<00:17:10.560> Then<00:17:10.760> the\nupper bound of the subset X. Then the\nupper bound of the subset X. Then the least<00:17:11.280> upper<00:17:11.520> bound<00:17:12.280> is<00:17:12.959> sort<00:17:13.160> of<00:17:13.280> the<00:17:13.640> the\nleast upper bound is sort of the the\nleast upper bound is sort of the the notion<00:17:14.199> of<00:17:14.319> the<00:17:14.800> thing<00:17:15.079> that<00:17:15.280> we<00:17:15.400> would<00:17:15.640> most\nnotion of the thing that we would most\nnotion of the thing that we would most intuitively<00:17:16.920> associate<00:17:17.400> with<00:17:17.520> an<00:17:17.600> upper\nintuitively associate with an upper\nintuitively associate with an upper bound,<00:17:18.600> and<00:17:18.760> it<00:17:18.880> is<00:17:19.120> the<00:17:19.640> element<00:17:20.000> in<00:17:20.120> P<00:17:20.600> which\nbound, and it is the element in P which\nbound, and it is the element in P which is<00:17:20.839> an<00:17:20.959> upper<00:17:21.120> bound<00:17:21.319> of<00:17:21.480> X<00:17:22.079> and<00:17:22.400> is<00:17:22.520> contained\nis an upper bound of X and is contained\nis an upper bound of X and is contained in<00:17:23.160> every<00:17:23.439> other<00:17:23.720> upper<00:17:23.920> bound<00:17:24.199> of<00:17:24.280> the\nin every other upper bound of the\nin every other upper bound of the subset.<00:17:24.800> So,<00:17:25.040> it<00:17:25.199> is,<00:17:25.800> as<00:17:25.920> the<00:17:26.040> name<00:17:26.199> suggests,\nsubset. So, it is, as the name suggests,\nsubset. So, it is, as the name suggests, the\nthe\nthe the<00:17:27.680> lowest<00:17:28.400> of<00:17:28.560> all<00:17:28.680> the<00:17:28.840> upper<00:17:29.000> bounds.\nthe lowest of all the upper bounds.\nthe lowest of all the upper bounds. And<00:17:31.160> uh<00:17:31.240> dually,<00:17:31.520> we<00:17:31.600> can<00:17:31.760> also<00:17:32.040> define<00:17:32.440> the\nAnd uh dually, we can also define the\nAnd uh dually, we can also define the greatest<00:17:32.880> lower<00:17:33.120> bound,<00:17:33.480> which<00:17:33.680> is<00:17:33.760> simply\ngreatest lower bound, which is simply\ngreatest lower bound, which is simply the<00:17:34.200> lower<00:17:34.480> bound<00:17:34.680> that<00:17:34.800> contains<00:17:35.520> all<00:17:35.760> other\nthe lower bound that contains all other\nthe lower bound that contains all other lower<00:17:36.480> bounds.<00:17:36.840> We<00:17:36.920> must<00:17:37.120> invert<00:17:37.480> the\nlower bounds. We must invert the\nlower bounds. We must invert the containment<00:17:38.160> operation.<00:17:39.240> And<00:17:39.320> a<00:17:39.360> lattice<00:17:39.680> is\ncontainment operation. And a lattice is\ncontainment operation. And a lattice is simply<00:17:40.040> a<00:17:40.120> poset<00:17:40.560> for<00:17:40.760> which<00:17:40.960> the<00:17:41.040> least<00:17:41.640> and\nsimply a poset for which the least and\nsimply a poset for which the least and upper<00:17:42.400> bound<00:17:42.720> and<00:17:42.800> the<00:17:42.880> greatest<00:17:43.160> lower<00:17:43.360> bound\nupper bound and the greatest lower bound\nupper bound and the greatest lower bound exist<00:17:44.000> for<00:17:44.160> all<00:17:44.360> pairs<00:17:44.600> of<00:17:44.720> elements<00:17:45.600> in<00:17:45.720> the\nexist for all pairs of elements in the\nexist for all pairs of elements in the set.\nThe<00:17:50.320> lower<00:17:50.720> upper<00:17:50.960> bound<00:17:51.160> between<00:17:51.560> A<00:17:51.640> and<00:17:51.760> B<00:17:52.000> is\nThe lower upper bound between A and B is\nThe lower upper bound between A and B is commonly<00:17:52.400> denoted<00:17:52.840> as,<00:17:53.760> again,<00:17:54.080> with<00:17:54.280> this\ncommonly denoted as, again, with this\ncommonly denoted as, again, with this kind<00:17:54.600> of<00:17:54.720> valley<00:17:54.960> notation,<00:17:55.400> which<00:17:55.520> is<00:17:55.600> called\nkind of valley notation, which is called\nkind of valley notation, which is called the<00:17:55.880> join<00:17:56.200> operation,<00:17:57.160> whereas<00:17:57.480> the<00:17:57.520> greatest\nthe join operation, whereas the greatest\nthe join operation, whereas the greatest lower<00:17:58.080> bound<00:17:58.560> is<00:17:59.040> this<00:17:59.280> hat,<00:18:00.200> meet,<00:18:00.560> and<00:18:00.960> these\nlower bound is this hat, meet, and these\nlower bound is this hat, meet, and these symbols<00:18:01.720> mirror<00:18:02.040> those<00:18:02.360> used<00:18:02.560> in<00:18:02.680> Boolean\nsymbols mirror those used in Boolean\nsymbols mirror those used in Boolean algebra,\nalgebra,\nalgebra, uh<00:18:04.560> because\nuh because\nuh because uh<00:18:06.240> when<00:18:06.480> you<00:18:06.720> treat<00:18:07.240> propositions<00:18:07.840> as\nuh when you treat propositions as\nuh when you treat propositions as ordered<00:18:08.160> by<00:18:08.240> implication,<00:18:08.840> the<00:18:09.000> logical<00:18:09.440> or\nordered by implication, the logical or\nordered by implication, the logical or and<00:18:09.960> logical<00:18:10.360> and<00:18:10.560> operation<00:18:11.360> act<00:18:11.720> exactly<00:18:12.400> as\nand logical and operation act exactly as\nand logical and operation act exactly as the<00:18:12.760> join<00:18:13.120> and<00:18:13.280> the<00:18:13.360> meet<00:18:13.560> operation.\nthe join and the meet operation.\nthe join and the meet operation. So,<00:18:14.880> there's<00:18:15.160> this<00:18:15.600> uh<00:18:15.720> nice<00:18:15.920> correspondence\nSo, there's this uh nice correspondence\nSo, there's this uh nice correspondence that<00:18:17.040> also<00:18:17.240> makes<00:18:17.440> the<00:18:17.480> notation<00:18:18.080> much<00:18:18.680> uh\nthat also makes the notation much uh\nthat also makes the notation much uh nicer<00:18:19.160> for<00:18:19.280> us<00:18:19.400> to<00:18:19.520> use.\nnicer for us to use.\nnicer for us to use. Uh<00:18:21.160> then<00:18:21.360> also<00:18:21.640> we<00:18:21.760> speak<00:18:22.040> of<00:18:22.160> a<00:18:22.200> distributive\nUh then also we speak of a distributive\nUh then also we speak of a distributive lattice<00:18:23.080> if<00:18:23.280> we<00:18:23.400> have<00:18:23.560> some<00:18:23.680> kind<00:18:23.840> of\nlattice if we have some kind of\nlattice if we have some kind of distributivity<00:18:25.160> property\ndistributivity property\ndistributivity property uh<00:18:26.760> of<00:18:26.960> the\nuh of the\nuh of the and<00:18:28.240> over<00:18:28.400> the<00:18:28.520> or.\nAnd<00:18:32.160> there's<00:18:32.480> also<00:18:32.800> an<00:18:33.200> even<00:18:33.440> more\nAnd there's also an even more\nAnd there's also an even more restrictive<00:18:34.000> class<00:18:34.280> of<00:18:34.400> lattices<00:18:34.920> which<00:18:35.080> are\nrestrictive class of lattices which are\nrestrictive class of lattices which are called<00:18:35.440> Boolean<00:18:35.800> lattices<00:18:36.280> for<00:18:36.520> which<00:18:37.000> each\ncalled Boolean lattices for which each\ncalled Boolean lattices for which each element<00:18:37.600> has<00:18:37.720> a<00:18:37.800> complement.<00:18:38.880> And<00:18:39.080> a\nelement has a complement. And a\nelement has a complement. And a complement<00:18:39.600> is<00:18:39.720> simply<00:18:40.160> an<00:18:40.320> element<00:18:41.400> um\ncomplement is simply an element um\ncomplement is simply an element um for<00:18:42.920> which<00:18:43.600> the<00:18:43.800> join<00:18:44.760> is<00:18:44.920> the<00:18:45.000> top<00:18:45.240> element\nfor which the join is the top element\nfor which the join is the top element and<00:18:46.000> the<00:18:46.280> the<00:18:46.400> meet<00:18:46.720> is<00:18:46.920> the<00:18:47.280> most<00:18:47.480> bottom\nand the the meet is the most bottom\nand the the meet is the most bottom element.<00:18:48.280> That's<00:18:48.440> simply<00:18:48.680> what<00:18:48.800> that<00:18:48.920> means.\nelement. That's simply what that means.\nelement. That's simply what that means. However,<00:18:50.200> in<00:18:50.320> this<00:18:50.520> derivation,<00:18:51.120> we're<00:18:51.240> not\nHowever, in this derivation, we're not\nHowever, in this derivation, we're not going<00:18:51.680> to<00:18:51.760> need<00:18:52.240> Boolean<00:18:52.480> lattices.\ngoing to need Boolean lattices.\ngoing to need Boolean lattices. Distributive<00:18:53.680> lattices<00:18:54.160> are<00:18:54.360> completely\nDistributive lattices are completely\nDistributive lattices are completely sufficient.\nsufficient.\nsufficient. Uh<00:18:56.240> which<00:18:56.440> is<00:18:57.080> has<00:18:57.280> some<00:18:57.480> practical\nUh which is has some practical\nUh which is has some practical implication,<00:18:58.280> but<00:18:58.880> this<00:18:59.080> is<00:18:59.800> not<00:18:59.960> important\nimplication, but this is not important\nimplication, but this is not important right<00:19:00.400> now.<00:19:01.000> So,<00:19:01.520> now<00:19:01.800> we're<00:19:01.920> going<00:19:02.080> to<00:19:02.120> define\nright now. So, now we're going to define\nright now. So, now we're going to define the<00:19:03.200> objective<00:19:03.840> of<00:19:03.960> this<00:19:04.080> derivation.<00:19:05.080> We\nthe objective of this derivation. We\nthe objective of this derivation. We want<00:19:05.760> to<00:19:06.240> basically<00:19:06.640> generalize<00:19:07.320> the<00:19:07.720> zeta\nwant to basically generalize the zeta\nwant to basically generalize the zeta function.<00:19:08.360> The<00:19:08.440> zeta<00:19:08.680> function\nfunction. The zeta function\nfunction. The zeta function uh<00:19:10.000> Uh,<00:19:10.200> is<00:19:10.880> in<00:19:11.080> its<00:19:11.360> classical<00:19:11.800> form<00:19:12.200> just<00:19:12.720> an\nuh Uh, is in its classical form just an\nuh Uh, is in its classical form just an indicator<00:19:13.680> that<00:19:13.800> tells<00:19:14.080> us<00:19:14.200> if<00:19:14.320> an<00:19:14.440> element<00:19:14.840> is\nindicator that tells us if an element is\nindicator that tells us if an element is below<00:19:16.360> uh<00:19:16.440> or<00:19:16.600> equal<00:19:16.800> to<00:19:16.920> another<00:19:17.240> and<00:19:17.400> zero\nbelow uh or equal to another and zero\nbelow uh or equal to another and zero otherwise.\notherwise.\notherwise. Uh,<00:19:19.080> in<00:19:19.200> our<00:19:19.320> context,<00:19:19.760> we're<00:19:19.880> looking<00:19:20.160> to\nUh, in our context, we're looking to\nUh, in our context, we're looking to kind<00:19:20.480> of<00:19:20.600> generalize<00:19:21.120> the<00:19:21.240> inverse,<00:19:21.880> so<00:19:22.200> the\nkind of generalize the inverse, so the\nkind of generalize the inverse, so the one<00:19:22.520> that<00:19:22.640> tells<00:19:22.960> us<00:19:23.680> um\none that tells us um\none that tells us um if<00:19:24.920> a<00:19:25.000> proposition<00:19:25.440> is<00:19:25.640> above.<00:19:26.560> However,<00:19:27.000> we\nif a proposition is above. However, we\nif a proposition is above. However, we want<00:19:27.560> it<00:19:27.720> to<00:19:27.800> be<00:19:27.920> not<00:19:28.120> only<00:19:28.280> just<00:19:28.440> a<00:19:28.520> binary\nwant it to be not only just a binary\nwant it to be not only just a binary indicator,<00:19:29.600> but<00:19:29.920> to<00:19:30.080> also<00:19:30.320> have<00:19:30.440> some<00:19:30.560> kind<00:19:30.840> of\nindicator, but to also have some kind of\nindicator, but to also have some kind of um\num\num some<00:19:32.760> continuity,<00:19:33.440> meaning<00:19:33.720> we<00:19:33.840> have<00:19:34.160> some\nsome continuity, meaning we have some\nsome continuity, meaning we have some degree<00:19:35.280> of<00:19:35.440> implication.<00:19:36.160> This<00:19:36.280> is<00:19:36.360> what\ndegree of implication. This is what\ndegree of implication. This is what we're<00:19:36.560> looking<00:19:36.800> for.<00:19:37.040> So,<00:19:37.240> we're<00:19:37.360> looking<00:19:37.600> to\nwe're looking for. So, we're looking to\nwe're looking for. So, we're looking to have<00:19:38.000> something<00:19:38.320> like<00:19:38.480> this<00:19:38.640> function<00:19:38.960> Z\nhave something like this function Z\nhave something like this function Z such<00:19:40.400> that<00:19:40.640> it<00:19:40.760> is<00:19:41.000> one<00:19:41.400> if<00:19:41.560> the<00:19:41.680> element<00:19:42.040> X<00:19:42.280> is\nsuch that it is one if the element X is\nsuch that it is one if the element X is above<00:19:42.840> T.\nabove T.\nabove T. It<00:19:44.000> is<00:19:44.160> zero<00:19:44.920> if<00:19:45.160> the<00:19:45.320> two<00:19:45.720> meet<00:19:46.000> at<00:19:46.080> the<00:19:46.160> bottom\nIt is zero if the two meet at the bottom\nIt is zero if the two meet at the bottom of<00:19:46.600> the<00:19:46.680> lattice,\nof the lattice,\nof the lattice, meaning\nmeaning\nmeaning uh<00:19:49.120> they<00:19:49.280> don't<00:19:49.520> imply<00:19:49.800> each<00:19:50.000> other<00:19:50.240> at<00:19:50.400> all.\nuh they don't imply each other at all.\nuh they don't imply each other at all. And<00:19:51.600> we<00:19:51.720> have<00:19:51.880> some<00:19:52.040> value<00:19:52.280> between<00:19:52.600> zero<00:19:52.880> and\nAnd we have some value between zero and\nAnd we have some value between zero and one<00:19:53.480> otherwise.\none otherwise.\none otherwise. And<00:19:56.160> this<00:19:56.640> generally<00:19:57.200> this<00:19:57.480> generalization\nAnd this generally this generalization\nAnd this generally this generalization of<00:19:58.160> the<00:19:58.400> inverse<00:19:58.720> zeta<00:19:58.920> function<00:19:59.280> is<00:19:59.480> then\nof the inverse zeta function is then\nof the inverse zeta function is then what<00:19:59.800> we're<00:19:59.880> going<00:20:00.000> to<00:20:00.080> call<00:20:00.320> probability.<00:20:01.480> It\nwhat we're going to call probability. It\nwhat we're going to call probability. It respects<00:20:02.200> the<00:20:02.240> ordering<00:20:02.480> of<00:20:02.560> the<00:20:02.640> zeta\nrespects the ordering of the zeta\nrespects the ordering of the zeta function,<00:20:03.160> but<00:20:03.320> allows<00:20:03.800> for<00:20:04.200> incomplete\nfunction, but allows for incomplete\nfunction, but allows for incomplete information.\ninformation.\ninformation. And<00:20:07.760> we're<00:20:07.920> going<00:20:08.080> to<00:20:08.600> derive<00:20:09.480> the<00:20:09.640> rules<00:20:09.800> of\nAnd we're going to derive the rules of\nAnd we're going to derive the rules of probability<00:20:10.480> by<00:20:10.640> looking<00:20:10.960> at<00:20:11.120> some\nprobability by looking at some\nprobability by looking at some symmetries<00:20:12.360> in<00:20:12.480> these<00:20:12.640> lattices.<00:20:13.520> The<00:20:13.600> most\nsymmetries in these lattices. The most\nsymmetries in these lattices. The most the<00:20:14.160> first<00:20:14.400> one<00:20:14.600> is<00:20:14.840> not<00:20:15.000> really<00:20:15.200> a<00:20:15.240> symmetry,\nthe first one is not really a symmetry,\nthe first one is not really a symmetry, it's<00:20:15.800> more<00:20:15.920> of<00:20:16.000> a<00:20:16.080> convention,<00:20:17.080> and<00:20:17.400> it's\nit's more of a convention, and it's\nit's more of a convention, and it's simply<00:20:17.880> that<00:20:18.280> elements<00:20:18.720> that<00:20:18.880> are<00:20:19.080> higher<00:20:19.520> up\nsimply that elements that are higher up\nsimply that elements that are higher up in<00:20:20.120> the<00:20:20.200> order<00:20:20.640> in<00:20:20.760> the<00:20:20.880> hierarchy<00:20:21.880> uh<00:20:22.240> are\nin the order in the hierarchy uh are\nin the order in the hierarchy uh are just<00:20:23.320> evaluated<00:20:24.000> by<00:20:24.480> higher<00:20:25.000> real<00:20:25.280> numbers.\njust evaluated by higher real numbers.\njust evaluated by higher real numbers. That's<00:20:26.400> all<00:20:26.520> it<00:20:26.600> means.<00:20:26.880> And<00:20:27.160> in<00:20:27.320> general,<00:20:27.760> for\nThat's all it means. And in general, for\nThat's all it means. And in general, for the<00:20:28.000> rest<00:20:28.280> of<00:20:28.640> the<00:20:28.760> presentation,<00:20:29.880> uh\nthe rest of the presentation, uh\nthe rest of the presentation, uh the<00:20:30.680> capital<00:20:31.040> letters<00:20:31.480> will<00:20:31.960> represent\nthe capital letters will represent\nthe capital letters will represent lattice<00:20:32.720> elements,\nlattice elements,\nlattice elements, and<00:20:34.600> small\nand small\nand small letters<00:20:36.120> will<00:20:36.240> represent<00:20:36.760> the<00:20:36.880> real<00:20:37.120> numbers,\nletters will represent the real numbers,\nletters will represent the real numbers, which<00:20:37.600> correspond<00:20:38.040> to<00:20:38.120> their<00:20:38.760> evaluations.\nNow,<00:20:41.960> the<00:20:42.080> first<00:20:42.320> symmetry<00:20:42.760> is<00:20:43.000> that<00:20:43.240> the\nNow, the first symmetry is that the\nNow, the first symmetry is that the combination<00:20:44.640> uh<00:20:45.200> preserves<00:20:45.760> order<00:20:46.000> from<00:20:46.160> the\ncombination uh preserves order from the\ncombination uh preserves order from the right<00:20:46.520> and<00:20:46.640> from<00:20:46.840> the<00:20:46.920> left.<00:20:47.280> So,<00:20:47.560> if<00:20:47.720> we<00:20:47.800> have\nright and from the left. So, if we have\nright and from the left. So, if we have two<00:20:48.520> elements,<00:20:49.000> one<00:20:49.120> is<00:20:49.200> strictly<00:20:49.880> above<00:20:50.120> the\ntwo elements, one is strictly above the\ntwo elements, one is strictly above the other,<00:20:51.040> then<00:20:52.000> uh\nother, then uh\nother, then uh the<00:20:52.680> join<00:20:52.960> operation\nthe join operation\nthe join operation uh<00:20:54.400> makes<00:20:54.680> it<00:20:54.800> so<00:20:55.080> that<00:20:55.600> kind<00:20:55.760> of<00:20:56.160> the<00:20:56.280> compound\nuh makes it so that kind of the compound\nuh makes it so that kind of the compound statements<00:20:57.840> uh<00:20:57.920> also<00:20:58.240> have<00:20:58.560> the<00:20:59.000> preserve<00:20:59.320> the\nstatements uh also have the preserve the\nstatements uh also have the preserve the order,<00:20:59.720> and<00:21:00.000> no<00:21:00.160> matter<00:21:00.480> from<00:21:00.640> which<00:21:00.840> side<00:21:01.240> you\norder, and no matter from which side you\norder, and no matter from which side you add<00:21:01.680> new<00:21:01.920> element.<00:21:02.920> And<00:21:03.560> by<00:21:03.720> extension,<00:21:04.280> this\nadd new element. And by extension, this\nadd new element. And by extension, this must<00:21:04.680> also<00:21:04.920> hold<00:21:05.240> for<00:21:05.360> the<00:21:05.480> operations<00:21:06.280> that\nmust also hold for the operations that\nmust also hold for the operations that quantifies<00:21:07.560> the<00:21:07.640> join<00:21:07.960> operation<00:21:08.680> of<00:21:08.920> these\nquantifies the join operation of these\nquantifies the join operation of these disjoint<00:21:09.800> elements.<00:21:10.240> So,<00:21:10.520> if<00:21:10.720> you<00:21:10.800> have<00:21:11.080> this\ndisjoint elements. So, if you have this\ndisjoint elements. So, if you have this uh<00:21:12.720> state<00:21:13.080> this<00:21:13.360> proposition<00:21:13.880> D,<00:21:14.600> which<00:21:14.760> is<00:21:14.920> A\nuh state this proposition D, which is A\nuh state this proposition D, which is A or<00:21:15.160> C,<00:21:16.000> then<00:21:16.120> the<00:21:16.160> valuation<00:21:16.680> of<00:21:16.800> D<00:21:16.960> must<00:21:17.200> be\nor C, then the valuation of D must be\nor C, then the valuation of D must be somehow<00:21:17.640> a<00:21:17.720> combination<00:21:18.240> of<00:21:18.360> the<00:21:18.440> valuation\nsomehow a combination of the valuation\nsomehow a combination of the valuation of<00:21:19.000> A<00:21:19.120> and<00:21:19.200> the<00:21:19.280> combination<00:21:19.600> of<00:21:19.680> C<00:21:19.880> for<00:21:20.000> this\nof A and the combination of C for this\nof A and the combination of C for this uh\nuh\nuh kind<00:21:21.920> of<00:21:22.040> plus<00:21:22.240> operator<00:21:22.600> that<00:21:22.760> we<00:21:22.920> will<00:21:23.080> see\nkind of plus operator that we will see\nkind of plus operator that we will see is<00:21:23.880> going<00:21:24.000> to<00:21:24.040> turn<00:21:24.240> out<00:21:24.360> to<00:21:24.440> be<00:21:24.560> the<00:21:24.640> sum.<00:21:25.360> And\nis going to turn out to be the sum. And\nis going to turn out to be the sum. And so,<00:21:25.560> here<00:21:25.760> we<00:21:25.840> have<00:21:25.960> the<00:21:26.040> same<00:21:26.280> symmetry\nso, here we have the same symmetry\nso, here we have the same symmetry reflected<00:21:27.560> uh<00:21:27.800> with<00:21:27.960> it.\nreflected uh with it.\nreflected uh with it. And<00:21:29.880> this<00:21:30.040> basically<00:21:30.360> means<00:21:30.760> that<00:21:31.040> the\nAnd this basically means that the\nAnd this basically means that the ordering<00:21:31.920> has<00:21:32.120> to<00:21:32.200> survive<00:21:32.560> a<00:21:32.640> combination\nordering has to survive a combination\nordering has to survive a combination with<00:21:33.680> any<00:21:33.880> arbitrary<00:21:34.360> context.<00:21:34.880> Otherwise,\nwith any arbitrary context. Otherwise,\nwith any arbitrary context. Otherwise, it<00:21:35.800> it's<00:21:36.240> basically<00:21:36.640> useless<00:21:37.200> for<00:21:37.600> any<00:21:37.760> kind\nit it's basically useless for any kind\nit it's basically useless for any kind of<00:21:38.000> reasoning.\nof reasoning.\nof reasoning. Uh<00:21:39.480> to<00:21:39.560> put<00:21:39.720> it<00:21:39.800> in<00:21:39.920> set<00:21:40.160> theoretical\nUh to put it in set theoretical\nUh to put it in set theoretical language,<00:21:41.000> if<00:21:41.360> X<00:21:41.560> is<00:21:41.920> strictly<00:21:42.440> in<00:21:42.560> contained\nlanguage, if X is strictly in contained\nlanguage, if X is strictly in contained in<00:21:43.000> Y,<00:21:43.480> then<00:21:44.320> if<00:21:44.680> you<00:21:44.960> uh\nin Y, then if you uh\nin Y, then if you uh if<00:21:45.840> you<00:21:46.160> add<00:21:46.400> another<00:21:46.720> set<00:21:47.200> to<00:21:47.320> both<00:21:47.640> sides,\nif you add another set to both sides,\nif you add another set to both sides, this<00:21:48.680> kind<00:21:48.880> of<00:21:49.240> ordering<00:21:49.600> relation,<00:21:50.080> this\nthis kind of ordering relation, this\nthis kind of ordering relation, this containment,<00:21:50.760> does<00:21:50.920> not<00:21:51.080> change.<00:21:52.000> And<00:21:52.120> to<00:21:52.240> put\ncontainment, does not change. And to put\ncontainment, does not change. And to put it<00:21:52.560> into<00:21:52.800> a<00:21:52.880> more<00:21:53.160> practical<00:21:53.600> example,\nit into a more practical example,\nit into a more practical example, um<00:21:55.280> we<00:21:55.480> all<00:21:55.880> we<00:21:56.000> know<00:21:56.200> that<00:21:56.480> all<00:21:56.640> dogs<00:21:56.920> are\num we all we know that all dogs are\num we all we know that all dogs are mammals,<00:21:57.600> but<00:21:57.760> not<00:21:57.960> all<00:21:58.080> mammals<00:21:58.440> are<00:21:58.520> dogs.\nmammals, but not all mammals are dogs.\nmammals, but not all mammals are dogs. Therefore,<00:21:59.200> being<00:21:59.440> a<00:21:59.480> dog<00:22:00.040> implies<00:22:00.480> being<00:22:00.720> a\nTherefore, being a dog implies being a\nTherefore, being a dog implies being a mammal,<00:22:01.520> which<00:22:01.720> we<00:22:01.800> could<00:22:02.000> write<00:22:02.320> as<00:22:02.720> uh\nmammal, which we could write as uh\nmammal, which we could write as uh like<00:22:03.840> this.\nlike this.\nlike this. Now,<00:22:06.040> if<00:22:06.160> combination<00:22:06.680> didn't<00:22:06.960> preserve\nNow, if combination didn't preserve\nNow, if combination didn't preserve order,<00:22:07.560> we<00:22:07.680> could<00:22:07.880> encounter<00:22:08.280> situations<00:22:09.440> uh\norder, we could encounter situations uh\norder, we could encounter situations uh such<00:22:10.000> that<00:22:10.680> dog<00:22:11.040> or<00:22:11.200> fish<00:22:11.800> is<00:22:11.960> somehow<00:22:12.800> implied\nsuch that dog or fish is somehow implied\nsuch that dog or fish is somehow implied by<00:22:13.760> mammal<00:22:14.120> or<00:22:14.200> fish,<00:22:14.480> which<00:22:14.680> is<00:22:14.760> of<00:22:14.880> course\nby mammal or fish, which is of course\nby mammal or fish, which is of course absurd.<00:22:16.120> Um\nabsurd. Um\nabsurd. Um because<00:22:17.600> uh\nbecause uh\nbecause uh these<00:22:19.560> additional<00:22:20.000> possibilities<00:22:20.760> for<00:22:20.880> what\nthese additional possibilities for what\nthese additional possibilities for what it<00:22:21.360> the<00:22:21.560> the<00:22:21.680> animal<00:22:21.960> really<00:22:22.160> is<00:22:22.400> shouldn't\nit the the animal really is shouldn't\nit the the animal really is shouldn't change<00:22:23.080> the<00:22:23.440> the<00:22:23.520> order<00:22:23.880> of<00:22:24.240> dog<00:22:24.440> and<00:22:24.560> mammal.\nAnd<00:22:27.640> then<00:22:27.800> there's<00:22:27.920> another<00:22:28.120> symmetry,<00:22:28.520> which\nAnd then there's another symmetry, which\nAnd then there's another symmetry, which is<00:22:28.840> simply<00:22:29.120> that<00:22:29.400> the<00:22:29.520> combination<00:22:30.240> is\nis simply that the combination is\nis simply that the combination is associative.\nassociative.\nassociative. Um\nUm\nUm And<00:22:33.520> this<00:22:33.680> follows<00:22:33.960> simply<00:22:34.160> from<00:22:34.320> the<00:22:34.400> fact\nAnd this follows simply from the fact\nAnd this follows simply from the fact that<00:22:35.000> the<00:22:35.240> the\nthat the the\nthat the the join<00:22:36.520> operator<00:22:37.040> is<00:22:37.280> defined<00:22:37.680> as<00:22:37.800> the<00:22:37.920> lower\njoin operator is defined as the lower\njoin operator is defined as the lower upper<00:22:38.400> bound,<00:22:39.120> which<00:22:39.360> just<00:22:39.600> takes<00:22:39.880> in<00:22:40.120> a<00:22:40.200> set\nupper bound, which just takes in a set\nupper bound, which just takes in a set of<00:22:40.520> elements,<00:22:41.080> and<00:22:41.280> so<00:22:41.480> it<00:22:41.640> doesn't<00:22:42.120> care<00:22:42.400> what\nof elements, and so it doesn't care what\nof elements, and so it doesn't care what grouping<00:22:42.840> at<00:22:42.960> all.\ngrouping at all.\ngrouping at all. Uh<00:22:44.000> it<00:22:44.080> doesn't<00:22:44.280> matter<00:22:44.480> which<00:22:44.720> order<00:22:45.480> uh<00:22:45.520> we\nUh it doesn't matter which order uh we\nUh it doesn't matter which order uh we do<00:22:45.800> it.\ndo it.\ndo it. And<00:22:47.000> so,<00:22:47.160> we're<00:22:47.280> going<00:22:47.400> to<00:22:47.520> use<00:22:47.840> the<00:22:47.960> most\nAnd so, we're going to use the most\nAnd so, we're going to use the most important<00:22:48.520> theorem<00:22:48.800> of<00:22:48.920> this<00:22:49.040> whole\nimportant theorem of this whole\nimportant theorem of this whole derivation,<00:22:49.720> which<00:22:49.880> is<00:22:50.000> the<00:22:50.320> associativity\nderivation, which is the associativity\nderivation, which is the associativity theorem.\ntheorem.\ntheorem. Uh\nUh\nUh we<00:22:52.960> have<00:22:53.440> two<00:22:53.520> disjoint<00:22:53.880> elements<00:22:54.560> of<00:22:54.680> a\nwe have two disjoint elements of a\nwe have two disjoint elements of a distributive<00:22:55.280> lattice<00:22:55.680> with<00:22:56.120> their\ndistributive lattice with their\ndistributive lattice with their respective<00:22:56.960> valuations.\nrespective valuations.\nrespective valuations. Now,<00:22:58.640> the<00:22:58.800> two<00:22:59.000> symmetries\nNow, the two symmetries\nNow, the two symmetries uh<00:23:00.480> it<00:23:00.560> translates<00:23:01.000> to<00:23:01.120> axioms<00:23:01.720> imply<00:23:02.360> that\nuh it translates to axioms imply that\nuh it translates to axioms imply that there<00:23:03.200> is<00:23:03.320> some<00:23:03.760> order<00:23:03.960> preserving<00:23:04.360> mapping\nthere is some order preserving mapping\nthere is some order preserving mapping for<00:23:04.920> which<00:23:05.760> this<00:23:06.320> kind<00:23:06.560> of<00:23:07.160> combination\nfor which this kind of combination\nfor which this kind of combination operator<00:23:08.800> uh,<00:23:09.320> is<00:23:10.040> addition.\noperator uh, is addition.\noperator uh, is addition. And<00:23:11.560> the<00:23:11.640> beauty<00:23:11.960> is<00:23:12.080> that<00:23:12.280> since<00:23:12.680> this<00:23:12.960> is<00:23:13.200> the\nAnd the beauty is that since this is the\nAnd the beauty is that since this is the first<00:23:13.960> operator<00:23:14.440> that<00:23:14.680> we<00:23:14.880> are<00:23:15.000> kind<00:23:15.200> of\nfirst operator that we are kind of\nfirst operator that we are kind of looking<00:23:15.560> at,<00:23:16.320> we<00:23:16.680> can<00:23:16.880> also<00:23:17.120> just<00:23:17.440> immediately\nlooking at, we can also just immediately\nlooking at, we can also just immediately throw<00:23:18.480> away<00:23:19.040> these<00:23:19.280> original<00:23:19.760> valuations\nthrow away these original valuations\nthrow away these original valuations that<00:23:20.520> we<00:23:20.640> had<00:23:20.880> for<00:23:21.040> all<00:23:21.200> of<00:23:21.280> these\nthat we had for all of these\nthat we had for all of these propositions<00:23:22.120> and<00:23:22.320> just<00:23:22.560> use<00:23:23.320> sort<00:23:23.600> of<00:23:24.040> theta\npropositions and just use sort of theta\npropositions and just use sort of theta theta<00:23:25.200> of<00:23:25.680> the<00:23:25.760> original<00:23:26.080> valuation<00:23:26.560> and<00:23:26.640> just\ntheta of the original valuation and just\ntheta of the original valuation and just use<00:23:27.120> addition<00:23:27.480> with<00:23:27.640> them.\nuse addition with them.\nuse addition with them. So,<00:23:28.880> we<00:23:29.000> can<00:23:29.640> use\nSo, we can use\nSo, we can use uh,\nuh,\nuh, sum<00:23:33.440> for<00:23:33.600> the<00:23:33.680> combination<00:23:34.240> of<00:23:34.400> disjoint\nsum for the combination of disjoint\nsum for the combination of disjoint elements<00:23:35.200> without<00:23:35.560> any<00:23:35.720> loss<00:23:35.920> of<00:23:36.000> generality.\nelements without any loss of generality.\nelements without any loss of generality. Now,<00:23:37.160> the<00:23:37.280> proof<00:23:37.520> of<00:23:37.640> this<00:23:37.960> is<00:23:38.640> too<00:23:38.760> long<00:23:39.040> for\nNow, the proof of this is too long for\nNow, the proof of this is too long for me<00:23:39.280> to<00:23:39.400> present<00:23:39.720> it<00:23:39.840> here.<00:23:40.000> It's<00:23:40.120> also<00:23:40.320> kind<00:23:40.440> of\nme to present it here. It's also kind of\nme to present it here. It's also kind of annoying,<00:23:41.480> uh,<00:23:41.640> but<00:23:42.080> uh,<00:23:42.400> you<00:23:42.560> can<00:23:42.760> find<00:23:42.960> it<00:23:43.080> in\nannoying, uh, but uh, you can find it in\nannoying, uh, but uh, you can find it in this<00:23:43.400> paper<00:23:44.040> by<00:23:44.320> Kevin<00:23:44.680> Knuth<00:23:45.000> and<00:23:45.120> John\nthis paper by Kevin Knuth and John\nthis paper by Kevin Knuth and John Skilling.<00:23:45.760> It<00:23:45.840> was<00:23:45.960> published<00:23:46.320> in<00:23:46.520> 2012.\nSkilling. It was published in 2012.\nSkilling. It was published in 2012. And<00:23:48.360> uh,\nAnd uh,\nAnd uh, there<00:23:49.840> there<00:23:50.040> the<00:23:50.160> paper<00:23:50.440> is<00:23:50.560> super\nthere there the paper is super\nthere there the paper is super interesting<00:23:51.280> and<00:23:51.400> it<00:23:51.520> rests<00:23:51.800> on<00:23:51.960> like<00:23:52.120> a\ninteresting and it rests on like a\ninteresting and it rests on like a decade<00:23:52.480> of<00:23:52.640> work<00:23:53.120> by<00:23:54.160> uh,<00:23:54.640> those<00:23:54.880> two.\ndecade of work by uh, those two.\ndecade of work by uh, those two. The<00:23:56.680> only<00:23:56.880> thing<00:23:57.120> that<00:23:57.880> I<00:23:58.000> would<00:23:58.160> recommend\nThe only thing that I would recommend\nThe only thing that I would recommend that<00:23:58.680> I<00:23:58.720> would<00:23:59.080> say<00:23:59.480> if<00:23:59.640> you<00:23:59.720> ever<00:23:59.960> read<00:24:00.160> this\nthat I would say if you ever read this\nthat I would say if you ever read this is<00:24:00.600> to<00:24:01.240> be<00:24:01.320> very<00:24:01.520> careful<00:24:01.840> because<00:24:02.200> they<00:24:02.440> are<00:24:02.600> a\nis to be very careful because they are a\nis to be very careful because they are a bit<00:24:02.880> rough<00:24:03.080> sometimes,<00:24:03.520> but<00:24:03.720> it's<00:24:03.960> absolutely\nbit rough sometimes, but it's absolutely\nbit rough sometimes, but it's absolutely a<00:24:04.600> gorgeous<00:24:05.400> paper.\nAnd<00:24:09.840> now<00:24:10.120> we<00:24:10.240> can<00:24:10.440> already<00:24:10.720> find<00:24:11.000> one<00:24:11.120> of<00:24:11.160> the\nAnd now we can already find one of the\nAnd now we can already find one of the familiar<00:24:12.320> rules<00:24:12.600> of<00:24:12.640> probability\nfamiliar rules of probability\nfamiliar rules of probability where<00:24:14.120> we<00:24:14.200> can<00:24:14.680> kind<00:24:14.800> of<00:24:14.880> get<00:24:15.000> a<00:24:15.040> hint<00:24:15.480> of<00:24:15.600> what\nwhere we can kind of get a hint of what\nwhere we can kind of get a hint of what it's<00:24:15.800> going<00:24:16.000> to<00:24:16.080> be.<00:24:16.880> Um,<00:24:17.520> we<00:24:17.640> can<00:24:17.880> send<00:24:18.120> out\nit's going to be. Um, we can send out\nit's going to be. Um, we can send out kind<00:24:18.440> of<00:24:18.520> see<00:24:18.640> the<00:24:18.720> combination<00:24:19.200> of<00:24:19.360> arbitrary\nkind of see the combination of arbitrary\nkind of see the combination of arbitrary arguments<00:24:20.160> because<00:24:20.480> before<00:24:20.920> we<00:24:21.080> only\narguments because before we only\narguments because before we only saw<00:24:22.400> that<00:24:22.520> we<00:24:22.600> could<00:24:22.760> use<00:24:22.960> addition<00:24:23.480> for\nsaw that we could use addition for\nsaw that we could use addition for disjoint<00:24:24.440> elements,<00:24:25.320> but<00:24:25.520> imagine<00:24:25.880> that<00:24:25.960> we\ndisjoint elements, but imagine that we\ndisjoint elements, but imagine that we have<00:24:26.280> X<00:24:26.440> and<00:24:26.600> Y<00:24:26.880> which<00:24:27.080> are<00:24:27.160> kind<00:24:27.400> of<00:24:27.680> have<00:24:27.920> some\nhave X and Y which are kind of have some\nhave X and Y which are kind of have some elements<00:24:28.960> in<00:24:29.080> common,<00:24:29.400> so<00:24:29.480> to<00:24:29.560> speak.\nelements in common, so to speak.\nelements in common, so to speak. And<00:24:31.400> then<00:24:31.760> if<00:24:31.880> we<00:24:31.960> want<00:24:32.120> to<00:24:32.200> have<00:24:32.360> their\nAnd then if we want to have their\nAnd then if we want to have their valuations,<00:24:33.760> uh,<00:24:34.080> we<00:24:34.280> would<00:24:34.720> could<00:24:34.880> of<00:24:34.960> course\nvaluations, uh, we would could of course\nvaluations, uh, we would could of course just<00:24:35.600> sum<00:24:36.480> their<00:24:36.720> their<00:24:36.920> components\njust sum their their components\njust sum their their components together,<00:24:37.920> but<00:24:38.200> if<00:24:38.280> you<00:24:38.360> want<00:24:38.520> to<00:24:38.560> express<00:24:38.960> it\ntogether, but if you want to express it\ntogether, but if you want to express it as<00:24:39.520> a<00:24:39.760> function<00:24:40.120> of<00:24:40.240> the<00:24:40.320> elements<00:24:41.400> of<00:24:41.640> X,<00:24:42.080> uh,\nas a function of the elements of X, uh,\nas a function of the elements of X, uh, sorry,<00:24:42.920> of<00:24:43.040> the\nsorry, of the\nsorry, of the the<00:24:44.400> valuation<00:24:44.760> of<00:24:44.880> X<00:24:45.080> and<00:24:45.200> the<00:24:45.240> valuation<00:24:45.640> of\nthe valuation of X and the valuation of\nthe valuation of X and the valuation of Y,<00:24:46.680> then<00:24:47.240> we<00:24:47.360> find<00:24:47.600> that<00:24:47.720> we<00:24:47.800> need<00:24:47.920> to<00:24:48.000> subtract\nY, then we find that we need to subtract\nY, then we find that we need to subtract the<00:24:48.440> valuation<00:24:48.960> of<00:24:49.080> the<00:24:49.160> element<00:24:49.520> that<00:24:49.640> they\nthe valuation of the element that they\nthe valuation of the element that they have<00:24:49.920> in<00:24:50.000> common.<00:24:50.720> And<00:24:50.880> this<00:24:51.120> leads<00:24:51.400> to<00:24:51.520> the\nhave in common. And this leads to the\nhave in common. And this leads to the general<00:24:52.160> inclusion-exclusion<00:24:53.640> uh,<00:24:53.720> sum<00:24:54.000> rule\ngeneral inclusion-exclusion uh, sum rule\ngeneral inclusion-exclusion uh, sum rule that<00:24:54.920> is<00:24:55.040> usually<00:24:55.480> taught<00:24:55.800> with<00:24:56.640> uh,<00:24:56.720> Venn\nthat is usually taught with uh, Venn\nthat is usually taught with uh, Venn diagrams<00:24:57.480> that<00:24:57.560> are<00:24:57.640> overlapping<00:24:58.360> stuff<00:24:58.560> like\ndiagrams that are overlapping stuff like\ndiagrams that are overlapping stuff like this.<00:24:58.920> I'm<00:24:59.040> sure<00:24:59.240> you've<00:24:59.400> all<00:24:59.520> seen<00:24:59.760> those.\nthis. I'm sure you've all seen those.\nthis. I'm sure you've all seen those. Um<00:25:01.280> and<00:25:01.440> interestingly,<00:25:02.160> we<00:25:02.320> also<00:25:02.520> can\nUm and interestingly, we also can\nUm and interestingly, we also can immediately\nimmediately\nimmediately find<00:25:04.720> out<00:25:04.960> that<00:25:05.840> this<00:25:06.240> um\nuh<00:25:09.800> the<00:25:09.920> commit<00:25:10.280> the<00:25:10.760> join<00:25:11.040> and<00:25:11.160> meet\nuh the commit the join and meet\nuh the commit the join and meet operations<00:25:12.200> are<00:25:12.440> associative<00:25:13.080> because<00:25:13.640> the\noperations are associative because the\noperations are associative because the commit<00:25:14.400> these<00:25:14.640> the<00:25:14.760> real<00:25:15.000> numbers<00:25:15.480> are<00:25:15.680> commit\ncommit these the real numbers are commit\ncommit these the real numbers are commit No,<00:25:16.400> sorry.<00:25:16.800> Did<00:25:16.960> I<00:25:17.000> say<00:25:17.120> associative?<00:25:17.600> I\nNo, sorry. Did I say associative? I\nNo, sorry. Did I say associative? I meant<00:25:17.840> commutative,<00:25:18.400> sorry.<00:25:19.240> Because<00:25:19.440> the\nmeant commutative, sorry. Because the\nmeant commutative, sorry. Because the real<00:25:19.720> numbers<00:25:20.040> are<00:25:20.160> of<00:25:20.280> course<00:25:20.440> commutative.\nreal numbers are of course commutative.\nreal numbers are of course commutative. So,<00:25:21.120> since<00:25:21.320> the<00:25:21.440> valuations<00:25:22.360> have<00:25:22.520> this\nSo, since the valuations have this\nSo, since the valuations have this property,\nproperty,\nproperty, uh<00:25:24.320> also<00:25:24.560> the<00:25:25.280> algebra<00:25:25.520> of<00:25:25.560> the<00:25:25.640> individual\nuh also the algebra of the individual\nuh also the algebra of the individual elements<00:25:26.400> must<00:25:26.600> have<00:25:26.760> it.\nThen<00:25:29.960> we're<00:25:30.040> going<00:25:30.160> to<00:25:30.240> introduce<00:25:30.520> a<00:25:30.560> new\nThen we're going to introduce a new\nThen we're going to introduce a new symmetry,\nsymmetry,\nsymmetry, uh<00:25:32.000> which<00:25:32.200> is<00:25:32.520> Imagine<00:25:32.840> we<00:25:32.920> have<00:25:33.080> two\nuh which is Imagine we have two\nuh which is Imagine we have two independent<00:25:33.760> systems,<00:25:34.400> S1<00:25:34.720> S2,<00:25:35.200> that<00:25:35.440> we<00:25:35.560> want\nindependent systems, S1 S2, that we want\nindependent systems, S1 S2, that we want to<00:25:35.840> combine.<00:25:36.440> We<00:25:36.520> sort<00:25:36.680> of<00:25:36.760> want<00:25:36.960> to<00:25:37.440> consider\nto combine. We sort of want to consider\nto combine. We sort of want to consider them<00:25:37.920> together.<00:25:38.880> And<00:25:39.200> these<00:25:39.360> two<00:25:39.480> systems\nthem together. And these two systems\nthem together. And these two systems would<00:25:40.000> then<00:25:40.160> be<00:25:40.360> two<00:25:40.520> different<00:25:40.840> lattices.<00:25:41.840> Um\nwould then be two different lattices. Um\nwould then be two different lattices. Um And<00:25:42.480> this<00:25:42.640> forms<00:25:42.840> a<00:25:42.880> new<00:25:43.080> space<00:25:43.800> with<00:25:44.080> the<00:25:44.160> type\nAnd this forms a new space with the type\nAnd this forms a new space with the type uh\nuh\nuh defined<00:25:45.840> by<00:25:45.960> some<00:25:46.080> kind<00:25:46.240> of<00:25:46.320> cross<00:25:46.600> product<00:25:46.880> of\ndefined by some kind of cross product of\ndefined by some kind of cross product of their<00:25:47.120> elements.\ntheir elements.\ntheir elements. And<00:25:48.600> the<00:25:49.080> direct<00:25:49.400> product<00:25:49.800> operator<00:25:51.040> uh<00:25:51.400> is\nAnd the direct product operator uh is\nAnd the direct product operator uh is right<00:25:52.160> distributive\nright distributive\nright distributive over<00:25:53.880> the<00:25:54.560> join<00:25:54.840> operation.\nTherefore,<00:25:58.480> the<00:25:59.120> operator<00:25:59.440> that<00:25:59.560> we<00:25:59.640> choose\nTherefore, the operator that we choose\nTherefore, the operator that we choose for<00:26:00.080> this<00:26:00.600> this<00:26:00.800> kind<00:26:01.000> of<00:26:01.560> uh\nfor this this kind of uh\nfor this this kind of uh crossed<00:26:02.480> out<00:26:02.920> o\ncrossed out o\ncrossed out o must<00:26:04.000> also<00:26:04.240> satisfy<00:26:04.560> this<00:26:04.720> property.<00:26:05.160> And<00:26:05.400> why\nmust also satisfy this property. And why\nmust also satisfy this property. And why do<00:26:05.720> we<00:26:05.800> think<00:26:05.960> this<00:26:06.120> is<00:26:06.280> reasonable?\ndo we think this is reasonable?\ndo we think this is reasonable? Uh<00:26:08.000> because<00:26:08.800> um\nUh because um\nUh because um if<00:26:09.760> we<00:26:09.840> imagine<00:26:10.160> that<00:26:10.240> we<00:26:10.320> want<00:26:10.520> to<00:26:10.640> analyze\nif we imagine that we want to analyze\nif we imagine that we want to analyze coin<00:26:11.400> tosses<00:26:11.800> and<00:26:11.960> die<00:26:12.200> tosses<00:26:12.560> at<00:26:12.680> the<00:26:12.760> same\ncoin tosses and die tosses at the same\ncoin tosses and die tosses at the same time,\ntime,\ntime, um<00:26:15.240> let's<00:26:15.480> propositions<00:26:16.480> refer<00:26:16.680> to<00:26:16.800> the\num let's propositions refer to the\num let's propositions refer to the outcome<00:26:17.160> of<00:26:17.240> the<00:26:17.320> coin<00:26:17.560> toss<00:26:17.720> just<00:26:17.880> be<00:26:18.160> H<00:26:18.360> and\noutcome of the coin toss just be H and\noutcome of the coin toss just be H and T,<00:26:18.800> and<00:26:18.960> then<00:26:19.120> we<00:26:19.240> have<00:26:19.440> D<00:26:20.280> K<00:26:20.920> with<00:26:21.120> K<00:26:21.440> just\nT, and then we have D K with K just\nT, and then we have D K with K just being<00:26:22.120> one<00:26:22.240> of<00:26:22.320> six,\nbeing one of six,\nbeing one of six, not<00:26:23.480> outcomes<00:26:23.840> of<00:26:23.880> the<00:26:23.960> die.\nnot outcomes of the die.\nnot outcomes of the die. Um<00:26:25.560> and<00:26:25.680> we<00:26:25.800> treat<00:26:26.000> them<00:26:26.120> as<00:26:26.280> independent.<00:26:27.440> Um\nUm and we treat them as independent. Um\nUm and we treat them as independent. Um if<00:26:28.480> we<00:26:28.600> had<00:26:28.840> it<00:26:28.920> that<00:26:29.080> somehow\nif we had it that somehow\nif we had it that somehow this<00:26:30.920> uh\nthis uh\nthis uh the<00:26:31.600> quality<00:26:31.840> before<00:26:32.120> was<00:26:32.240> not<00:26:32.440> true<00:26:32.720> as<00:26:32.840> shown\nthe quality before was not true as shown\nthe quality before was not true as shown here,<00:26:33.800> then<00:26:33.960> it<00:26:34.040> would<00:26:34.240> mean<00:26:34.520> that<00:26:34.680> somehow\nhere, then it would mean that somehow\nhere, then it would mean that somehow considering<00:26:35.840> the<00:26:35.960> outcomes<00:26:36.440> of<00:26:36.480> the<00:26:36.600> coin\nconsidering the outcomes of the coin\nconsidering the outcomes of the coin toss<00:26:37.560> would<00:26:37.960> be<00:26:38.200> different<00:26:38.640> if<00:26:38.800> you<00:26:38.880> paired\ntoss would be different if you paired\ntoss would be different if you paired them<00:26:39.280> with<00:26:39.400> the<00:26:39.520> outcomes<00:26:39.800> of<00:26:39.880> the<00:26:40.000> die,<00:26:40.960> uh\nthem with the outcomes of the die, uh\nthem with the outcomes of the die, uh which<00:26:41.400> would<00:26:41.720> kind<00:26:41.920> of<00:26:42.000> mean<00:26:42.200> that<00:26:42.320> there<00:26:42.400> was\nwhich would kind of mean that there was\nwhich would kind of mean that there was some<00:26:42.800> kind<00:26:43.000> of<00:26:43.280> entanglement,<00:26:43.840> some<00:26:43.960> kind<00:26:44.120> of\nsome kind of entanglement, some kind of\nsome kind of entanglement, some kind of relationship<00:26:45.480> uh<00:26:45.560> that<00:26:46.360> uh<00:26:46.560> directly\nrelationship uh that uh directly\nrelationship uh that uh directly contradicts<00:26:47.520> our<00:26:47.640> assumption<00:26:48.000> of\ncontradicts our assumption of\ncontradicts our assumption of independence.\nindependence.\nindependence. And<00:26:49.840> this<00:26:50.000> This<00:26:50.200> us<00:26:50.280> to<00:26:50.360> a<00:26:50.440> very<00:26:50.680> important\nAnd this This us to a very important\nAnd this This us to a very important point.<00:26:52.160> Uh<00:26:52.720> this<00:26:52.920> is<00:26:53.040> more<00:26:53.160> like<00:26:53.280> a\npoint. Uh this is more like a\npoint. Uh this is more like a parenthesis,<00:26:53.840> but<00:26:54.000> it's<00:26:54.120> very<00:26:54.280> important.\nparenthesis, but it's very important.\nparenthesis, but it's very important. Independence\nIndependence\nIndependence is<00:26:57.920> classically<00:26:58.400> seen<00:26:59.000> or<00:26:59.520> in<00:26:59.640> the<00:26:59.720> orthodox\nis classically seen or in the orthodox\nis classically seen or in the orthodox methodology<00:27:00.520> somehow<00:27:01.160> a<00:27:01.240> property<00:27:01.920> of<00:27:02.040> the\nmethodology somehow a property of the\nmethodology somehow a property of the experiment.<00:27:02.760> You<00:27:02.880> have<00:27:03.200> independent<00:27:03.680> trials,\nexperiment. You have independent trials,\nexperiment. You have independent trials, independent<00:27:05.240> draws<00:27:05.520> from<00:27:05.640> distribution.\nindependent draws from distribution.\nindependent draws from distribution. However,<00:27:07.720> the<00:27:07.800> better<00:27:08.080> way<00:27:08.160> to<00:27:08.280> reason<00:27:08.520> about\nHowever, the better way to reason about\nHowever, the better way to reason about it<00:27:08.840> is<00:27:09.120> independence<00:27:09.680> is<00:27:09.920> epistemological,\nit is independence is epistemological,\nit is independence is epistemological, meaning<00:27:11.400> it's<00:27:11.600> just<00:27:11.800> a<00:27:11.840> property<00:27:12.640> of<00:27:12.960> our\nmeaning it's just a property of our\nmeaning it's just a property of our knowledge.\nFormally,<00:27:16.400> independence<00:27:17.000> simply<00:27:17.280> means<00:27:17.560> that\nFormally, independence simply means that\nFormally, independence simply means that learning<00:27:18.680> about<00:27:19.160> one<00:27:19.400> system<00:27:20.000> doesn't<00:27:20.360> tell\nlearning about one system doesn't tell\nlearning about one system doesn't tell us<00:27:20.720> anything<00:27:21.040> about<00:27:21.240> the<00:27:21.360> other.\nus anything about the other.\nus anything about the other. Now,<00:27:22.960> to<00:27:23.120> illustrate<00:27:23.960> why<00:27:24.480> in<00:27:24.600> my<00:27:24.720> opinion\nNow, to illustrate why in my opinion\nNow, to illustrate why in my opinion it's<00:27:25.200> better<00:27:25.400> to<00:27:25.520> say<00:27:25.680> this<00:27:25.840> is\nit's better to say this is\nit's better to say this is epistemological<00:27:27.240> is<00:27:27.880> imagine<00:27:28.320> the<00:27:28.800> classical\nepistemological is imagine the classical\nepistemological is imagine the classical Bernoulli<00:27:29.920> urn,<00:27:30.200> right?<00:27:30.440> You<00:27:30.520> have<00:27:30.960> red<00:27:31.200> balls\nBernoulli urn, right? You have red balls\nBernoulli urn, right? You have red balls and<00:27:31.680> white<00:27:31.920> balls.\nand white balls.\nand white balls. You<00:27:33.480> draw<00:27:33.880> one<00:27:34.440> and<00:27:34.600> you<00:27:34.840> record<00:27:35.120> the<00:27:35.160> result\nYou draw one and you record the result\nYou draw one and you record the result and<00:27:35.560> you<00:27:35.600> put<00:27:35.760> it<00:27:35.840> back<00:27:36.040> in,<00:27:36.200> you<00:27:36.320> shake<00:27:36.600> it<00:27:36.680> and\nand you put it back in, you shake it and\nand you put it back in, you shake it and whatnot.\nwhatnot.\nwhatnot. Those<00:27:39.160> this<00:27:39.320> kind<00:27:39.440> of<00:27:39.520> experiment<00:27:40.200> is\nThose this kind of experiment is\nThose this kind of experiment is typically<00:27:40.600> described<00:27:41.160> as<00:27:42.080> IID<00:27:42.560> random\ntypically described as IID random\ntypically described as IID random variables,<00:27:43.360> right?<00:27:43.600> You<00:27:43.760> have<00:27:44.560> always<00:27:44.800> the\nvariables, right? You have always the\nvariables, right? You have always the same<00:27:45.040> distribution<00:27:45.600> of<00:27:45.760> the<00:27:45.880> same<00:27:46.120> balls<00:27:46.840> and\nsame distribution of the same balls and\nsame distribution of the same balls and all<00:27:47.160> the<00:27:47.280> draws<00:27:47.600> are<00:27:47.680> independent,<00:27:48.240> meaning\nall the draws are independent, meaning\nall the draws are independent, meaning seeing<00:27:49.280> one<00:27:49.600> doesn't<00:27:50.080> tell<00:27:50.120> you<00:27:50.240> anything\nseeing one doesn't tell you anything\nseeing one doesn't tell you anything about<00:27:50.760> the<00:27:50.840> other.<00:27:51.280> However,<00:27:52.040> physically\nabout the other. However, physically\nabout the other. However, physically speaking,<00:27:53.400> they<00:27:53.600> very<00:27:53.920> much<00:27:54.280> depend<00:27:54.640> on<00:27:54.760> each\nspeaking, they very much depend on each\nspeaking, they very much depend on each other.\nother.\nother. I<00:27:56.400> mean,<00:27:56.840> the<00:27:57.280> the<00:27:57.360> next<00:27:57.600> result<00:27:57.960> definitely\nI mean, the the next result definitely\nI mean, the the next result definitely depends<00:27:58.680> on<00:27:58.760> which<00:27:58.920> color<00:27:59.160> was<00:27:59.280> previously\ndepends on which color was previously\ndepends on which color was previously drawn<00:28:00.000> and<00:28:00.120> then<00:28:00.360> from<00:28:01.040> where<00:28:01.240> this<00:28:01.400> ball<00:28:01.640> was\ndrawn and then from where this ball was\ndrawn and then from where this ball was placed<00:28:02.120> again<00:28:02.360> into<00:28:02.560> the<00:28:02.640> ball,\nplaced again into the ball,\nplaced again into the ball, how<00:28:03.800> the<00:28:03.920> ball<00:28:04.200> was<00:28:04.400> shaken<00:28:05.120> and<00:28:05.320> of<00:28:05.440> course\nhow the ball was shaken and of course\nhow the ball was shaken and of course and\nand\nand all<00:28:06.480> the<00:28:06.600> attrition<00:28:07.120> coefficients<00:28:07.800> between\nall the attrition coefficients between\nall the attrition coefficients between the<00:28:08.200> walls\nthe walls\nthe walls of<00:28:09.600> the<00:28:09.960> and<00:28:10.080> the<00:28:10.160> balls<00:28:10.560> and<00:28:10.680> between<00:28:10.920> the\nof the and the balls and between the\nof the and the balls and between the balls<00:28:11.240> themselves,<00:28:11.640> etc.<00:28:12.200> But\nballs themselves, etc. But\nballs themselves, etc. But unfortunately,<00:28:12.800> all<00:28:12.960> of<00:28:13.040> these<00:28:13.240> influences\nunfortunately, all of these influences\nunfortunately, all of these influences are<00:28:13.920> so<00:28:14.080> incredibly<00:28:14.960> complicated<00:28:15.560> to<00:28:15.640> track\nare so incredibly complicated to track\nare so incredibly complicated to track that\nthat\nthat our<00:28:18.000> attempts<00:28:18.440> to<00:28:18.720> extract<00:28:19.200> any<00:28:19.360> kind<00:28:19.520> of\nour attempts to extract any kind of\nour attempts to extract any kind of information<00:28:20.080> from<00:28:20.240> them<00:28:20.480> would<00:28:20.640> be<00:28:20.800> almost\ninformation from them would be almost\ninformation from them would be almost futile.\nfutile.\nfutile. And<00:28:22.520> so<00:28:22.680> the<00:28:22.800> draws<00:28:23.160> are<00:28:23.280> definitely<00:28:23.680> not\nAnd so the draws are definitely not\nAnd so the draws are definitely not physically<00:28:24.680> independent.<00:28:25.320> However,\nphysically independent. However,\nphysically independent. However, we<00:28:27.920> just<00:28:28.280> because<00:28:28.560> we<00:28:28.680> don't<00:28:28.920> know<00:28:29.080> how<00:28:29.200> to<00:28:29.320> do\nwe just because we don't know how to do\nwe just because we don't know how to do any<00:28:29.600> better,<00:28:29.840> we<00:28:30.000> ignore<00:28:30.480> all<00:28:30.640> of<00:28:30.760> these\nany better, we ignore all of these\nany better, we ignore all of these non-linear<00:28:31.760> dependencies.<00:28:32.640> And<00:28:32.760> in<00:28:32.840> fact,\nnon-linear dependencies. And in fact,\nnon-linear dependencies. And in fact, confusing<00:28:34.240> physical<00:28:34.680> dependence<00:28:35.200> with\nconfusing physical dependence with\nconfusing physical dependence with logical<00:28:35.840> independence<00:28:36.800> has<00:28:37.040> led<00:28:37.280> to<00:28:37.640> leads<00:28:37.880> to\nlogical independence has led to leads to\nlogical independence has led to leads to all<00:28:38.320> sorts<00:28:38.520> of<00:28:38.640> paradoxes<00:28:39.200> in<00:28:39.280> practice.<00:28:39.960> Um\nall sorts of paradoxes in practice. Um\nall sorts of paradoxes in practice. Um Famously,\nFamously,\nFamously, in<00:28:43.480> quantum<00:28:43.760> mechanics,<00:28:44.480> this<00:28:44.920> this<00:28:45.080> is\nin quantum mechanics, this this is\nin quantum mechanics, this this is uh\nuh\nuh you<00:28:46.800> have<00:28:46.960> some<00:28:47.080> paradoxes<00:28:47.520> because<00:28:47.800> of<00:28:47.920> this,\nyou have some paradoxes because of this,\nyou have some paradoxes because of this, but<00:28:48.200> in<00:28:48.320> all<00:28:48.440> sorts<00:28:48.640> of<00:28:48.760> other<00:28:48.920> fields<00:28:49.200> as\nbut in all sorts of other fields as\nbut in all sorts of other fields as well.\nwell.\nwell. So,<00:28:50.080> this<00:28:50.240> is<00:28:50.360> distinction<00:28:50.760> is<00:28:50.880> very\nSo, this is distinction is very\nSo, this is distinction is very important<00:28:51.440> to<00:28:51.480> keep<00:28:51.680> in<00:28:51.760> mind.\nThen<00:28:54.400> we<00:28:54.480> have<00:28:54.600> a<00:28:54.640> new<00:28:54.800> symmetry.\nThen we have a new symmetry.\nThen we have a new symmetry. Uh<00:28:56.200> we<00:28:56.320> assume<00:28:56.880> or<00:28:57.200> we<00:28:57.360> see<00:28:57.480> that<00:28:57.600> the<00:28:58.000> the<00:28:58.360> the\nUh we assume or we see that the the the\nUh we assume or we see that the the the direct<00:28:58.840> product<00:28:59.240> is<00:28:59.480> associative.\ndirect product is associative.\ndirect product is associative. And<00:29:00.960> therefore<00:29:01.200> also<00:29:01.360> the<00:29:01.480> operator.<00:29:02.280> And<00:29:03.160> um\nAnd therefore also the operator. And um\nAnd therefore also the operator. And um again,<00:29:03.560> this<00:29:03.720> would<00:29:03.920> mean<00:29:04.240> that<00:29:04.400> if<00:29:04.520> this<00:29:04.720> were\nagain, this would mean that if this were\nagain, this would mean that if this were not<00:29:05.040> true,\nnot true,\nnot true, then<00:29:06.880> the<00:29:07.000> nature<00:29:07.520> of<00:29:07.680> the<00:29:07.800> elements<00:29:08.160> of<00:29:08.240> a\nthen the nature of the elements of a\nthen the nature of the elements of a composite<00:29:08.800> system<00:29:09.160> would<00:29:09.280> somehow<00:29:09.560> depend<00:29:10.760> on\ncomposite system would somehow depend on\ncomposite system would somehow depend on the<00:29:10.960> order<00:29:11.200> in<00:29:11.280> which<00:29:11.400> the<00:29:11.800> independent\nthe order in which the independent\nthe order in which the independent contributions<00:29:12.800> were<00:29:12.960> added.\ncontributions were added.\ncontributions were added. Uh<00:29:14.120> meaning,<00:29:14.560> if<00:29:14.760> we<00:29:14.840> somehow<00:29:15.680> combined<00:29:16.600> coins\nUh meaning, if we somehow combined coins\nUh meaning, if we somehow combined coins and<00:29:17.280> die<00:29:17.480> first<00:29:17.880> and<00:29:18.000> then<00:29:18.240> added<00:29:18.480> somehow<00:29:18.760> an\nand die first and then added somehow an\nand die first and then added somehow an analysis<00:29:19.480> on<00:29:19.880> the<00:29:20.000> weather,<00:29:20.720> just<00:29:20.880> some<00:29:21.280> uh\nanalysis on the weather, just some uh\nanalysis on the weather, just some uh random<00:29:22.640> example,<00:29:23.280> would<00:29:23.480> yield<00:29:23.680> a<00:29:23.720> different\nrandom example, would yield a different\nrandom example, would yield a different uh\nuh\nuh joint<00:29:26.040> space,<00:29:26.320> different<00:29:26.560> probability\njoint space, different probability\njoint space, different probability space,<00:29:27.360> than<00:29:27.520> if<00:29:27.640> we<00:29:27.720> combined<00:29:28.400> die<00:29:28.640> and<00:29:28.760> the\nspace, than if we combined die and the\nspace, than if we combined die and the weather<00:29:29.080> first<00:29:29.400> and<00:29:29.520> then<00:29:29.760> added<00:29:29.960> the<00:29:30.040> coin,\nweather first and then added the coin,\nweather first and then added the coin, which<00:29:30.480> makes<00:29:30.680> no<00:29:30.840> sense,<00:29:31.080> of<00:29:31.200> course.\nAnd<00:29:34.600> we<00:29:34.720> can<00:29:34.920> also<00:29:35.720> to<00:29:35.880> apply<00:29:36.280> our\nAnd we can also to apply our\nAnd we can also to apply our associativity<00:29:37.040> theorem,<00:29:37.320> we<00:29:37.440> also<00:29:37.720> need\nassociativity theorem, we also need\nassociativity theorem, we also need symmetry<00:29:39.440> one.<00:29:39.840> We<00:29:39.920> need<00:29:40.040> to<00:29:40.160> show<00:29:40.320> it<00:29:40.440> for\nsymmetry one. We need to show it for\nsymmetry one. We need to show it for this<00:29:40.960> new<00:29:41.520> operator.\nthis new operator.\nthis new operator. The<00:29:42.840> direct<00:29:43.120> product<00:29:43.480> also<00:29:43.760> preserves<00:29:44.160> order.\nThe direct product also preserves order.\nThe direct product also preserves order. Uh<00:29:46.160> meaning\nUh meaning\nUh meaning um\num\num that<00:29:48.560> if<00:29:48.680> we<00:29:48.760> have<00:29:48.880> some<00:29:49.000> kind<00:29:49.160> of<00:29:49.480> uh\nthat if we have some kind of uh\nthat if we have some kind of uh uh<00:29:51.480> proposition<00:29:51.960> Y<00:29:52.560> that<00:29:52.760> can<00:29:52.920> be<00:29:53.400> decomposed\nuh proposition Y that can be decomposed\nuh proposition Y that can be decomposed into\ninto\ninto uh<00:29:55.600> X<00:29:55.840> or<00:29:55.960> D,\nuh X or D,\nuh X or D, if<00:29:57.720> we<00:29:57.800> apply<00:29:58.320> the<00:29:58.520> direct<00:29:58.880> product,<00:29:59.280> we\nif we apply the direct product, we\nif we apply the direct product, we obtain<00:30:00.200> uh<00:30:00.360> this.<00:30:01.200> But<00:30:01.360> since<00:30:01.960> neither<00:30:02.440> X\nobtain uh this. But since neither X\nobtain uh this. But since neither X nor<00:30:04.280> U<00:30:04.840> nor<00:30:05.480> D<00:30:05.600> times<00:30:05.920> X<00:30:06.840> are<00:30:07.080> the<00:30:07.200> bottom\nnor U nor D times X are the bottom\nnor U nor D times X are the bottom elements<00:30:08.120> here<00:30:08.360> by<00:30:08.480> assumption.<00:30:09.040> Uh<00:30:09.080> this<00:30:09.320> is\nelements here by assumption. Uh this is\nelements here by assumption. Uh this is definitely<00:30:09.760> possible.\ndefinitely possible.\ndefinitely possible. Uh<00:30:11.240> then<00:30:11.480> it<00:30:11.600> must<00:30:11.880> mean<00:30:12.320> that<00:30:12.480> somehow<00:30:13.600> Y\nUh then it must mean that somehow Y\nUh then it must mean that somehow Y times<00:30:14.600> U\ntimes U\ntimes U must<00:30:15.880> have<00:30:16.120> a<00:30:16.200> higher<00:30:16.400> valuation\nmust have a higher valuation\nmust have a higher valuation than<00:30:17.640> X<00:30:17.880> times<00:30:18.160> U<00:30:18.360> because<00:30:19.000> in<00:30:19.120> order<00:30:19.280> to\nthan X times U because in order to\nthan X times U because in order to obtain<00:30:19.880> this<00:30:20.080> one,<00:30:20.240> we<00:30:20.320> must<00:30:20.480> have<00:30:20.880> combined<00:30:21.280> X\nobtain this one, we must have combined X\nobtain this one, we must have combined X times<00:30:21.680> U<00:30:21.960> with<00:30:22.680> another<00:30:22.920> valuation.<00:30:23.480> And<00:30:23.600> so,\ntimes U with another valuation. And so,\ntimes U with another valuation. And so, we<00:30:24.040> can\nwe can\nwe can write<00:30:25.360> this<00:30:25.480> as<00:30:25.600> such.\nwrite this as such.\nwrite this as such. To<00:30:26.760> see<00:30:26.880> that<00:30:27.040> X<00:30:27.240> times<00:30:27.480> U<00:30:27.800> is<00:30:28.080> smaller<00:30:28.400> than,\nTo see that X times U is smaller than,\nTo see that X times U is smaller than, well,<00:30:29.240> this<00:30:29.880> uh\nwell, this uh\nwell, this uh combined<00:30:31.440> statements,<00:30:32.080> which<00:30:32.320> is<00:30:32.440> smaller\ncombined statements, which is smaller\ncombined statements, which is smaller than<00:30:33.560> the<00:30:33.640> top<00:30:33.880> one.\nthan the top one.\nthan the top one. And<00:30:34.880> so,<00:30:35.040> we<00:30:35.160> have<00:30:35.320> symmetry<00:30:35.720> one<00:30:36.480> again<00:30:36.680> also\nAnd so, we have symmetry one again also\nAnd so, we have symmetry one again also for<00:30:37.040> this<00:30:37.640> uh\nfor this uh\nfor this uh product.\nproduct.\nproduct. Now,<00:30:39.840> interestingly,<00:30:40.280> we<00:30:40.440> can<00:30:41.080> uh<00:30:41.160> apply<00:30:41.400> the\nNow, interestingly, we can uh apply the\nNow, interestingly, we can uh apply the associ-<00:30:42.120> associativity<00:30:42.840> theorem<00:30:43.280> again,\nassoci- associativity theorem again,\nassoci- associativity theorem again, which<00:30:43.920> is<00:30:44.000> very<00:30:44.200> convenient,<00:30:44.560> of<00:30:44.680> course.\nwhich is very convenient, of course.\nwhich is very convenient, of course. Um<00:30:46.200> so<00:30:46.320> again,<00:30:46.920> we<00:30:47.040> have\nUm so again, we have\nUm so again, we have two<00:30:48.560> propositions\ntwo propositions\ntwo propositions um<00:30:50.640> of<00:30:50.800> the<00:30:51.440> that<00:30:51.640> are<00:30:51.760> elements<00:30:52.080> of<00:30:52.200> the\num of the that are elements of the\num of the that are elements of the lattice<00:30:53.280> and<00:30:53.400> with<00:30:53.600> the<00:30:53.720> evaluations<00:30:54.520> and<00:30:54.720> we\nlattice and with the evaluations and we\nlattice and with the evaluations and we must<00:30:55.240> and<00:30:55.400> we<00:30:55.480> know<00:30:55.720> that<00:30:55.880> somehow<00:30:56.160> they<00:30:56.280> must\nmust and we know that somehow they must\nmust and we know that somehow they must satisfy<00:30:57.080> this.<00:30:57.360> However,\nsatisfy this. However,\nsatisfy this. However, while<00:30:58.920> before<00:30:59.680> with<00:30:59.840> the<00:30:59.920> combination,<00:31:00.560> we\nwhile before with the combination, we\nwhile before with the combination, we could<00:31:00.800> just<00:31:00.960> simply<00:31:01.160> throw<00:31:01.440> away\ncould just simply throw away\ncould just simply throw away the<00:31:02.680> old<00:31:03.120> valuations<00:31:03.800> and<00:31:03.960> simply<00:31:04.520> use<00:31:04.920> the\nthe old valuations and simply use the\nthe old valuations and simply use the the<00:31:06.440> the<00:31:06.560> new<00:31:06.800> ones<00:31:07.120> given<00:31:07.320> us<00:31:07.440> by<00:31:07.520> the\nthe the new ones given us by the\nthe the new ones given us by the mapping,<00:31:08.000> here<00:31:08.240> we<00:31:08.320> cannot<00:31:08.560> do<00:31:08.640> this<00:31:08.880> because\nmapping, here we cannot do this because\nmapping, here we cannot do this because the<00:31:09.280> plus<00:31:09.640> operation<00:31:10.120> was<00:31:10.280> already<00:31:11.080> chosen\nthe plus operation was already chosen\nthe plus operation was already chosen uh<00:31:12.520> for<00:31:12.760> the<00:31:12.840> combination<00:31:13.280> of<00:31:13.400> the<00:31:13.480> joint\nuh for the combination of the joint\nuh for the combination of the joint elements.<00:31:14.080> So,<00:31:14.320> we<00:31:14.480> must<00:31:14.960> find<00:31:15.680> uh<00:31:15.800> something\nelements. So, we must find uh something\nelements. So, we must find uh something else.\nelse.\nelse. And<00:31:17.240> to<00:31:17.320> do<00:31:17.520> that,<00:31:18.240> uh<00:31:18.400> we<00:31:18.520> must<00:31:19.000> find<00:31:19.320> a<00:31:19.360> form\nAnd to do that, uh we must find a form\nAnd to do that, uh we must find a form of<00:31:20.600> theta<00:31:20.920> tilde.\nof theta tilde.\nof theta tilde. To<00:31:22.600> do<00:31:22.760> that,<00:31:23.000> we<00:31:23.600> um\nTo do that, we um\nTo do that, we um set<00:31:24.520> up<00:31:24.640> some<00:31:24.960> functional<00:31:25.111> [snorts]<00:31:25.280> equation\nset up some functional [snorts] equation\nset up some functional [snorts] equation uh<00:31:26.560> because\nuh because\nuh because uh<00:31:28.240> theta<00:31:28.480> tilde<00:31:28.720> needs<00:31:28.880> to<00:31:28.960> be<00:31:29.040> consistent\nuh theta tilde needs to be consistent\nuh theta tilde needs to be consistent with<00:31:29.960> symmetry<00:31:30.360> three<00:31:31.240> and<00:31:31.360> with<00:31:31.440> the<00:31:31.520> fact\nwith symmetry three and with the fact\nwith symmetry three and with the fact that<00:31:31.960> the<00:31:32.040> combination<00:31:32.600> operation<00:31:33.040> is\nthat the combination operation is\nthat the combination operation is addition.<00:31:33.960> So,<00:31:34.040> we<00:31:34.120> can<00:31:34.280> set<00:31:34.440> up<00:31:34.560> this<00:31:34.760> kind<00:31:35.000> of\naddition. So, we can set up this kind of\naddition. So, we can set up this kind of uh<00:31:35.800> equation.\nuh equation.\nuh equation. Uh\nUh\nUh and<00:31:37.840> if<00:31:38.000> we<00:31:38.080> define<00:31:38.960> phi<00:31:39.200> tilde<00:31:39.680> as<00:31:40.120> the\nand if we define phi tilde as the\nand if we define phi tilde as the opposite<00:31:40.760> of<00:31:40.880> theta\nopposite of theta\nopposite of theta with<00:31:42.160> the<00:31:42.240> inverse<00:31:42.520> mapping,<00:31:43.280> we<00:31:43.400> can<00:31:43.600> write\nwith the inverse mapping, we can write\nwith the inverse mapping, we can write uh\nuh\nuh this<00:31:45.400> equation<00:31:46.360> where\nthis equation where\nthis equation where the<00:31:47.840> the<00:31:47.960> plus<00:31:48.200> sign<00:31:48.400> here\nthe the plus sign here\nthe the plus sign here uh<00:31:50.120> is<00:31:50.280> used<00:31:51.280> uh\nuh is used uh\nuh is used uh this<00:31:53.400> because<00:31:53.760> it's<00:31:53.880> between<00:31:54.760> the<00:31:54.840> different\nthis because it's between the different\nthis because it's between the different elements<00:31:55.920> of<00:31:56.280> theta<00:31:57.040> and<00:31:57.200> here<00:31:57.520> the<00:31:57.600> different\nelements of theta and here the different\nelements of theta and here the different valuations<00:31:58.360> are<00:31:58.440> just<00:31:58.600> summed<00:31:58.800> together\nvaluations are just summed together\nvaluations are just summed together because<00:31:59.440> those<00:31:59.640> are<00:31:59.960> disjoint<00:32:00.400> elements\nbecause those are disjoint elements\nbecause those are disjoint elements being<00:32:00.960> combined<00:32:02.120> uh<00:32:02.400> by<00:32:02.520> the<00:32:02.600> first<00:32:02.880> operation\nbeing combined uh by the first operation\nbeing combined uh by the first operation that<00:32:03.440> we<00:32:03.560> saw.\nUm<00:32:06.600> and<00:32:06.760> once<00:32:06.920> again,<00:32:07.280> there<00:32:07.520> is<00:32:07.760> a<00:32:08.040> nice\nUm and once again, there is a nice\nUm and once again, there is a nice little<00:32:08.480> proof<00:32:08.680> for<00:32:08.800> this\nlittle proof for this\nlittle proof for this and<00:32:10.480> we<00:32:10.800> find<00:32:11.240> that<00:32:11.920> phi<00:32:12.440> is<00:32:12.640> actually<00:32:12.920> c<00:32:13.480> times\nand we find that phi is actually c times\nand we find that phi is actually c times e<00:32:13.880> to<00:32:14.000> the<00:32:14.120> ax,<00:32:15.200> which<00:32:15.400> in<00:32:15.520> turn<00:32:15.800> means<00:32:16.040> that<00:32:16.120> we\ne to the ax, which in turn means that we\ne to the ax, which in turn means that we can<00:32:16.400> write<00:32:16.920> uh<00:32:17.240> this<00:32:17.440> beautiful<00:32:17.760> equation\ncan write uh this beautiful equation\ncan write uh this beautiful equation where<00:32:19.160> 1<00:32:19.440> over<00:32:19.720> a<00:32:19.800> simplifies<00:32:20.920> and<00:32:21.560> we<00:32:21.720> are\nwhere 1 over a simplifies and we are\nwhere 1 over a simplifies and we are also<00:32:22.160> free<00:32:22.360> to<00:32:22.520> choose<00:32:23.360> c<00:32:23.880> equals<00:32:24.160> to<00:32:24.280> 1<00:32:24.680> with\nalso free to choose c equals to 1 with\nalso free to choose c equals to 1 with this<00:32:25.000> just<00:32:25.200> a<00:32:25.240> constant<00:32:26.200> with<00:32:26.360> no<00:32:26.480> loss<00:32:26.680> of\nthis just a constant with no loss of\nthis just a constant with no loss of generality.\ngenerality.\ngenerality. And\nAnd\nAnd this<00:32:29.640> in<00:32:29.760> turn<00:32:30.000> tells<00:32:30.320> us<00:32:30.880> um\nthis in turn tells us um\nthis in turn tells us um that<00:32:32.040> this<00:32:32.440> kind<00:32:32.600> of<00:32:32.680> product<00:32:32.960> operator<00:32:33.560> is\nthat this kind of product operator is\nthat this kind of product operator is just<00:32:34.160> multiplication,\njust multiplication,\njust multiplication, uh<00:32:35.720> which<00:32:35.840> is<00:32:35.960> of<00:32:36.080> course<00:32:36.360> the<00:32:37.040> product<00:32:37.360> rule\nuh which is of course the product rule\nuh which is of course the product rule that<00:32:37.640> we<00:32:37.760> all<00:32:37.880> know<00:32:38.080> from<00:32:38.240> probability.<00:32:38.920> If<00:32:39.080> we\nthat we all know from probability. If we\nthat we all know from probability. If we have\nhave\nhave uh,<00:32:40.560> the<00:32:40.680> joint<00:32:40.880> probability<00:32:41.400> of<00:32:41.560> two\nuh, the joint probability of two\nuh, the joint probability of two independent<00:32:43.200> uh,<00:32:43.280> random<00:32:43.560> variables<00:32:43.920> is<00:32:44.040> just\nindependent uh, random variables is just\nindependent uh, random variables is just the<00:32:44.480> product<00:32:44.840> of<00:32:44.960> the<00:32:45.040> individual\nthe product of the individual\nthe product of the individual probabilities.<00:32:46.200> This<00:32:46.400> is<00:32:46.520> essentially<00:32:46.880> what\nprobabilities. This is essentially what\nprobabilities. This is essentially what we<00:32:47.120> have<00:32:47.240> derived<00:32:47.600> here.\nwe have derived here.\nwe have derived here. Now<00:32:49.840> we're<00:32:49.960> going<00:32:50.080> to<00:32:50.200> talk<00:32:50.720> uh,<00:32:50.840> we're<00:32:50.960> going\nNow we're going to talk uh, we're going\nNow we're going to talk uh, we're going to<00:32:51.160> go<00:32:51.280> back<00:32:51.600> to<00:32:51.800> our<00:32:52.000> idea<00:32:52.280> of<00:32:52.400> the<00:32:52.520> zeta\nto go back to our idea of the zeta\nto go back to our idea of the zeta function<00:32:53.160> because<00:32:53.520> if<00:32:53.680> you<00:32:53.760> remember\nfunction because if you remember\nfunction because if you remember zeta<00:32:55.240> function<00:32:55.680> actually<00:32:56.000> always<00:32:56.320> took<00:32:56.560> two\nzeta function actually always took two\nzeta function actually always took two elements.<00:32:57.840> Uh,<00:32:58.120> it<00:32:58.240> compared<00:32:59.080> sort<00:32:59.320> of<00:32:59.960> it\nelements. Uh, it compared sort of it\nelements. Uh, it compared sort of it always<00:33:00.840> was<00:33:01.000> evaluation<00:33:01.600> of<00:33:01.760> some<00:33:01.920> kind<00:33:02.080> of\nalways was evaluation of some kind of\nalways was evaluation of some kind of range<00:33:03.880> uh,<00:33:04.400> of<00:33:04.520> the<00:33:04.640> lattice.<00:33:05.240> So<00:33:05.400> we\nrange uh, of the lattice. So we\nrange uh, of the lattice. So we essentially<00:33:06.040> are<00:33:06.200> trying<00:33:06.520> to<00:33:06.640> build<00:33:06.880> a\nessentially are trying to build a\nessentially are trying to build a b-valuation,<00:33:07.680> not<00:33:07.840> evaluation<00:33:08.360> of<00:33:08.480> single\nb-valuation, not evaluation of single\nb-valuation, not evaluation of single elements<00:33:09.120> of<00:33:09.240> the<00:33:09.320> lattice.\nelements of the lattice.\nelements of the lattice. Um,\nUm,\nUm, in<00:33:12.240> general,<00:33:12.760> we're<00:33:12.880> going<00:33:13.080> to<00:33:13.400> just<00:33:13.800> describe\nin general, we're going to just describe\nin general, we're going to just describe this<00:33:14.320> as<00:33:14.560> the<00:33:14.920> b-valuation<00:33:15.760> with<00:33:16.000> let<00:33:16.200> b<00:33:16.960> x<00:33:17.200> and\nthis as the b-valuation with let b x and\nthis as the b-valuation with let b x and t<00:33:18.120> where<00:33:18.320> the<00:33:18.440> order<00:33:18.840> inside<00:33:19.400> this<00:33:20.000> uh,\nt where the order inside this uh,\nt where the order inside this uh, matters<00:33:20.480> a<00:33:20.520> lot.<00:33:21.000> So<00:33:21.360> the<00:33:21.520> right<00:33:21.760> one<00:33:21.920> is<00:33:22.080> the\nmatters a lot. So the right one is the\nmatters a lot. So the right one is the top<00:33:22.480> one<00:33:22.600> and<00:33:22.720> the<00:33:22.760> left<00:33:22.960> one<00:33:23.080> is<00:33:23.200> the<00:33:23.280> bottom\ntop one and the left one is the bottom\ntop one and the left one is the bottom one.\none.\none. Um,<00:33:25.080> x<00:33:25.280> is<00:33:25.400> commonly<00:33:25.640> called<00:33:25.960> predicate\nUm, x is commonly called predicate\nUm, x is commonly called predicate and<00:33:27.440> t<00:33:27.640> is<00:33:27.800> called<00:33:28.160> the<00:33:28.240> context.\nand t is called the context.\nand t is called the context. Uh,<00:33:30.720> and<00:33:30.920> note<00:33:31.160> that<00:33:31.400> t<00:33:31.600> does<00:33:31.840> not<00:33:32.640> need<00:33:32.920> to<00:33:33.000> be\nUh, and note that t does not need to be\nUh, and note that t does not need to be the<00:33:33.240> top<00:33:33.480> element<00:33:33.840> of<00:33:33.960> the<00:33:34.040> lattice.<00:33:34.560> Uh,<00:33:34.600> t\nthe top element of the lattice. Uh, t\nthe top element of the lattice. Uh, t can<00:33:35.040> just<00:33:35.160> be<00:33:35.320> any<00:33:35.560> element.<00:33:36.240> Uh,<00:33:36.360> this<00:33:36.560> is<00:33:37.080> we\ncan just be any element. Uh, this is we\ncan just be any element. Uh, this is we want<00:33:37.400> this<00:33:37.560> to<00:33:37.640> be<00:33:37.760> a<00:33:37.840> very<00:33:38.080> general<00:33:38.480> kind<00:33:38.680> of\nwant this to be a very general kind of\nwant this to be a very general kind of operation<00:33:39.360> that<00:33:39.440> we<00:33:39.520> can<00:33:39.640> do.\nNow<00:33:43.000> these<00:33:43.480> b-valuations,<00:33:44.720> because<00:33:45.280> they<00:33:45.480> are\nNow these b-valuations, because they are\nNow these b-valuations, because they are essentially<00:33:46.080> just<00:33:46.840> ranges<00:33:47.720> o-<00:33:47.920> over<00:33:48.480> sort<00:33:48.680> of\nessentially just ranges o- over sort of\nessentially just ranges o- over sort of connect<00:33:49.480> arrays<00:33:50.120> of<00:33:50.280> the<00:33:50.400> elements<00:33:50.760> of<00:33:50.880> the\nconnect arrays of the elements of the\nconnect arrays of the elements of the lattice,\nlattice,\nlattice, um,<00:33:52.720> they<00:33:52.920> can<00:33:53.160> be<00:33:53.360> ordered<00:33:53.760> by<00:33:53.960> set\num, they can be ordered by set\num, they can be ordered by set inclusion.<00:33:55.080> And<00:33:55.400> consistency<00:33:56.000> requires<00:33:56.520> that\ninclusion. And consistency requires that\ninclusion. And consistency requires that um,\num,\num, smaller\nsmaller\nsmaller uh,<00:34:00.320> smaller<00:34:00.760> ranges<00:34:01.320> have<00:34:02.120> uh,<00:34:02.560> higher\nuh, smaller ranges have uh, higher\nuh, smaller ranges have uh, higher valuations<00:34:03.720> and<00:34:03.960> sort<00:34:04.160> of<00:34:04.400> valuations<00:34:04.920> of<00:34:05.040> two\nvaluations and sort of valuations of two\nvaluations and sort of valuations of two elements<00:34:05.520> that<00:34:05.640> are<00:34:05.760> further<00:34:06.120> apart<00:34:07.200> uh,<00:34:07.440> have\nelements that are further apart uh, have\nelements that are further apart uh, have smaller<00:34:08.040> valuations.\nsmaller valuations.\nsmaller valuations. The<00:34:09.440> reason<00:34:10.200> is<00:34:10.480> that<00:34:11.200> if<00:34:11.399> we<00:34:11.520> have<00:34:11.679> the<00:34:11.800> same\nThe reason is that if we have the same\nThe reason is that if we have the same context,<00:34:13.320> then<00:34:13.879> an<00:34:14.040> element<00:34:14.320> that<00:34:14.440> is<00:34:14.560> further\ncontext, then an element that is further\ncontext, then an element that is further down<00:34:15.120> the<00:34:15.200> lattice\ndown the lattice\ndown the lattice um,\num,\num, needs<00:34:17.960> to<00:34:18.080> have<00:34:18.560> uh,<00:34:18.840> a<00:34:18.960> lower<00:34:19.520> valuation<00:34:20.080> than\nneeds to have uh, a lower valuation than\nneeds to have uh, a lower valuation than an<00:34:20.399> element<00:34:20.720> that<00:34:20.840> is<00:34:20.960> implied<00:34:21.679> by<00:34:21.840> the<00:34:21.919> same\nan element that is implied by the same\nan element that is implied by the same element<00:34:22.639> x,<00:34:23.040> right?<00:34:23.879> Uh,\nelement x, right? Uh,\nelement x, right? Uh, for<00:34:26.159> example,<00:34:26.639> if<00:34:27.320> uh,<00:34:27.399> what's<00:34:27.600> a<00:34:27.639> good\nfor example, if uh, what's a good\nfor example, if uh, what's a good example<00:34:28.120> to<00:34:28.240> illustrate<00:34:28.679> this?\nexample to illustrate this?\nexample to illustrate this? Uh,<00:34:30.159> well<00:34:30.240> I<00:34:30.280> think<00:34:30.440> this<00:34:30.560> one's<00:34:30.720> kind<00:34:30.879> of\nUh, well I think this one's kind of\nUh, well I think this one's kind of obvious.<00:34:31.280> it's<00:34:31.399> the<00:34:31.480> opposite<00:34:31.879> that<00:34:31.960> needs<00:34:32.120> to\nobvious. it's the opposite that needs to\nobvious. it's the opposite that needs to be<00:34:32.320> explained.<00:34:33.240> Um,\nbe explained. Um,\nbe explained. Um, but<00:34:35.240> given<00:34:35.480> the<00:34:35.560> same<00:34:35.800> predicate,<00:34:36.399> also<00:34:36.679> we\nbut given the same predicate, also we\nbut given the same predicate, also we know<00:34:37.040> that<00:34:37.240> a<00:34:37.280> more<00:34:37.679> diluted<00:34:38.200> context<00:34:38.679> gives<00:34:38.919> a\nknow that a more diluted context gives a\nknow that a more diluted context gives a lower<00:34:39.200> valuation.<00:34:39.760> So,<00:34:39.879> this<00:34:40.080> one<00:34:40.399> needs<00:34:40.600> an\nlower valuation. So, this one needs an\nlower valuation. So, this one needs an example.<00:34:41.200> So,<00:34:41.360> imagine<00:34:41.840> that<00:34:42.000> you<00:34:42.120> want<00:34:42.360> to\nexample. So, imagine that you want to\nexample. So, imagine that you want to define<00:34:42.840> the<00:34:42.919> probability<00:34:43.480> that<00:34:43.639> a<00:34:43.679> person<00:34:44.080> is\ndefine the probability that a person is\ndefine the probability that a person is in<00:34:44.480> a<00:34:44.679> city,<00:34:45.240> let's<00:34:45.480> say<00:34:45.679> Paris<00:34:46.040> for<00:34:46.159> example,\nin a city, let's say Paris for example,\nin a city, let's say Paris for example, right?\nright?\nright? So,<00:34:47.840> the<00:34:48.000> probability<00:34:48.560> that<00:34:48.639> the<00:34:48.760> person<00:34:49.080> is\nSo, the probability that the person is\nSo, the probability that the person is in<00:34:49.360> Paris<00:34:50.000> given<00:34:50.360> that<00:34:50.520> you<00:34:50.639> know<00:34:50.800> the<00:34:50.879> person\nin Paris given that you know the person\nin Paris given that you know the person is<00:34:51.320> in<00:34:51.440> France<00:34:52.480> is<00:34:52.679> definitely<00:34:53.040> much<00:34:53.280> bigger\nis in France is definitely much bigger\nis in France is definitely much bigger than<00:34:53.679> the<00:34:53.760> probability<00:34:54.360> that<00:34:54.520> they<00:34:54.639> are<00:34:54.800> in<00:34:55.040> in\nthan the probability that they are in in\nthan the probability that they are in in Paris<00:34:56.159> knowing<00:34:56.639> only<00:34:56.840> that<00:34:57.000> they<00:34:57.120> are\nParis knowing only that they are\nParis knowing only that they are somewhere<00:34:58.040> in<00:34:58.200> Europe.<00:34:59.200> Uh,<00:34:59.560> the<00:35:00.040> the<00:35:00.120> second\nsomewhere in Europe. Uh, the the second\nsomewhere in Europe. Uh, the the second context<00:35:00.920> is<00:35:01.080> much<00:35:01.440> much<00:35:02.120> uh<00:35:02.240> more<00:35:02.400> diluted.\ncontext is much much uh more diluted.\ncontext is much much uh more diluted. It's<00:35:03.000> much<00:35:03.560> Being<00:35:03.800> in<00:35:03.880> France<00:35:04.240> implies<00:35:04.680> being\nIt's much Being in France implies being\nIt's much Being in France implies being in<00:35:04.920> Europe,<00:35:05.720> hence<00:35:06.440> uh<00:35:06.600> France<00:35:06.920> is<00:35:07.040> strictly\nin Europe, hence uh France is strictly\nin Europe, hence uh France is strictly below<00:35:07.880> Europe<00:35:08.240> in<00:35:08.320> kind<00:35:08.480> of<00:35:08.560> this<00:35:08.760> ordering<00:35:09.120> by\nbelow Europe in kind of this ordering by\nbelow Europe in kind of this ordering by implication.<00:35:10.320> Uh,<00:35:10.480> and<00:35:10.640> Europe<00:35:10.840> is<00:35:10.920> just<00:35:11.240> so\nimplication. Uh, and Europe is just so\nimplication. Uh, and Europe is just so much<00:35:11.680> more<00:35:11.800> diluted.<00:35:12.160> It<00:35:12.520> allows<00:35:12.800> for<00:35:12.920> so<00:35:13.160> many\nmuch more diluted. It allows for so many\nmuch more diluted. It allows for so many more<00:35:13.600> possibilities.\nmore possibilities.\nmore possibilities. Uh,<00:35:15.480> and<00:35:15.560> this<00:35:15.720> is<00:35:15.960> what<00:35:16.120> we're<00:35:16.520> essentially\nUh, and this is what we're essentially\nUh, and this is what we're essentially reconstructing<00:35:17.600> here,<00:35:17.760> what<00:35:17.920> we're\nreconstructing here, what we're\nreconstructing here, what we're obtaining.\nobtaining.\nobtaining. And<00:35:20.640> this<00:35:20.880> B-valuation<00:35:21.680> is<00:35:21.880> then<00:35:22.080> what<00:35:22.240> we<00:35:22.720> uh\nAnd this B-valuation is then what we uh\nAnd this B-valuation is then what we uh call<00:35:23.080> probability,<00:35:23.840> where<00:35:24.000> again<00:35:24.240> here<00:35:24.840> this\ncall probability, where again here this\ncall probability, where again here this B-valuation<00:35:25.560> of<00:35:25.640> the<00:35:25.720> range<00:35:26.040> X<00:35:26.840> uh<00:35:26.920> and<00:35:27.080> T\nB-valuation of the range X uh and T\nB-valuation of the range X uh and T is<00:35:28.280> the<00:35:28.360> probability<00:35:28.800> of<00:35:28.960> X<00:35:29.560> given<00:35:29.880> T.\nUm,<00:35:34.200> and<00:35:34.400> the<00:35:34.480> interesting<00:35:34.800> thing<00:35:35.000> is\nUm, and the interesting thing is\nUm, and the interesting thing is uh<00:35:36.440> given<00:35:37.480> a<00:35:37.520> context,<00:35:38.680> then<00:35:38.840> the<00:35:38.960> valuation\nuh given a context, then the valuation\nuh given a context, then the valuation of<00:35:39.560> all<00:35:39.720> the<00:35:39.840> elements<00:35:40.200> sort<00:35:40.320> of<00:35:40.440> below<00:35:40.840> it\nof all the elements sort of below it\nof all the elements sort of below it just<00:35:41.200> reduces<00:35:41.680> to<00:35:41.840> the<00:35:42.560> valuation<00:35:43.160> of<00:35:43.480> those\njust reduces to the valuation of those\njust reduces to the valuation of those single<00:35:44.040> elements,<00:35:44.920> which<00:35:45.120> is<00:35:45.240> why<00:35:45.520> we<00:35:45.640> can\nsingle elements, which is why we can\nsingle elements, which is why we can apply<00:35:46.160> those<00:35:46.760> uh<00:35:46.840> results<00:35:47.240> that<00:35:47.320> we<00:35:47.400> saw\napply those uh results that we saw\napply those uh results that we saw before<00:35:47.920> is<00:35:48.000> only<00:35:48.200> caring<00:35:48.520> about<00:35:49.120> valuation<00:35:49.600> of\nbefore is only caring about valuation of\nbefore is only caring about valuation of single<00:35:49.960> elements.\nsingle elements.\nsingle elements. And<00:35:51.520> this<00:35:51.680> allows<00:35:52.040> us<00:35:52.160> to<00:35:52.640> write<00:35:53.080> the<00:35:53.320> the<00:35:53.400> sum\nAnd this allows us to write the the sum\nAnd this allows us to write the the sum rule,<00:35:53.960> which<00:35:54.120> is<00:35:54.240> very<00:35:54.400> nice.\nrule, which is very nice.\nrule, which is very nice. Uh,<00:35:55.600> so<00:35:55.720> we<00:35:55.760> already<00:35:56.000> have<00:35:56.160> this.<00:35:56.960> And<00:35:57.480> we<00:35:57.600> can\nUh, so we already have this. And we can\nUh, so we already have this. And we can also<00:35:58.000> use<00:35:58.160> the<00:35:58.280> product<00:35:58.600> rule<00:35:58.760> for\nalso use the product rule for\nalso use the product rule for independently<00:35:59.600> treated<00:35:59.960> systems<00:36:00.880> uh<00:36:01.200> like\nindependently treated systems uh like\nindependently treated systems uh like so,\nso,\nso, uh<00:36:02.600> where<00:36:03.080> the<00:36:03.240> top<00:36:03.480> element<00:36:04.360> uh<00:36:04.760> so<00:36:04.880> that<00:36:05.040> the\nuh where the top element uh so that the\nuh where the top element uh so that the combined<00:36:05.600> context<00:36:06.080> is<00:36:06.280> again<00:36:06.840> T<00:36:07.840> uh<00:36:07.920> context\ncombined context is again T uh context\ncombined context is again T uh context one<00:36:08.480> times<00:36:08.680> context<00:36:09.040> two.\none times context two.\none times context two. And<00:36:10.240> just<00:36:10.400> to<00:36:10.520> illustrate<00:36:11.000> what<00:36:11.120> this<00:36:11.280> would\nAnd just to illustrate what this would\nAnd just to illustrate what this would look<00:36:11.640> like<00:36:12.000> with<00:36:12.160> some<00:36:12.320> kind<00:36:12.520> of<00:36:12.920> uh\nlook like with some kind of uh\nlook like with some kind of uh with<00:36:13.720> the<00:36:13.800> lattice<00:36:14.120> case,\nwith the lattice case,\nwith the lattice case, uh<00:36:15.360> so<00:36:15.480> a<00:36:15.520> quick<00:36:15.720> example,<00:36:16.200> let's<00:36:16.440> take<00:36:16.760> these\nuh so a quick example, let's take these\nuh so a quick example, let's take these two<00:36:17.560> simple<00:36:17.920> lattices<00:36:18.360> with<00:36:18.520> just<00:36:18.800> two<00:36:19.080> atoms\ntwo simple lattices with just two atoms\ntwo simple lattices with just two atoms on<00:36:20.000> top<00:36:20.200> element<00:36:20.480> and<00:36:20.560> bottom<00:36:20.840> element,<00:36:21.680> and\non top element and bottom element, and\non top element and bottom element, and we<00:36:22.000> want<00:36:22.280> to<00:36:22.800> say<00:36:23.400> find<00:36:24.120> the<00:36:24.240> valuation<00:36:24.880> of<00:36:25.720> uh\nwe want to say find the valuation of uh\nwe want to say find the valuation of uh A<00:36:26.200> times<00:36:26.520> X.<00:36:27.280> Um,<00:36:27.800> then<00:36:27.960> the<00:36:28.080> top<00:36:28.360> element<00:36:28.800> here\nA times X. Um, then the top element here\nA times X. Um, then the top element here becomes<00:36:29.920> t1<00:36:30.200> times<00:36:30.360> t2.<00:36:31.200> We<00:36:31.320> can<00:36:31.520> use<00:36:31.880> the\nbecomes t1 times t2. We can use the\nbecomes t1 times t2. We can use the distributivity<00:36:32.840> property<00:36:33.560> to<00:36:33.720> obtain<00:36:34.080> that\ndistributivity property to obtain that\ndistributivity property to obtain that this<00:36:34.360> is<00:36:34.480> the<00:36:34.720> top<00:36:35.000> element<00:36:35.360> of<00:36:35.480> the<00:36:35.600> new\nthis is the top element of the new\nthis is the top element of the new lattice<00:36:36.120> that<00:36:36.240> we're<00:36:36.320> going<00:36:36.480> to<00:36:37.080> uh<00:36:37.120> get.\nlattice that we're going to uh get.\nlattice that we're going to uh get. And<00:36:39.000> again,<00:36:39.360> note<00:36:39.520> that<00:36:39.680> neither<00:36:40.160> t1<00:36:40.480> or<00:36:40.560> t2\nAnd again, note that neither t1 or t2\nAnd again, note that neither t1 or t2 need<00:36:41.160> to<00:36:41.200> be<00:36:41.320> the<00:36:41.440> top<00:36:41.680> element<00:36:42.120> of<00:36:42.200> their\nneed to be the top element of their\nneed to be the top element of their respective<00:36:42.800> lattices.<00:36:43.120> This<00:36:43.320> could<00:36:43.480> just<00:36:43.680> be\nrespective lattices. This could just be\nrespective lattices. This could just be These<00:36:44.400> two<00:36:44.600> could<00:36:44.760> just<00:36:44.920> be<00:36:45.480> sub-lattices<00:36:46.160> of\nThese two could just be sub-lattices of\nThese two could just be sub-lattices of some<00:36:46.480> kind<00:36:46.640> of<00:36:46.720> bigger<00:36:46.960> structure.<00:36:47.400> It\nsome kind of bigger structure. It\nsome kind of bigger structure. It doesn't<00:36:47.760> matter.\ndoesn't matter.\ndoesn't matter. And<00:36:49.160> the<00:36:49.280> combination<00:36:49.880> of<00:36:49.960> them<00:36:50.720> uh<00:36:50.800> results\nAnd the combination of them uh results\nAnd the combination of them uh results in<00:36:51.200> this.<00:36:51.800> Um\nin this. Um\nin this. Um And<00:36:52.800> this<00:36:52.960> is<00:36:53.400> what<00:36:53.560> we're<00:36:53.640> doing<00:36:54.000> when<00:36:54.160> we<00:36:54.320> are\nAnd this is what we're doing when we are\nAnd this is what we're doing when we are combining<00:36:55.080> two<00:36:55.960> uh<00:36:56.040> systems<00:36:56.400> that<00:36:56.520> we<00:36:56.600> treat\ncombining two uh systems that we treat\ncombining two uh systems that we treat independently.<00:36:57.640> We<00:36:58.040> kind<00:36:58.240> of<00:36:58.320> create<00:36:58.560> this\nindependently. We kind of create this\nindependently. We kind of create this new<00:36:59.120> bigger<00:36:59.400> structure\nnew bigger structure\nnew bigger structure that<00:37:01.160> has<00:37:01.400> all<00:37:01.520> these<00:37:02.040> uh<00:37:02.360> cross<00:37:02.600> product\nthat has all these uh cross product\nthat has all these uh cross product points.\nUm<00:37:07.440> yeah.\nUm yeah.\nUm yeah. Now,<00:37:09.240> the<00:37:09.360> next<00:37:09.600> thing<00:37:09.760> that<00:37:09.880> we<00:37:10.000> would<00:37:10.240> that\nNow, the next thing that we would that\nNow, the next thing that we would that we<00:37:10.480> need<00:37:10.680> to<00:37:10.760> do<00:37:10.960> to<00:37:11.080> have<00:37:11.240> some<00:37:11.360> kind<00:37:11.560> of\nwe need to do to have some kind of\nwe need to do to have some kind of complete<00:37:12.480> uh\ncomplete uh\ncomplete uh reasoning<00:37:13.520> apparatus<00:37:14.000> is<00:37:14.200> that<00:37:14.320> we<00:37:14.400> need<00:37:14.560> to\nreasoning apparatus is that we need to\nreasoning apparatus is that we need to somehow<00:37:14.920> quantify<00:37:15.440> the<00:37:15.560> degree<00:37:15.800> of\nsomehow quantify the degree of\nsomehow quantify the degree of implication<00:37:16.440> between<00:37:16.760> two<00:37:16.880> elements<00:37:17.960> that\nimplication between two elements that\nimplication between two elements that are<00:37:18.840> not<00:37:19.160> directly<00:37:19.560> one<00:37:19.680> above<00:37:20.040> each<00:37:20.240> other.\nare not directly one above each other.\nare not directly one above each other. Because<00:37:21.440> if<00:37:21.560> they<00:37:21.720> are<00:37:21.800> above<00:37:22.120> each<00:37:22.320> other,<00:37:23.040> uh\nBecause if they are above each other, uh\nBecause if they are above each other, uh you<00:37:23.240> can<00:37:23.440> somehow<00:37:23.800> just<00:37:24.560> uh<00:37:24.640> combine<00:37:25.200> the\nyou can somehow just uh combine the\nyou can somehow just uh combine the the<00:37:26.440> all<00:37:26.600> the<00:37:26.720> elements<00:37:27.800> uh<00:37:27.880> with<00:37:28.040> the<00:37:28.120> join\nthe all the elements uh with the join\nthe all the elements uh with the join operation,<00:37:29.200> with<00:37:29.360> the<00:37:29.720> sum.<00:37:30.120> But<00:37:30.320> if<00:37:30.480> they're\noperation, with the sum. But if they're\noperation, with the sum. But if they're not<00:37:30.800> directly<00:37:31.080> above<00:37:31.320> each<00:37:31.440> other,<00:37:31.600> what<00:37:31.760> do\nnot directly above each other, what do\nnot directly above each other, what do you<00:37:31.920> do?\nyou do?\nyou do? Um\nUm\nUm So,<00:37:34.640> for<00:37:34.760> example,<00:37:35.040> imagine<00:37:35.360> the<00:37:35.480> chain<00:37:36.040> where\nSo, for example, imagine the chain where\nSo, for example, imagine the chain where all<00:37:36.640> these<00:37:36.800> elements<00:37:37.160> are<00:37:37.280> directly<00:37:38.240> one<00:37:38.440> the\nall these elements are directly one the\nall these elements are directly one the superior<00:37:39.000> of<00:37:39.120> the<00:37:39.240> other.<00:37:40.080> Then<00:37:40.240> we<00:37:40.360> somehow\nsuperior of the other. Then we somehow\nsuperior of the other. Then we somehow need<00:37:41.040> to<00:37:41.280> obtain\nneed to obtain\nneed to obtain the<00:37:42.840> valuation<00:37:43.400> of<00:37:43.600> the<00:37:44.000> in<00:37:44.440> of<00:37:44.520> this\nthe valuation of the in of this\nthe valuation of the in of this generalization<00:37:45.280> of<00:37:45.400> the<00:37:45.480> inverse<00:37:45.800> zeta\ngeneralization of the inverse zeta\ngeneralization of the inverse zeta function\nfunction\nfunction uh<00:37:47.360> of\nuh of\nuh of over<00:37:48.680> the<00:37:48.760> whole<00:37:49.040> range<00:37:49.320> of<00:37:49.520> x<00:37:50.200> to<00:37:50.360> t.\nover the whole range of x to t.\nover the whole range of x to t. Um<00:37:52.160> and<00:37:52.320> we<00:37:52.440> can<00:37:52.640> have<00:37:52.880> to<00:37:53.000> find<00:37:53.320> this<00:37:53.480> from<00:37:53.680> all\nUm and we can have to find this from all\nUm and we can have to find this from all the<00:37:54.000> sub-intervals,<00:37:55.040> x<00:37:55.280> to<00:37:55.400> y,<00:37:55.880> y<00:37:56.120> to<00:37:56.280> z,<00:37:56.640> and\nthe sub-intervals, x to y, y to z, and\nthe sub-intervals, x to y, y to z, and then<00:37:56.920> finally<00:37:57.120> z<00:37:57.320> to<00:37:57.400> t.<00:37:57.560> We<00:37:57.640> need<00:37:57.760> to<00:37:57.840> somehow\nthen finally z to t. We need to somehow\nthen finally z to t. We need to somehow be<00:37:59.120> able<00:37:59.360> to<00:37:59.480> combine<00:38:00.000> those<00:38:00.360> to<00:38:00.440> get<00:38:00.760> the<00:38:01.520> the\nbe able to combine those to get the the\nbe able to combine those to get the the bigger<00:38:01.880> valuation.<00:38:03.000> And<00:38:03.240> this,<00:38:03.640> mind<00:38:04.000> you,<00:38:04.120> is\nbigger valuation. And this, mind you, is\nbigger valuation. And this, mind you, is an<00:38:04.400> entirely<00:38:04.960> different<00:38:05.320> operation\nan entirely different operation\nan entirely different operation than<00:38:07.920> adding<00:38:08.320> independent<00:38:08.800> systems\nthan adding independent systems\nthan adding independent systems together,<00:38:10.080> but<00:38:10.320> it<00:38:10.440> turns<00:38:10.760> out<00:38:10.880> that<00:38:11.000> this\ntogether, but it turns out that this\ntogether, but it turns out that this will<00:38:11.400> also<00:38:11.680> be<00:38:12.320> uh<00:38:12.520> a<00:38:12.600> product<00:38:13.080> rule.\nNow,<00:38:15.640> we<00:38:15.760> have<00:38:16.000> the<00:38:16.440> fifth<00:38:16.720> and<00:38:16.840> last<00:38:17.080> symmetry\nNow, we have the fifth and last symmetry\nNow, we have the fifth and last symmetry that<00:38:17.840> we<00:38:17.920> are<00:38:18.040> going<00:38:18.200> to<00:38:18.440> look<00:38:18.640> at.\nthat we are going to look at.\nthat we are going to look at. The<00:38:19.840> chaining<00:38:20.480> of<00:38:20.680> these<00:38:21.360> uh<00:38:21.760> intervals<00:38:22.320> in\nThe chaining of these uh intervals in\nThe chaining of these uh intervals in the<00:38:22.560> lattice<00:38:22.960> is<00:38:23.280> associative.\nthe lattice is associative.\nthe lattice is associative. Meaning<00:38:25.160> it<00:38:25.400> doesn't<00:38:25.720> really<00:38:25.880> matter<00:38:26.240> in<00:38:26.800> what\nMeaning it doesn't really matter in what\nMeaning it doesn't really matter in what order\norder\norder we<00:38:28.280> do<00:38:28.400> the<00:38:28.520> chaining<00:38:28.840> operation.\nwe do the chaining operation.\nwe do the chaining operation. This<00:38:30.160> simply<00:38:30.400> follows<00:38:30.640> from<00:38:30.760> the<00:38:30.840> fact<00:38:31.160> that\nThis simply follows from the fact that\nThis simply follows from the fact that implication<00:38:32.280> is<00:38:32.400> a<00:38:32.440> property<00:38:33.000> of<00:38:33.200> the\nimplication is a property of the\nimplication is a property of the intervals<00:38:33.800> as<00:38:33.920> a<00:38:33.960> whole.<00:38:34.680> And<00:38:34.840> so<00:38:35.080> it's<00:38:35.280> absurd\nintervals as a whole. And so it's absurd\nintervals as a whole. And so it's absurd to<00:38:35.760> make<00:38:35.920> it<00:38:36.040> somehow<00:38:36.320> depend<00:38:36.800> on<00:38:37.000> our<00:38:37.440> way<00:38:37.600> of\nto make it somehow depend on our way of\nto make it somehow depend on our way of chaining<00:38:38.080> them<00:38:38.200> together.\nchaining them together.\nchaining them together. As<00:38:39.480> long<00:38:39.640> as<00:38:39.760> of<00:38:39.840> course\nAs long as of course\nAs long as of course we<00:38:41.240> don't<00:38:41.520> just<00:38:42.080> skip<00:38:42.400> elements<00:38:42.760> and<00:38:42.880> won't<00:38:43.040> do\nwe don't just skip elements and won't do\nwe don't just skip elements and won't do anything<00:38:43.480> do<00:38:43.640> anything<00:38:43.920> stupid.<00:38:44.280> But<00:38:44.640> given\nanything do anything stupid. But given\nanything do anything stupid. But given that<00:38:45.000> this<00:38:45.120> is<00:38:45.880> this<00:38:46.000> is<00:38:46.080> a<00:38:46.120> valid<00:38:46.720> you<00:38:46.800> know\nthat this is this is a valid you know\nthat this is this is a valid you know choice<00:38:49.040> of<00:38:49.480> elements\nchoice of elements\nchoice of elements it<00:38:50.920> doesn't<00:38:51.120> matter<00:38:51.320> how<00:38:51.440> we<00:38:51.520> combine<00:38:51.840> them.\nit doesn't matter how we combine them.\nit doesn't matter how we combine them. And<00:38:53.720> then<00:38:53.840> of<00:38:53.920> course<00:38:54.400> the<00:38:54.600> operator<00:38:55.080> which<00:38:55.240> is\nAnd then of course the operator which is\nAnd then of course the operator which is kind<00:38:55.440> of<00:38:55.520> this<00:38:56.040> bull's-eye<00:38:57.080> bull's-eye\nkind of this bull's-eye bull's-eye\nkind of this bull's-eye bull's-eye symbol\nsymbol\nsymbol must<00:39:01.080> also<00:39:01.560> satisfy<00:39:02.040> this.<00:39:02.320> This<00:39:02.520> is<00:39:02.640> just\nmust also satisfy this. This is just\nmust also satisfy this. This is just written<00:39:03.400> down<00:39:03.640> with<00:39:03.800> the<00:39:04.120> probability\nwritten down with the probability\nwritten down with the probability notation.\nnotation.\nnotation. And<00:39:07.480> we<00:39:08.080> in<00:39:08.240> order<00:39:08.400> to<00:39:08.920> use<00:39:09.240> our<00:39:09.360> beautiful\nAnd we in order to use our beautiful\nAnd we in order to use our beautiful associativity<00:39:10.640> theorem<00:39:10.960> once<00:39:11.160> again<00:39:11.440> we\nassociativity theorem once again we\nassociativity theorem once again we unfortunately<00:39:12.080> need<00:39:12.400> to<00:39:12.600> show<00:39:12.800> that<00:39:12.960> it\nunfortunately need to show that it\nunfortunately need to show that it satisfies<00:39:13.560> symmetry<00:39:13.960> one<00:39:14.800> yet<00:39:14.920> again.<00:39:15.480> I'm\nsatisfies symmetry one yet again. I'm\nsatisfies symmetry one yet again. I'm sorry.\nsorry.\nsorry. And<00:39:17.560> we<00:39:17.640> show<00:39:17.840> that<00:39:18.120> chaining<00:39:18.800> of<00:39:18.960> intervals\nAnd we show that chaining of intervals\nAnd we show that chaining of intervals preserves<00:39:20.040> the<00:39:20.160> order<00:39:21.120> from<00:39:21.760> the<00:39:21.880> left<00:39:22.320> and\npreserves the order from the left and\npreserves the order from the left and the<00:39:22.520> right.<00:39:23.160> So<00:39:23.400> again<00:39:23.680> if<00:39:23.800> you<00:39:23.880> have<00:39:24.040> this\nthe right. So again if you have this\nthe right. So again if you have this element<00:39:24.640> XYZT\nelement XYZT\nelement XYZT then<00:39:26.880> given<00:39:27.280> the<00:39:27.400> same<00:39:27.640> context\nthen given the same context\nthen given the same context the<00:39:29.280> probability<00:39:29.680> of<00:39:29.840> X<00:39:30.240> is<00:39:30.600> strictly<00:39:30.880> lower\nthe probability of X is strictly lower\nthe probability of X is strictly lower than<00:39:31.080> probability<00:39:31.440> of<00:39:31.560> Y<00:39:32.360> because<00:39:33.080> it's\nthan probability of Y because it's\nthan probability of Y because it's further<00:39:33.480> apart<00:39:33.720> from<00:39:33.840> the<00:39:33.920> context.<00:39:34.440> It's<00:39:34.600> a\nfurther apart from the context. It's a\nfurther apart from the context. It's a smaller<00:39:35.520> it's<00:39:35.720> a<00:39:35.760> smaller<00:39:36.040> thing.<00:39:36.240> It's<00:39:36.320> a\nsmaller it's a smaller thing. It's a\nsmaller it's a smaller thing. It's a smaller<00:39:36.640> subset<00:39:36.960> of<00:39:37.040> the<00:39:37.120> possibilities.\nsmaller subset of the possibilities.\nsmaller subset of the possibilities. Um\nUm\nUm then<00:39:40.200> if<00:39:40.440> we<00:39:40.560> kind<00:39:40.720> of<00:39:41.040> chain<00:39:41.440> them<00:39:42.160> so<00:39:42.320> that\nthen if we kind of chain them so that\nthen if we kind of chain them so that both<00:39:43.120> both<00:39:43.360> have<00:39:43.520> kind<00:39:43.680> of<00:39:44.080> a<00:39:44.120> larger<00:39:44.440> context\nboth both have kind of a larger context\nboth both have kind of a larger context we<00:39:45.480> know<00:39:45.800> from<00:39:46.000> the<00:39:46.080> property<00:39:46.840> that<00:39:47.000> these<00:39:47.160> B\nwe know from the property that these B\nwe know from the property that these B valuations<00:39:48.000> are<00:39:48.160> just<00:39:49.120> um\nvaluations are just um\nvaluations are just um sort<00:39:50.040> of<00:39:50.160> these<00:39:50.400> these<00:39:50.640> these<00:39:50.920> ranges\nsort of these these these ranges\nsort of these these these ranges we<00:39:52.320> know<00:39:52.520> that<00:39:52.680> this<00:39:53.280> still<00:39:53.520> preserves<00:39:53.880> the\nwe know that this still preserves the\nwe know that this still preserves the order.<00:39:54.400> And<00:39:54.960> dually<00:39:55.320> for<00:39:55.440> the<00:39:55.560> other<00:39:55.760> case<00:39:56.560> if\norder. And dually for the other case if\norder. And dually for the other case if we<00:39:57.040> have<00:39:57.200> the<00:39:57.840> two<00:39:58.120> propositions<00:39:58.920> that<00:39:59.440> have\nwe have the two propositions that have\nwe have the two propositions that have different<00:40:00.000> contexts\ndifferent contexts\ndifferent contexts if<00:40:01.560> we<00:40:01.680> somehow<00:40:02.840> um\nif we somehow um\nif we somehow um chain<00:40:04.200> them<00:40:04.280> together<00:40:04.560> such<00:40:04.840> that<00:40:05.000> they<00:40:05.120> have\nchain them together such that they have\nchain them together such that they have another<00:40:06.200> new<00:40:06.480> common<00:40:06.760> proposition<00:40:07.160> that<00:40:07.320> is\nanother new common proposition that is\nanother new common proposition that is further<00:40:08.160> up<00:40:08.360> the<00:40:08.480> order<00:40:09.240> then<00:40:09.440> this<00:40:09.720> also\nfurther up the order then this also\nfurther up the order then this also preserves\npreserves\npreserves the<00:40:12.440> original<00:40:12.840> order.\nAnd<00:40:16.760> now<00:40:16.880> we<00:40:16.960> can<00:40:17.120> once<00:40:17.360> again<00:40:17.760> apply<00:40:18.160> a\nAnd now we can once again apply a\nAnd now we can once again apply a beautiful<00:40:18.760> associativity<00:40:19.480> theorem<00:40:19.880> yet\nbeautiful associativity theorem yet\nbeautiful associativity theorem yet again.\nagain.\nagain. Um<00:40:21.320> I<00:40:21.400> just<00:40:21.600> gave<00:40:21.760> it<00:40:21.840> a<00:40:21.920> slightly<00:40:22.200> different\nUm I just gave it a slightly different\nUm I just gave it a slightly different name<00:40:22.840> with<00:40:23.240> theta<00:40:23.560> hat<00:40:23.840> because<00:40:24.160> it<00:40:24.520> me<00:40:24.840> it\nname with theta hat because it me it\nname with theta hat because it me it always<00:40:25.200> pisses<00:40:25.480> me<00:40:25.640> off<00:40:25.920> when<00:40:26.440> authors<00:40:26.840> reuse\nalways pisses me off when authors reuse\nalways pisses me off when authors reuse the<00:40:27.240> symbols<00:40:27.800> in<00:40:27.920> different<00:40:28.160> contexts,<00:40:28.600> so\nthe symbols in different contexts, so\nthe symbols in different contexts, so I<00:40:29.640> thought<00:40:29.800> I'd<00:40:29.880> do<00:40:29.960> the<00:40:30.040> courtesy<00:40:30.440> of<00:40:30.600> not\nI thought I'd do the courtesy of not\nI thought I'd do the courtesy of not doing<00:40:31.040> it.<00:40:31.680> And<00:40:31.800> once<00:40:31.920> again,<00:40:32.200> we<00:40:32.360> know<00:40:32.600> that\ndoing it. And once again, we know that\ndoing it. And once again, we know that somehow<00:40:33.480> some<00:40:34.360> ins-<00:40:34.720> for<00:40:34.880> some<00:40:35.920> uh\nsomehow some ins- for some uh\nsomehow some ins- for some uh theta<00:40:37.760> hat<00:40:37.920> for<00:40:38.000> some<00:40:38.360> mapping,<00:40:39.160> we<00:40:39.280> have\ntheta hat for some mapping, we have\ntheta hat for some mapping, we have addition<00:40:39.960> with\naddition with\naddition with this<00:40:41.640> operation<00:40:42.040> is<00:40:42.120> going<00:40:42.240> to<00:40:42.320> be<00:40:42.400> addition\nthis operation is going to be addition\nthis operation is going to be addition again.\nagain.\nagain. Um\nUm\nUm once<00:40:45.520> again,<00:40:45.880> we<00:40:46.040> have<00:40:46.240> already<00:40:46.480> chosen\nonce again, we have already chosen\nonce again, we have already chosen addition<00:40:47.120> for<00:40:47.480> our<00:40:48.040> first<00:40:48.320> operator,<00:40:49.400> and<00:40:49.560> so\naddition for our first operator, and so\naddition for our first operator, and so we<00:40:50.040> can<00:40:50.360> just<00:40:50.520> simply<00:40:50.800> recycle<00:40:51.200> the<00:40:51.280> proof\nwe can just simply recycle the proof\nwe can just simply recycle the proof from<00:40:52.360> our<00:40:53.080> previous<00:40:53.440> case.<00:40:54.200> And<00:40:54.880> so<00:40:55.520> this\nfrom our previous case. And so this\nfrom our previous case. And so this obviously<00:40:56.400> gave<00:40:56.640> you<00:40:56.720> already<00:40:56.960> the<00:40:57.040> hint<00:40:57.280> that\nobviously gave you already the hint that\nobviously gave you already the hint that it's<00:40:57.480> going<00:40:57.600> to<00:40:57.640> be<00:40:57.720> another<00:40:57.920> product<00:40:58.280> rule.\nit's going to be another product rule.\nit's going to be another product rule. Uh<00:40:59.680> we<00:41:00.080> set<00:41:00.320> up<00:41:00.440> kind<00:41:00.680> of<00:41:00.800> the<00:41:01.600> very<00:41:02.040> similar\nUh we set up kind of the very similar\nUh we set up kind of the very similar kind<00:41:02.880> of\nkind of\nkind of um\nconsistency<00:41:06.760> check<00:41:07.320> that<00:41:07.480> must<00:41:07.640> be\nconsistency check that must be\nconsistency check that must be consistent<00:41:08.200> with<00:41:08.360> the<00:41:08.920> operation<00:41:09.360> of<00:41:09.480> joining\nconsistent with the operation of joining\nconsistent with the operation of joining to<00:41:10.400> disjoint<00:41:10.920> elements<00:41:11.720> that<00:41:11.880> we<00:41:11.960> just<00:41:12.680> for\nto disjoint elements that we just for\nto disjoint elements that we just for which<00:41:12.960> we<00:41:13.080> know<00:41:13.720> for<00:41:13.840> which<00:41:14.000> we<00:41:14.080> have<00:41:14.240> chosen\nwhich we know for which we have chosen\nwhich we know for which we have chosen addition.\naddition.\naddition. Um<00:41:16.520> we<00:41:16.640> again<00:41:16.880> define<00:41:17.160> the<00:41:17.280> opposite.<00:41:18.320> We<00:41:18.440> find\nUm we again define the opposite. We find\nUm we again define the opposite. We find this<00:41:19.160> very<00:41:19.400> similar<00:41:19.720> structure<00:41:20.000> to<00:41:20.120> before,\nthis very similar structure to before,\nthis very similar structure to before, and<00:41:20.920> through<00:41:21.120> the<00:41:21.200> same<00:41:21.480> proof<00:41:21.800> in<00:41:21.920> fact,<00:41:22.200> we\nand through the same proof in fact, we\nand through the same proof in fact, we find<00:41:22.640> that<00:41:23.040> V<00:41:23.240> is<00:41:23.400> once<00:41:23.560> again<00:41:23.840> C<00:41:24.040> to<00:41:24.160> the<00:41:24.360> AX.\nfind that V is once again C to the AX.\nfind that V is once again C to the AX. And<00:41:25.880> meaning<00:41:26.120> once<00:41:26.320> again,<00:41:26.880> we<00:41:27.120> see<00:41:27.360> that<00:41:27.600> this\nAnd meaning once again, we see that this\nAnd meaning once again, we see that this uh<00:41:28.520> chaining<00:41:28.920> operator<00:41:29.400> is<00:41:29.640> also\nuh chaining operator is also\nuh chaining operator is also multiplication.<00:41:30.840> So<00:41:31.040> the<00:41:31.160> probability<00:41:31.840> of<00:41:32.120> A\nmultiplication. So the probability of A\nmultiplication. So the probability of A given<00:41:32.600> C\ngiven C\ngiven C is<00:41:33.760> equal<00:41:33.920> to<00:41:34.000> the<00:41:34.080> probability<00:41:34.600> of<00:41:34.880> A<00:41:35.160> given<00:41:35.480> B\nis equal to the probability of A given B\nis equal to the probability of A given B and<00:41:36.320> times<00:41:36.600> the<00:41:36.680> probability<00:41:37.120> of<00:41:37.360> B<00:41:37.640> given<00:41:37.960> C.\nand times the probability of B given C.\nand times the probability of B given C. Um\nUm\nUm and<00:41:40.400> there<00:41:40.520> we<00:41:40.600> go.<00:41:40.680> Now<00:41:40.840> we<00:41:40.960> have<00:41:41.160> derived\nand there we go. Now we have derived\nand there we go. Now we have derived basically<00:41:42.440> uh<00:41:42.960> all<00:41:43.120> the<00:41:43.200> probability\nbasically uh all the probability\nbasically uh all the probability calculus<00:41:44.280> of<00:41:44.480> the<00:41:44.720> Bayesian<00:41:45.160> inference\ncalculus of the Bayesian inference\ncalculus of the Bayesian inference machine.\nmachine.\nmachine. And<00:41:47.160> we<00:41:47.400> just<00:41:48.040> need<00:41:48.280> to<00:41:48.400> find<00:41:48.840> the<00:41:49.120> kind<00:41:49.360> of\nAnd we just need to find the kind of\nAnd we just need to find the kind of more<00:41:50.160> general<00:41:50.680> form<00:41:51.400> of<00:41:51.560> the<00:41:51.640> product<00:41:52.040> rule.\nmore general form of the product rule.\nmore general form of the product rule. Uh<00:41:54.040> and<00:41:54.200> we<00:41:54.320> can<00:41:54.600> do<00:41:54.760> that<00:41:54.960> by<00:41:55.080> simply<00:41:55.400> noting\nUh and we can do that by simply noting\nUh and we can do that by simply noting that<00:41:56.400> probability<00:41:57.160> of<00:41:57.560> X<00:41:57.800> and<00:41:58.000> Y<00:41:58.480> given<00:41:58.880> Y<00:41:59.320> is\nthat probability of X and Y given Y is\nthat probability of X and Y given Y is equal<00:41:59.680> to<00:41:59.760> the<00:41:59.840> probability<00:42:00.320> of<00:42:00.520> X<00:42:00.800> given<00:42:01.080> Y<00:42:01.560> if\nequal to the probability of X given Y if\nequal to the probability of X given Y if X<00:42:02.240> is<00:42:02.640> strictly<00:42:03.000> below<00:42:03.240> Y.<00:42:04.160> Uh<00:42:04.400> if<00:42:04.560> X<00:42:04.680> implies\nX is strictly below Y. Uh if X implies\nX is strictly below Y. Uh if X implies Y.<00:42:05.840> Uh<00:42:05.920> that's<00:42:06.200> simply<00:42:06.480> because<00:42:07.360> you<00:42:07.480> have<00:42:07.600> to\nY. Uh that's simply because you have to\nY. Uh that's simply because you have to remember<00:42:08.000> that<00:42:08.240> these<00:42:08.520> valuations<00:42:09.360> are<00:42:09.480> just\nremember that these valuations are just\nremember that these valuations are just uh,<00:42:10.360> these<00:42:10.960> sort<00:42:11.160> of<00:42:11.400> ranges<00:42:12.200> on<00:42:12.400> on<00:42:12.480> the\nuh, these sort of ranges on on the\nuh, these sort of ranges on on the lattice<00:42:13.000> and<00:42:13.320> obviously<00:42:14.240> these<00:42:14.520> two\nlattice and obviously these two\nlattice and obviously these two describe<00:42:16.600> the<00:42:16.680> same<00:42:17.160> because<00:42:18.000> X<00:42:18.320> implies<00:42:18.800> Y\ndescribe the same because X implies Y\ndescribe the same because X implies Y already.\nalready.\nalready. So,<00:42:20.720> these<00:42:20.880> two<00:42:21.000> are<00:42:21.080> equivalent.<00:42:22.000> And<00:42:22.160> so,<00:42:22.560> if\nSo, these two are equivalent. And so, if\nSo, these two are equivalent. And so, if we<00:42:22.840> have<00:42:22.960> a<00:42:23.040> probability<00:42:23.480> of<00:42:23.680> A<00:42:23.840> and<00:42:23.960> B<00:42:24.240> given\nwe have a probability of A and B given\nwe have a probability of A and B given C,<00:42:25.240> we<00:42:25.360> can<00:42:25.560> kind<00:42:25.760> of<00:42:26.000> expand<00:42:26.520> that<00:42:26.800> by<00:42:27.000> adding\nC, we can kind of expand that by adding\nC, we can kind of expand that by adding another<00:42:27.840> NC<00:42:28.320> at<00:42:28.440> the<00:42:28.560> end<00:42:29.000> because<00:42:29.640> this\nanother NC at the end because this\nanother NC at the end because this doesn't<00:42:30.040> change<00:42:30.240> our<00:42:30.320> valuation.<00:42:31.480> And<00:42:31.640> then\ndoesn't change our valuation. And then\ndoesn't change our valuation. And then we<00:42:31.960> can<00:42:32.120> kind<00:42:32.360> of<00:42:32.480> split<00:42:33.040> it<00:42:33.160> up<00:42:33.280> into<00:42:33.480> two\nwe can kind of split it up into two\nwe can kind of split it up into two parts<00:42:34.440> and<00:42:34.680> we<00:42:34.800> use<00:42:35.160> our<00:42:35.280> chaining\nparts and we use our chaining\nparts and we use our chaining uh,<00:42:37.040> like<00:42:37.200> this.<00:42:37.720> Here<00:42:37.960> we<00:42:38.280> sort<00:42:38.480> of<00:42:38.600> condition\nuh, like this. Here we sort of condition\nuh, like this. Here we sort of condition on<00:42:39.520> B<00:42:39.760> and<00:42:39.920> C<00:42:40.280> and<00:42:40.440> then<00:42:40.640> we<00:42:41.280> go<00:42:41.440> from<00:42:41.640> B<00:42:41.840> and<00:42:41.960> C\non B and C and then we go from B and C\non B and C and then we go from B and C to<00:42:43.040> just<00:42:43.360> C.\nto just C.\nto just C. But<00:42:44.720> these<00:42:45.040> two,<00:42:45.760> uh,<00:42:45.960> can<00:42:46.120> again<00:42:46.360> be\nBut these two, uh, can again be\nBut these two, uh, can again be simplified.<00:42:47.560> The<00:42:47.680> right<00:42:47.960> one<00:42:48.120> just<00:42:48.320> becomes\nsimplified. The right one just becomes\nsimplified. The right one just becomes probability<00:42:49.160> of<00:42:49.280> B<00:42:49.480> given<00:42:49.760> C<00:42:50.120> and<00:42:50.320> the<00:42:50.400> first\nprobability of B given C and the first\nprobability of B given C and the first one<00:42:51.280> just<00:42:51.400> becomes<00:42:51.760> probability<00:42:52.160> of<00:42:52.400> A<00:42:52.680> given\none just becomes probability of A given\none just becomes probability of A given B<00:42:53.080> and<00:42:53.200> C,<00:42:53.400> which<00:42:53.480> is<00:42:53.600> always<00:42:54.280> referring<00:42:54.720> to<00:42:55.080> to\nB and C, which is always referring to to\nB and C, which is always referring to to this<00:42:55.440> property<00:42:55.880> here,<00:42:56.320> right?\nthis property here, right?\nthis property here, right? And<00:42:59.040> very<00:42:59.320> interestingly,<00:42:59.800> we<00:42:59.920> can<00:43:00.120> also\nAnd very interestingly, we can also\nAnd very interestingly, we can also write<00:43:00.880> this<00:43:01.360> as<00:43:02.000> probability<00:43:02.560> of<00:43:02.840> B<00:43:03.080> given<00:43:03.400> AC\nwrite this as probability of B given AC\nwrite this as probability of B given AC times<00:43:04.000> probability<00:43:04.360> of<00:43:04.600> A<00:43:04.760> given<00:43:05.040> C.<00:43:05.240> So,\ntimes probability of A given C. So,\ntimes probability of A given C. So, this<00:43:06.440> probability\nthis probability\nthis probability of<00:43:08.080> A<00:43:08.200> and<00:43:08.320> B<00:43:08.480> given<00:43:08.760> C<00:43:09.560> can<00:43:09.760> be<00:43:09.880> written<00:43:10.080> in<00:43:10.160> two\nof A and B given C can be written in two\nof A and B given C can be written in two different<00:43:10.760> ways<00:43:11.320> and<00:43:11.440> we<00:43:11.560> can<00:43:11.720> exploit<00:43:12.280> this\ndifferent ways and we can exploit this\ndifferent ways and we can exploit this fact\nfact\nfact to<00:43:13.760> write<00:43:14.000> down<00:43:14.320> this<00:43:14.560> equivalence,<00:43:15.160> which<00:43:15.480> is\nto write down this equivalence, which is\nto write down this equivalence, which is just<00:43:16.280> Bayes'<00:43:16.600> rule.<00:43:17.200> So,<00:43:17.280> Bayes'<00:43:17.600> rule<00:43:17.760> is\njust Bayes' rule. So, Bayes' rule is\njust Bayes' rule. So, Bayes' rule is just<00:43:18.000> a<00:43:18.040> consequence\njust a consequence\njust a consequence of<00:43:19.720> this,<00:43:20.280> uh,<00:43:20.400> product<00:43:20.800> rule<00:43:20.920> for<00:43:21.040> chaining\nof this, uh, product rule for chaining\nof this, uh, product rule for chaining uh,<00:43:22.680> different<00:43:22.960> valuations.\nuh, different valuations.\nuh, different valuations. And<00:43:25.080> I'm<00:43:25.200> sure<00:43:25.400> you<00:43:25.480> all<00:43:25.640> know<00:43:26.000> just<00:43:26.200> how\nAnd I'm sure you all know just how\nAnd I'm sure you all know just how powerful<00:43:26.760> this<00:43:26.960> rule<00:43:27.160> is.\npowerful this rule is.\npowerful this rule is. Um,\nUm,\nUm, one<00:43:30.160> last<00:43:30.320> thing<00:43:30.480> that<00:43:30.600> we<00:43:30.680> need<00:43:30.800> to<00:43:30.920> show<00:43:31.480> is\none last thing that we need to show is\none last thing that we need to show is what<00:43:32.200> happens<00:43:32.680> when<00:43:33.480> we<00:43:33.640> choose<00:43:34.080> a<00:43:34.160> context,\nwhat happens when we choose a context,\nwhat happens when we choose a context, some<00:43:35.480> an<00:43:35.640> element<00:43:35.960> T<00:43:36.080> that<00:43:36.240> is<00:43:36.360> not<00:43:36.640> the<00:43:36.760> top\nsome an element T that is not the top\nsome an element T that is not the top element<00:43:37.440> of<00:43:37.560> its<00:43:37.720> lattice.<00:43:38.120> What<00:43:38.440> happens\nelement of its lattice. What happens\nelement of its lattice. What happens when<00:43:38.880> we<00:43:38.920> somehow<00:43:39.200> choose<00:43:39.440> something<00:43:39.760> else?\nwhen we somehow choose something else?\nwhen we somehow choose something else? Cuz<00:43:40.840> I<00:43:40.960> just<00:43:41.400> uh,<00:43:41.560> kept<00:43:41.880> saying<00:43:42.120> the<00:43:42.240> whole\nCuz I just uh, kept saying the whole\nCuz I just uh, kept saying the whole time,<00:43:42.680> \"Well,<00:43:43.000> well,<00:43:43.200> it<00:43:43.320> doesn't<00:43:43.600> really\ntime, \"Well, well, it doesn't really\ntime, \"Well, well, it doesn't really matter<00:43:44.320> what<00:43:44.480> it<00:43:44.640> is,<00:43:44.960> you<00:43:45.040> know,\nmatter what it is, you know,\nmatter what it is, you know, you<00:43:46.120> can<00:43:46.240> just<00:43:46.400> use<00:43:46.520> whatever.\"<00:43:47.320> But<00:43:47.480> how<00:43:47.640> does\nyou can just use whatever.\" But how does\nyou can just use whatever.\" But how does our<00:43:47.880> valuation<00:43:48.320> change?<00:43:49.160> Um,\nwe<00:43:51.840> can<00:43:52.000> just,<00:43:52.480> uh,\nwe can just, uh,\nwe can just, uh, go<00:43:54.120> evaluate<00:43:54.520> the<00:43:54.600> extreme<00:43:55.040> case<00:43:55.920> uh,<00:43:56.400> where\ngo evaluate the extreme case uh, where\ngo evaluate the extreme case uh, where we<00:43:56.680> have<00:43:56.840> this<00:43:57.120> this<00:43:57.280> is<00:43:57.360> obviously<00:43:57.880> the<00:43:58.280> valid\nwe have this this is obviously the valid\nwe have this this is obviously the valid form<00:43:58.760> that<00:43:58.880> we<00:43:58.960> explored<00:43:59.320> before.<00:44:00.160> But<00:44:00.320> what\nform that we explored before. But what\nform that we explored before. But what if<00:44:00.880> we<00:44:01.080> have<00:44:01.240> the<00:44:01.360> probability<00:44:02.000> of<00:44:02.240> the\nif we have the probability of the\nif we have the probability of the sort<00:44:03.640> of<00:44:03.720> the<00:44:03.800> context<00:44:04.480> being<00:44:04.720> itself?<00:44:05.800> The\nsort of the context being itself? The\nsort of the context being itself? The the<00:44:06.360> given<00:44:06.600> itself<00:44:07.240> and<00:44:07.880> obviously<00:44:08.280> the\nthe given itself and obviously the\nthe given itself and obviously the product<00:44:08.720> rule<00:44:08.840> must<00:44:09.040> still<00:44:09.240> apply<00:44:10.000> because\nproduct rule must still apply because\nproduct rule must still apply because it's<00:44:10.360> true<00:44:10.520> for<00:44:10.680> any<00:44:11.320> any<00:44:11.480> kind<00:44:11.680> of<00:44:11.800> range<00:44:12.680> or\nit's true for any any kind of range or\nit's true for any any kind of range or any<00:44:13.480> section<00:44:13.840> of<00:44:13.920> the<00:44:14.000> lattice.\nany section of the lattice.\nany section of the lattice. And<00:44:15.520> this<00:44:15.680> means<00:44:16.000> that<00:44:16.280> the<00:44:16.400> probability<00:44:17.040> of\nAnd this means that the probability of\nAnd this means that the probability of the<00:44:17.280> context<00:44:17.640> given<00:44:17.840> itself<00:44:18.200> must<00:44:18.480> be<00:44:18.640> equal\nthe context given itself must be equal\nthe context given itself must be equal to<00:44:19.000> itself<00:44:19.320> squared,<00:44:19.720> so<00:44:19.920> it<00:44:20.080> must<00:44:20.280> be<00:44:20.400> equal\nto itself squared, so it must be equal\nto itself squared, so it must be equal to<00:44:20.760> one.\nto one.\nto one. And<00:44:21.880> this<00:44:22.080> means<00:44:22.400> that<00:44:22.600> very<00:44:22.840> conveniently\nAnd this means that very conveniently\nAnd this means that very conveniently our<00:44:23.720> measure<00:44:24.080> is<00:44:24.360> just<00:44:24.720> always<00:44:25.080> normalized.\nour measure is just always normalized.\nour measure is just always normalized. It<00:44:25.960> uh\nIt uh\nIt uh it<00:44:27.480> always<00:44:27.760> sums<00:44:28.040> up<00:44:28.160> to<00:44:28.280> one.\nAnd<00:44:32.240> here<00:44:32.440> we<00:44:32.520> have<00:44:32.720> Bayes'<00:44:32.960> theorem<00:44:33.320> in<00:44:33.400> its\nAnd here we have Bayes' theorem in its\nAnd here we have Bayes' theorem in its full<00:44:33.760> power<00:44:34.640> um\nfull power um\nfull power um because<00:44:35.880> it<00:44:36.280> allows<00:44:36.720> us<00:44:36.920> to<00:44:37.040> do<00:44:37.240> something\nbecause it allows us to do something\nbecause it allows us to do something that<00:44:37.720> a<00:44:37.800> frequentist<00:44:38.320> methodology<00:44:38.800> just<00:44:39.120> not\nthat a frequentist methodology just not\nthat a frequentist methodology just not allow<00:44:39.640> us<00:44:39.760> to<00:44:39.840> do,<00:44:40.040> meaning<00:44:40.840> we<00:44:41.000> can<00:44:41.640> update\nallow us to do, meaning we can update\nallow us to do, meaning we can update the<00:44:42.240> probability<00:44:42.800> of<00:44:42.920> a<00:44:43.000> parameter\nthe probability of a parameter\nthe probability of a parameter rigorously.\nrigorously.\nrigorously. So,<00:44:45.200> imagine<00:44:45.600> you<00:44:45.680> have<00:44:45.800> some<00:44:46.000> context<00:44:46.760> for\nSo, imagine you have some context for\nSo, imagine you have some context for your<00:44:47.120> investigation,<00:44:47.840> your<00:44:47.960> experiment,\nyour investigation, your experiment,\nyour investigation, your experiment, whatever<00:44:48.880> it<00:44:48.960> is,<00:44:49.200> and<00:44:49.320> we<00:44:49.400> call<00:44:49.560> it<00:44:49.720> T.\nwhatever it is, and we call it T.\nwhatever it is, and we call it T. And<00:44:50.800> you<00:44:50.880> want<00:44:51.080> to<00:44:51.160> learn<00:44:51.320> the<00:44:51.440> parameter\nAnd you want to learn the parameter\nAnd you want to learn the parameter theta<00:44:52.960> from<00:44:53.200> data<00:44:53.880> D.\ntheta from data D.\ntheta from data D. Then<00:44:55.160> by<00:44:55.280> simply<00:44:55.560> applying<00:44:55.880> the<00:44:56.000> product<00:44:56.320> and\nThen by simply applying the product and\nThen by simply applying the product and the<00:44:56.520> sum<00:44:56.800> rules\nthe sum rules\nthe sum rules uh\nuh\nuh you<00:44:58.800> obtain<00:44:59.160> Bayes'<00:45:00.000> uh<00:45:00.080> theorem<00:45:00.560> and<00:45:00.720> you<00:45:00.800> can\nyou obtain Bayes' uh theorem and you can\nyou obtain Bayes' uh theorem and you can simply<00:45:01.600> write<00:45:01.800> it<00:45:01.880> down<00:45:02.040> as<00:45:02.160> this.\nsimply write it down as this.\nsimply write it down as this. Uh<00:45:03.160> you<00:45:03.320> have<00:45:03.600> the<00:45:03.680> probability<00:45:04.520> which<00:45:04.680> is\nUh you have the probability which is\nUh you have the probability which is called<00:45:05.040> the<00:45:05.120> posterior,\ncalled the posterior,\ncalled the posterior, the<00:45:06.720> probability<00:45:07.400> of<00:45:07.600> the<00:45:07.680> parameter<00:45:08.200> given\nthe probability of the parameter given\nthe probability of the parameter given the<00:45:08.520> data<00:45:08.760> and<00:45:08.880> the<00:45:09.000> context,<00:45:09.720> which<00:45:09.920> is<00:45:10.080> equal\nthe data and the context, which is equal\nthe data and the context, which is equal to<00:45:10.720> on<00:45:10.840> the<00:45:10.920> numerator<00:45:11.440> you<00:45:11.560> have<00:45:12.400> the<00:45:12.560> prior\nto on the numerator you have the prior\nto on the numerator you have the prior probability<00:45:13.760> of<00:45:13.920> the<00:45:14.040> parameter.\nprobability of the parameter.\nprobability of the parameter. Why<00:45:16.440> can't<00:45:16.720> I<00:45:16.760> highlight<00:45:17.080> anymore?\nWhy can't I highlight anymore?\nWhy can't I highlight anymore? Probability<00:45:18.680> of<00:45:18.760> the<00:45:18.840> parameter<00:45:19.320> given<00:45:19.600> the\nProbability of the parameter given the\nProbability of the parameter given the context<00:45:20.640> times<00:45:21.080> the<00:45:21.160> likelihood,<00:45:21.640> which<00:45:21.760> is\ncontext times the likelihood, which is\ncontext times the likelihood, which is the<00:45:21.920> probability<00:45:22.480> of<00:45:22.600> the<00:45:22.680> data<00:45:23.160> given<00:45:23.560> your\nthe probability of the data given your\nthe probability of the data given your uh<00:45:24.960> parameter.\nuh parameter.\nuh parameter. And<00:45:26.400> on<00:45:26.480> the<00:45:26.560> bottom<00:45:26.840> you<00:45:26.920> have<00:45:27.040> some<00:45:27.160> kind<00:45:27.360> of\nAnd on the bottom you have some kind of\nAnd on the bottom you have some kind of normalizing<00:45:28.440> constant<00:45:29.360> which<00:45:29.680> makes<00:45:29.920> it<00:45:30.040> so\nnormalizing constant which makes it so\nnormalizing constant which makes it so that<00:45:30.880> this<00:45:31.160> all<00:45:31.520> sums<00:45:31.840> up<00:45:31.960> to<00:45:32.080> one<00:45:32.760> and<00:45:32.920> it<00:45:33.080> is\nthat this all sums up to one and it is\nthat this all sums up to one and it is simply<00:45:33.480> obtained<00:45:33.960> by<00:45:34.600> um\nsimply obtained by um\nsimply obtained by um summing<00:45:36.520> over<00:45:36.680> all<00:45:36.880> possible<00:45:37.240> values<00:45:38.000> of\nsumming over all possible values of\nsumming over all possible values of theta\ntheta\ntheta and<00:45:39.240> then<00:45:39.440> having<00:45:39.720> all<00:45:39.880> possible<00:45:40.440> for<00:45:40.600> all\nand then having all possible for all\nand then having all possible for all possible<00:45:40.960> values<00:45:41.360> summing<00:45:41.760> over<00:45:41.960> this\npossible values summing over this\npossible values summing over this expression<00:45:42.920> in<00:45:43.040> the<00:45:43.120> numerator.\nAnd<00:45:47.440> again,<00:45:47.720> usually<00:45:48.080> for<00:45:48.240> gravity\nAnd again, usually for gravity\nAnd again, usually for gravity the<00:45:50.280> context<00:45:50.800> T<00:45:51.160> because<00:45:51.480> it's<00:45:51.680> common<00:45:52.000> in<00:45:52.120> our\nthe context T because it's common in our\nthe context T because it's common in our expression<00:45:52.760> is<00:45:52.920> omitted.<00:45:53.760> However,<00:45:54.080> remember\nexpression is omitted. However, remember\nexpression is omitted. However, remember that<00:45:54.600> in<00:45:54.800> this<00:45:55.160> view<00:45:55.680> of<00:45:56.160> probability\nthat in this view of probability\nthat in this view of probability we<00:45:57.760> all<00:45:58.080> it<00:45:58.160> is<00:45:58.360> always<00:45:59.400> a<00:45:59.480> measure<00:45:59.960> of<00:46:00.120> a\nwe all it is always a measure of a\nwe all it is always a measure of a degree<00:46:00.480> of<00:46:00.600> implication,<00:46:01.320> meaning<00:46:01.680> it's\ndegree of implication, meaning it's\ndegree of implication, meaning it's always<00:46:02.480> by<00:46:02.640> necessity<00:46:03.200> a<00:46:03.280> B<00:46:03.480> valuation.\nalways by necessity a B valuation.\nalways by necessity a B valuation. There's<00:46:06.120> no<00:46:06.240> such<00:46:06.400> thing<00:46:06.560> as<00:46:06.720> a<00:46:06.840> properly\nThere's no such thing as a properly\nThere's no such thing as a properly truly<00:46:08.400> in<00:46:08.520> conditional<00:46:08.920> probability.\ntruly in conditional probability.\ntruly in conditional probability. Somehow<00:46:09.880> floats<00:46:10.320> on<00:46:10.400> its<00:46:10.560> own<00:46:10.920> in<00:46:11.080> the\nSomehow floats on its own in the\nSomehow floats on its own in the no<00:46:12.360> kind<00:46:12.480> of<00:46:12.560> platforming<00:46:12.960> world.<00:46:13.280> That's<00:46:13.440> not\nno kind of platforming world. That's not\nno kind of platforming world. That's not a<00:46:13.640> thing.\na thing.\na thing. Um<00:46:16.280> then<00:46:16.680> we<00:46:16.840> of<00:46:17.000> course<00:46:17.320> need<00:46:17.480> to<00:46:17.560> spend<00:46:17.880> a\nUm then we of course need to spend a\nUm then we of course need to spend a little<00:46:18.200> bit<00:46:18.840> we<00:46:18.920> need<00:46:19.080> to<00:46:19.160> say<00:46:19.360> a<00:46:19.440> word<00:46:19.920> on\nlittle bit we need to say a word on\nlittle bit we need to say a word on prior<00:46:21.080> probabilities<00:46:21.600> because<00:46:21.840> that<00:46:22.000> is<00:46:22.120> a\nprior probabilities because that is a\nprior probabilities because that is a major\nmajor\nmajor uh<00:46:23.760> point<00:46:24.000> of<00:46:24.080> contention<00:46:25.120> and<00:46:25.240> is<00:46:25.400> also<00:46:25.680> why\nuh point of contention and is also why\nuh point of contention and is also why the<00:46:26.160> Bayesian<00:46:26.760> approach<00:46:27.240> gets<00:46:27.480> such<00:46:27.680> a<00:46:27.720> bad\nthe Bayesian approach gets such a bad\nthe Bayesian approach gets such a bad reputation\nreputation\nreputation uh<00:46:29.440> because\nuh because\nuh because where<00:46:30.840> do<00:46:30.920> you<00:46:31.000> get<00:46:31.160> those<00:46:31.320> prior\nwhere do you get those prior\nwhere do you get those prior probabilities?<00:46:32.160> I<00:46:32.200> mean<00:46:32.400> you<00:46:32.480> don't<00:46:32.600> you\nprobabilities? I mean you don't you\nprobabilities? I mean you don't you haven't<00:46:32.920> done<00:46:33.040> the<00:46:33.120> experiment<00:46:33.640> yet<00:46:33.800> by\nhaven't done the experiment yet by\nhaven't done the experiment yet by definition.<00:46:34.720> So,<00:46:35.040> what<00:46:35.200> the<00:46:35.280> hell<00:46:35.480> do<00:46:35.640> you\ndefinition. So, what the hell do you\ndefinition. So, what the hell do you know?<00:46:36.440> Um\nknow? Um\nknow? Um and\nand\nand unfortunately,<00:46:39.480> the<00:46:39.600> language<00:46:39.960> has<00:46:40.080> caught\nunfortunately, the language has caught\nunfortunately, the language has caught on<00:46:40.680> that<00:46:41.360> somehow<00:46:42.400> the<00:46:42.520> probability<00:46:43.120> the\non that somehow the probability the\non that somehow the probability the prior<00:46:43.440> probability<00:46:44.200> represents<00:46:44.760> the<00:46:44.880> belief\nprior probability represents the belief\nprior probability represents the belief of<00:46:45.840> the<00:46:45.960> researcher<00:46:46.520> and<00:46:46.640> this<00:46:46.760> makes<00:46:46.960> it\nof the researcher and this makes it\nof the researcher and this makes it sound<00:46:47.520> like<00:46:47.680> you<00:46:47.800> can<00:46:47.920> just<00:46:48.480> somehow<00:46:49.240> listen\nsound like you can just somehow listen\nsound like you can just somehow listen into<00:46:49.800> your<00:46:49.960> gut<00:46:50.280> and<00:46:50.480> and<00:46:50.600> choose<00:46:50.840> the<00:46:50.920> numbers\ninto your gut and and choose the numbers\ninto your gut and and choose the numbers yourself.<00:46:51.960> Uh<00:46:52.000> and<00:46:52.120> in<00:46:52.200> fact,<00:46:52.480> there<00:46:52.640> is<00:46:53.400> a\nyourself. Uh and in fact, there is a\nyourself. Uh and in fact, there is a school<00:46:53.920> of<00:46:54.040> probability<00:46:54.480> which<00:46:54.800> does\nschool of probability which does\nschool of probability which does something<00:46:55.640> like<00:46:55.880> this,<00:46:56.640> the<00:46:56.720> school<00:46:57.000> of<00:46:57.120> the\nsomething like this, the school of the\nsomething like this, the school of the de<00:46:57.320> Finetti.<00:46:58.240> But<00:46:58.960> uh<00:46:59.040> I<00:46:59.240> and<00:46:59.400> many<00:46:59.680> others,\nde Finetti. But uh I and many others,\nde Finetti. But uh I and many others, including<00:47:00.800> Edwin<00:47:01.080> Thompson<00:47:01.360> James<00:47:01.760> famously,\nincluding Edwin Thompson James famously,\nincluding Edwin Thompson James famously, uh<00:47:02.960> reject<00:47:03.360> this<00:47:03.520> and<00:47:03.640> find<00:47:03.840> it<00:47:03.920> totally\nuh reject this and find it totally\nuh reject this and find it totally absurd.\nabsurd.\nabsurd. And<00:47:05.720> uh<00:47:06.200> it<00:47:06.320> has<00:47:06.440> to<00:47:06.520> be<00:47:06.640> pointed<00:47:07.000> out<00:47:07.160> that\nAnd uh it has to be pointed out that\nAnd uh it has to be pointed out that while<00:47:08.080> the<00:47:08.160> word<00:47:08.360> subjective<00:47:09.040> is<00:47:09.240> usually<00:47:10.080> uh\nwhile the word subjective is usually uh\nwhile the word subjective is usually uh used<00:47:10.640> for<00:47:10.760> something<00:47:11.120> that<00:47:11.840> it<00:47:11.960> is<00:47:12.120> not<00:47:12.360> valid,\nused for something that it is not valid,\nused for something that it is not valid, actually<00:47:13.840> subjective<00:47:14.440> is<00:47:14.600> not<00:47:14.800> quite<00:47:15.080> the\nactually subjective is not quite the\nactually subjective is not quite the same<00:47:15.440> thing<00:47:15.680> as<00:47:16.000> arbitrary.\nsame thing as arbitrary.\nsame thing as arbitrary. Um\nUm\nUm because<00:47:18.280> probability<00:47:18.960> is<00:47:19.320> in<00:47:19.480> this<00:47:19.720> sense\nbecause probability is in this sense\nbecause probability is in this sense necessarily<00:47:20.720> subjective<00:47:21.200> because<00:47:21.560> it<00:47:21.760> always\nnecessarily subjective because it always\nnecessarily subjective because it always depends<00:47:22.440> on<00:47:22.560> what<00:47:22.760> knowledge<00:47:23.040> that<00:47:23.160> you<00:47:23.240> have.\ndepends on what knowledge that you have.\ndepends on what knowledge that you have. So,<00:47:23.920> people<00:47:24.160> with<00:47:24.360> very<00:47:24.600> different<00:47:24.960> knowledge\nSo, people with very different knowledge\nSo, people with very different knowledge might<00:47:25.640> assign\nmight assign\nmight assign very<00:47:27.080> different<00:47:27.280> probabilities<00:47:27.920> to<00:47:28.000> the<00:47:28.120> same\nvery different probabilities to the same\nvery different probabilities to the same uh<00:47:29.440> to<00:47:29.520> the<00:47:29.600> same<00:47:29.880> notion<00:47:30.240> to<00:47:30.360> the<00:47:30.440> same<00:47:31.080> uh\nuh to the same notion to the same uh\nuh to the same notion to the same uh proposition<00:47:31.640> to<00:47:31.720> the<00:47:31.800> same<00:47:31.960> event,<00:47:32.840> right?<00:47:33.520> Uh\nproposition to the same event, right? Uh\nproposition to the same event, right? Uh but<00:47:33.920> what<00:47:34.160> we<00:47:34.400> require<00:47:35.080> of<00:47:35.280> this<00:47:36.040> uh\nbut what we require of this uh\nbut what we require of this uh probability<00:47:36.560> calculus<00:47:37.240> is<00:47:37.400> simply<00:47:37.760> that\nprobability calculus is simply that\nprobability calculus is simply that two<00:47:38.880> different<00:47:39.400> researchers<00:47:40.080> with<00:47:41.000> uh<00:47:41.080> the\ntwo different researchers with uh the\ntwo different researchers with uh the same<00:47:41.560> information<00:47:42.640> will<00:47:42.760> give<00:47:42.960> the<00:47:43.040> exact\nsame information will give the exact\nsame information will give the exact same<00:47:43.840> prior<00:47:44.080> probability<00:47:44.640> and<00:47:44.760> this<00:47:44.920> is<00:47:45.040> what\nsame prior probability and this is what\nsame prior probability and this is what is<00:47:45.240> called<00:47:45.560> the<00:47:45.920> objective<00:47:46.880> uh<00:47:47.120> Bayesian\nis called the objective uh Bayesian\nis called the objective uh Bayesian school.\nAnd<00:47:51.960> now<00:47:52.200> we're<00:47:52.320> going<00:47:52.440> to<00:47:52.520> look<00:47:52.680> at<00:47:52.800> some<00:47:52.960> of\nAnd now we're going to look at some of\nAnd now we're going to look at some of the<00:47:53.160> amazing<00:47:53.600> things<00:47:53.880> you<00:47:53.960> can<00:47:54.120> do<00:47:54.360> with<00:47:54.560> this\nthe amazing things you can do with this\nthe amazing things you can do with this uh<00:47:55.520> with<00:47:55.640> this<00:47:56.040> cal-<00:47:56.400> of<00:47:56.800> this<00:47:56.960> reasoning\nuh with this cal- of this reasoning\nuh with this cal- of this reasoning apparatus,<00:47:57.760> this<00:47:58.000> common\napparatus, this common\napparatus, this common this<00:47:59.160> quantified<00:47:59.680> common<00:48:00.000> sense.\nthis quantified common sense.\nthis quantified common sense. Um,<00:48:01.440> if<00:48:01.600> you<00:48:01.680> want<00:48:01.840> to<00:48:02.200> know<00:48:02.440> more<00:48:02.800> as<00:48:02.960> to<00:48:03.120> why\nUm, if you want to know more as to why\nUm, if you want to know more as to why it's<00:48:03.720> quantified<00:48:04.160> common<00:48:04.400> sense,<00:48:04.760> once\nit's quantified common sense, once\nit's quantified common sense, once again,<00:48:05.240> I<00:48:05.320> recommend<00:48:05.840> you<00:48:05.920> the<00:48:06.040> book<00:48:06.920> um,<00:48:07.520> by\nagain, I recommend you the book um, by\nagain, I recommend you the book um, by Edwin<00:48:08.400> Thompson<00:48:08.680> Jaynes,<00:48:09.040> which<00:48:09.200> is<00:48:09.320> called\nEdwin Thompson Jaynes, which is called\nEdwin Thompson Jaynes, which is called Probability:<00:48:10.400> The<00:48:10.520> Logic<00:48:10.840> of<00:48:10.960> Science.\nProbability: The Logic of Science.\nProbability: The Logic of Science. Amazing<00:48:12.120> book.<00:48:12.360> Just<00:48:12.600> read<00:48:12.720> the<00:48:12.800> first<00:48:13.040> two\nAmazing book. Just read the first two\nAmazing book. Just read the first two chapters<00:48:13.560> and<00:48:13.760> you<00:48:13.920> immediately<00:48:14.440> see<00:48:14.800> what<00:48:14.920> he\nchapters and you immediately see what he\nchapters and you immediately see what he means.<00:48:15.360> Very<00:48:15.680> easy<00:48:15.920> and<00:48:16.200> quick<00:48:16.440> to<00:48:16.520> read<00:48:16.720> and\nmeans. Very easy and quick to read and\nmeans. Very easy and quick to read and super<00:48:17.040> interesting.\nBut<00:48:20.000> now,<00:48:20.160> assume<00:48:20.520> that<00:48:20.640> we<00:48:20.760> have<00:48:21.080> again<00:48:21.320> some\nBut now, assume that we have again some\nBut now, assume that we have again some context.<00:48:22.000> Oops.\ncontext. Oops.\ncontext. Oops. A<00:48:23.160> context<00:48:23.720> with<00:48:23.880> a<00:48:23.920> parameter<00:48:24.560> A\nA context with a parameter A\nA context with a parameter A uh,\nuh,\nuh, that<00:48:26.400> we<00:48:26.520> will<00:48:26.760> for<00:48:26.920> now,<00:48:27.200> just<00:48:27.360> for\nthat we will for now, just for\nthat we will for now, just for simplicity,<00:48:28.040> assume<00:48:28.360> to<00:48:28.480> be<00:48:28.560> discrete.<00:48:29.320> And\nsimplicity, assume to be discrete. And\nsimplicity, assume to be discrete. And this<00:48:29.680> parameter<00:48:30.160> can<00:48:30.320> take<00:48:30.600> values<00:48:31.360> 1,<00:48:31.800> A1,\nthis parameter can take values 1, A1,\nthis parameter can take values 1, A1, A2,<00:48:33.040> up<00:48:33.200> to<00:48:33.360> AN.\nA2, up to AN.\nA2, up to AN. Since<00:48:36.040> these<00:48:36.160> values<00:48:36.480> are<00:48:36.560> mutually\nSince these values are mutually\nSince these values are mutually exclusive<00:48:37.360> and<00:48:37.480> disjoint,<00:48:38.760> um,<00:48:39.400> and\nexclusive and disjoint, um, and\nexclusive and disjoint, um, and exhaustive,<00:48:40.240> we<00:48:40.440> know<00:48:40.760> that<00:48:41.440> the<00:48:41.840> the\nexhaustive, we know that the the\nexhaustive, we know that the the combination<00:48:42.480> of<00:48:42.600> all<00:48:42.760> of<00:48:42.880> them<00:48:43.720> uh,\ncombination of all of them uh,\ncombination of all of them uh, must<00:48:45.200> sum<00:48:45.400> up<00:48:45.520> to<00:48:45.640> one.<00:48:45.800> So,<00:48:46.200> the<00:48:46.320> the\nmust sum up to one. So, the the\nmust sum up to one. So, the the probability<00:48:46.920> of<00:48:47.040> their<00:48:47.160> combination<00:48:47.720> must\nprobability of their combination must\nprobability of their combination must sum<00:48:48.200> must<00:48:48.440> be<00:48:48.560> one.<00:48:49.320> Uh,<00:48:49.760> because<00:48:50.320> otherwise\nsum must be one. Uh, because otherwise\nsum must be one. Uh, because otherwise there's<00:48:51.080> possibilities<00:48:51.600> that<00:48:51.760> we<00:48:51.880> have<00:48:52.040> not\nthere's possibilities that we have not\nthere's possibilities that we have not considered.\nconsidered.\nconsidered. Uh,<00:48:53.680> and<00:48:53.840> since<00:48:54.160> this<00:48:54.360> notion<00:48:54.800> is<00:48:54.920> already<00:48:55.200> a\nUh, and since this notion is already a\nUh, and since this notion is already a consequence<00:48:55.760> of<00:48:55.840> the<00:48:55.920> context,<00:48:56.560> because<00:48:56.800> the\nconsequence of the context, because the\nconsequence of the context, because the context<00:48:57.320> tells<00:48:57.560> us,<00:48:57.680> \"Okay,<00:48:57.920> yes,<00:48:58.200> we<00:48:58.320> are\ncontext tells us, \"Okay, yes, we are\ncontext tells us, \"Okay, yes, we are looking<00:48:58.640> at<00:48:58.720> this<00:48:58.920> with<00:48:59.120> this<00:48:59.280> parameter.\"\nlooking at this with this parameter.\"\nlooking at this with this parameter.\" Uh,<00:49:01.600> then<00:49:01.840> conditioning<00:49:02.440> on<00:49:02.560> the<00:49:02.640> data<00:49:02.960> does\nUh, then conditioning on the data does\nUh, then conditioning on the data does not<00:49:03.280> change<00:49:03.560> it.<00:49:03.680> So,<00:49:03.880> also<00:49:04.200> the<00:49:04.480> probability\nnot change it. So, also the probability\nnot change it. So, also the probability of\nof\nof either<00:49:06.320> one<00:49:06.520> of<00:49:06.640> them<00:49:06.880> being<00:49:07.160> true,<00:49:07.520> given<00:49:07.800> the\neither one of them being true, given the\neither one of them being true, given the data<00:49:08.120> and<00:49:08.240> the<00:49:08.320> context,<00:49:08.760> is<00:49:08.880> still<00:49:09.160> one.<00:49:09.440> That\ndata and the context, is still one. That\ndata and the context, is still one. That doesn't<00:49:09.800> change.\ndoesn't change.\ndoesn't change. But<00:49:11.120> now,<00:49:11.600> what\nBut now, what\nBut now, what if<00:49:12.920> the<00:49:13.040> model<00:49:13.360> has<00:49:13.560> also<00:49:13.760> another<00:49:14.000> parameter,\nif the model has also another parameter,\nif the model has also another parameter, omega,\nomega,\nomega, which<00:49:15.720> is<00:49:15.880> the<00:49:15.960> one<00:49:16.080> that<00:49:16.200> we<00:49:16.280> are<00:49:16.320> actually\nwhich is the one that we are actually\nwhich is the one that we are actually interested<00:49:17.040> in,\ninterested in,\ninterested in, uh,\nuh,\nuh, that<00:49:18.640> we<00:49:18.720> want<00:49:18.880> to<00:49:19.000> estimate,<00:49:19.400> that<00:49:19.480> we<00:49:19.520> want\nthat we want to estimate, that we want\nthat we want to estimate, that we want to<00:49:19.760> learn<00:49:19.920> from<00:49:20.040> the<00:49:20.120> data,\nto learn from the data,\nto learn from the data, um,<00:49:21.760> meaning<00:49:22.040> we<00:49:22.120> want<00:49:22.280> to<00:49:22.360> find<00:49:22.520> the\num, meaning we want to find the\num, meaning we want to find the probability<00:49:23.000> of<00:49:23.080> omega<00:49:23.440> given<00:49:23.680> the<00:49:23.800> data,<00:49:24.520> and\nprobability of omega given the data, and\nprobability of omega given the data, and not<00:49:25.000> probability\nnot probability\nnot probability probability<00:49:26.320> of<00:49:26.480> omega<00:49:26.800> and<00:49:27.080> A<00:49:27.640> given<00:49:27.880> the\nprobability of omega and A given the\nprobability of omega and A given the data,<00:49:28.320> where<00:49:28.520> here<00:49:29.120> comma<00:49:29.440> simply<00:49:29.680> means<00:49:30.080> it's\ndata, where here comma simply means it's\ndata, where here comma simply means it's just<00:49:30.400> an<00:49:30.520> and<00:49:30.760> operation,<00:49:31.240> right?<00:49:31.800> You<00:49:31.920> want\njust an and operation, right? You want\njust an and operation, right? You want to<00:49:32.160> know<00:49:32.320> both<00:49:32.600> simultaneously.<00:49:33.760> But<00:49:33.840> in<00:49:33.920> this\nto know both simultaneously. But in this\nto know both simultaneously. But in this case,<00:49:34.200> we<00:49:34.280> don't<00:49:34.480> want<00:49:34.640> that.<00:49:35.240> So,<00:49:35.360> how<00:49:35.480> do<00:49:35.600> we\ncase, we don't want that. So, how do we\ncase, we don't want that. So, how do we get<00:49:35.840> rid<00:49:36.040> of<00:49:36.160> this<00:49:36.400> what<00:49:36.560> is<00:49:36.680> called<00:49:37.040> a\nget rid of this what is called a\nget rid of this what is called a nuisance<00:49:37.560> parameter,<00:49:38.200> A?\nnuisance parameter, A?\nnuisance parameter, A? The<00:49:39.720> way<00:49:39.840> we<00:49:39.960> do<00:49:40.160> this<00:49:40.360> is<00:49:40.600> we<00:49:40.720> apply<00:49:41.200> the<00:49:41.360> sum\nThe way we do this is we apply the sum\nThe way we do this is we apply the sum and<00:49:41.680> product<00:49:42.000> rules.\nand product rules.\nand product rules. Um,\nUm,\nUm, we<00:49:44.520> start<00:49:44.840> from<00:49:45.760> uh,<00:49:46.000> this<00:49:46.480> statement<00:49:46.800> that<00:49:46.880> we\nwe start from uh, this statement that we\nwe start from uh, this statement that we just<00:49:47.120> saw<00:49:47.240> before,<00:49:47.600> that<00:49:47.760> is<00:49:47.880> still<00:49:48.080> equal<00:49:48.320> to\njust saw before, that is still equal to\njust saw before, that is still equal to one,\none,\none, and<00:49:49.800> we\nand we\nand we uh,<00:49:50.600> we<00:49:50.720> split<00:49:50.920> it<00:49:51.000> up.<00:49:51.640> Uh,<00:49:51.840> we<00:49:52.000> apply<00:49:52.240> the\nuh, we split it up. Uh, we apply the\nuh, we split it up. Uh, we apply the product<00:49:52.680> rule<00:49:52.960> and<00:49:53.720> basically<00:49:54.240> we<00:49:54.360> have<00:49:55.040> omega\nproduct rule and basically we have omega\nproduct rule and basically we have omega given<00:49:56.240> all<00:49:56.400> of<00:49:56.480> this<00:49:56.600> combination<00:49:57.160> of<00:49:57.240> the\ngiven all of this combination of the\ngiven all of this combination of the parameter<00:49:58.480> and<00:49:58.720> times<00:49:59.080> the<00:49:59.200> probability<00:50:00.040> the\nparameter and times the probability the\nparameter and times the probability the prior<00:50:00.360> probability<00:50:01.280> of<00:50:02.000> all<00:50:02.160> these<00:50:02.320> possible\nprior probability of all these possible\nprior probability of all these possible values<00:50:02.920> of<00:50:03.000> the<00:50:03.080> parameter.<00:50:03.960> And<00:50:04.080> since<00:50:04.320> this\nvalues of the parameter. And since this\nvalues of the parameter. And since this is<00:50:04.640> equal<00:50:04.880> to<00:50:05.000> one<00:50:05.360> in<00:50:05.400> the<00:50:05.480> second<00:50:05.800> part\nis equal to one in the second part\nis equal to one in the second part um\num\num and<00:50:09.440> this<00:50:09.560> parameter<00:50:09.960> does<00:50:10.120> not<00:50:10.280> give<00:50:10.480> us<00:50:10.720> any\nand this parameter does not give us any\nand this parameter does not give us any additional<00:50:11.280> information<00:50:11.840> on<00:50:11.920> the<00:50:12.000> second\nadditional information on the second\nadditional information on the second parameter,\nparameter,\nparameter, uh<00:50:14.080> this<00:50:14.320> essentially<00:50:15.360> all<00:50:15.480> of<00:50:15.560> this<00:50:15.840> is<00:50:16.000> equal\nuh this essentially all of this is equal\nuh this essentially all of this is equal to<00:50:16.600> just<00:50:16.800> the<00:50:16.880> probability<00:50:17.680> of\nto just the probability of\nto just the probability of the<00:50:18.880> parameter<00:50:19.240> that<00:50:19.360> we<00:50:19.480> want.<00:50:20.160> So,<00:50:20.520> this<00:50:20.760> is\nthe parameter that we want. So, this is\nthe parameter that we want. So, this is the<00:50:20.920> expression<00:50:21.720> that<00:50:21.920> we're<00:50:22.040> looking<00:50:22.280> for.\nthe expression that we're looking for.\nthe expression that we're looking for. We<00:50:22.560> have<00:50:22.680> found<00:50:23.000> it.<00:50:23.760> Um\nAnd<00:50:26.360> this<00:50:26.480> is<00:50:26.560> again<00:50:26.760> a<00:50:26.800> consequence<00:50:27.360> of<00:50:27.480> the\nAnd this is again a consequence of the\nAnd this is again a consequence of the fact<00:50:27.880> that<00:50:28.280> this<00:50:28.440> kind<00:50:28.680> of<00:50:28.960> massive<00:50:29.800> compound\nfact that this kind of massive compound\nfact that this kind of massive compound statement<00:50:30.800> is<00:50:30.960> already<00:50:31.200> contained<00:50:31.560> in<00:50:31.640> the\nstatement is already contained in the\nstatement is already contained in the context.<00:50:32.560> And<00:50:32.680> so,<00:50:32.880> it<00:50:32.960> doesn't<00:50:33.240> bear<00:50:33.680> any\ncontext. And so, it doesn't bear any\ncontext. And so, it doesn't bear any additional<00:50:34.360> influence<00:50:35.280> on<00:50:35.400> the<00:50:35.480> knowledge<00:50:35.840> of\nadditional influence on the knowledge of\nadditional influence on the knowledge of omega<00:50:36.440> because<00:50:36.840> it<00:50:37.000> doesn't<00:50:37.280> add<00:50:37.520> anything\nomega because it doesn't add anything\nomega because it doesn't add anything that<00:50:37.960> is<00:50:38.080> not<00:50:38.280> already<00:50:38.520> contained<00:50:38.880> in<00:50:39.040> T.\nSo,<00:50:42.840> how<00:50:43.080> how<00:50:43.320> do<00:50:43.400> we<00:50:43.480> make<00:50:43.720> use<00:50:43.880> of<00:50:43.960> this?<00:50:44.720> Um\nSo, how how do we make use of this? Um\nSo, how how do we make use of this? Um we<00:50:46.080> make<00:50:46.320> use<00:50:46.560> of<00:50:46.720> the<00:50:46.800> distributivity<00:50:47.760> of\nwe make use of the distributivity of\nwe make use of the distributivity of logical<00:50:48.400> operations.<00:50:48.840> So,<00:50:48.960> we<00:50:49.080> have<00:50:49.200> seen\nlogical operations. So, we have seen\nlogical operations. So, we have seen that<00:50:49.680> this<00:50:50.160> expression\nthat this expression\nthat this expression gives<00:50:51.880> us<00:50:51.960> the<00:50:52.080> probability<00:50:52.600> that<00:50:52.760> we<00:50:52.840> want.\ngives us the probability that we want.\ngives us the probability that we want. This<00:50:53.520> is<00:50:53.600> the<00:50:53.640> thing<00:50:53.800> that<00:50:53.920> we<00:50:54.000> saw<00:50:54.160> just\nThis is the thing that we saw just\nThis is the thing that we saw just above.\nabove.\nabove. And<00:50:56.160> now<00:50:56.440> we<00:50:56.560> just<00:50:56.720> simply<00:50:57.000> make<00:50:57.160> use<00:50:57.320> of<00:50:57.400> the\nAnd now we just simply make use of the\nAnd now we just simply make use of the distributivity<00:50:58.360> of<00:50:58.600> the<00:50:58.760> and<00:50:59.360> over<00:50:59.560> the<00:50:59.680> or\ndistributivity of the and over the or\ndistributivity of the and over the or operations<00:51:00.440> and<00:51:00.560> we<00:51:00.680> obtain<00:51:00.960> kind<00:51:01.160> of<00:51:01.760> this\noperations and we obtain kind of this\noperations and we obtain kind of this massive<00:51:02.360> compound<00:51:02.640> statement<00:51:03.080> where<00:51:03.240> we<00:51:03.320> have\nmassive compound statement where we have\nmassive compound statement where we have omega\nomega\nomega and<00:51:05.120> A1<00:51:05.640> or<00:51:05.880> omega<00:51:06.240> and<00:51:06.440> A2<00:51:07.080> and<00:51:07.320> so<00:51:07.520> on<00:51:07.760> or\nand A1 or omega and A2 and so on or\nand A1 or omega and A2 and so on or omega<00:51:08.320> and<00:51:08.600> AN.\nomega and AN.\nomega and AN. And<00:51:10.440> since<00:51:10.920> the<00:51:11.040> or<00:51:11.240> operation<00:51:11.800> is<00:51:12.000> always\nAnd since the or operation is always\nAnd since the or operation is always a<00:51:13.320> sum,<00:51:14.000> we<00:51:14.120> can<00:51:14.280> split<00:51:14.560> this<00:51:14.720> up<00:51:14.920> into\na sum, we can split this up into\na sum, we can split this up into different<00:51:16.520> probabilities<00:51:17.640> that<00:51:17.840> we<00:51:17.920> just\ndifferent probabilities that we just\ndifferent probabilities that we just simply<00:51:18.600> sum<00:51:18.920> together.\nsimply sum together.\nsimply sum together. Um<00:51:20.720> and<00:51:20.840> this<00:51:21.080> gives<00:51:21.280> us<00:51:21.440> a<00:51:21.480> general<00:51:21.840> recipe\nUm and this gives us a general recipe\nUm and this gives us a general recipe for<00:51:22.640> essentially<00:51:23.200> what<00:51:23.320> is<00:51:23.400> called<00:51:23.640> averaging\nfor essentially what is called averaging\nfor essentially what is called averaging out<00:51:24.520> the<00:51:24.600> nuisance<00:51:24.920> parameter.<00:51:25.600> So,<00:51:25.800> in<00:51:25.960> a\nout the nuisance parameter. So, in a\nout the nuisance parameter. So, in a continuous<00:51:26.600> case,\ncontinuous case,\ncontinuous case, we<00:51:27.800> obtain<00:51:28.200> in<00:51:28.320> this<00:51:28.520> beautiful<00:51:28.880> simple\nwe obtain in this beautiful simple\nwe obtain in this beautiful simple formula\nformula\nformula that<00:51:30.480> we<00:51:30.600> can<00:51:30.720> just<00:51:31.000> integrate<00:51:31.520> out<00:51:31.800> the\nthat we can just integrate out the\nthat we can just integrate out the nuisance<00:51:32.160> parameter.<00:51:32.480> We<00:51:32.560> just<00:51:32.720> need<00:51:32.840> to\nnuisance parameter. We just need to\nnuisance parameter. We just need to sum<00:51:34.320> all<00:51:34.480> the<00:51:34.560> probabilities\nsum all the probabilities\nsum all the probabilities of<00:51:36.440> the<00:51:36.520> kind<00:51:36.680> of<00:51:36.760> the<00:51:36.880> value<00:51:37.200> of<00:51:37.320> omega<00:51:37.560> that\nof the kind of the value of omega that\nof the kind of the value of omega that we're<00:51:37.760> looking<00:51:38.000> for<00:51:38.800> over<00:51:39.120> all<00:51:39.360> possible\nwe're looking for over all possible\nwe're looking for over all possible values<00:51:40.200> of<00:51:40.440> A.\nvalues of A.\nvalues of A. And<00:51:42.360> this<00:51:42.520> is<00:51:42.640> very<00:51:43.160> very<00:51:43.400> very<00:51:43.640> powerful\nAnd this is very very very powerful\nAnd this is very very very powerful because<00:51:44.440> in<00:51:44.560> a<00:51:44.640> high<00:51:45.680> when<00:51:45.920> a<00:51:45.960> problem<00:51:46.240> is<00:51:46.360> very\nbecause in a high when a problem is very\nbecause in a high when a problem is very high<00:51:46.800> dimensional,\nhigh dimensional,\nhigh dimensional, trying<00:51:48.720> to<00:51:48.840> estimate\ntrying to estimate\ntrying to estimate all<00:51:50.400> the<00:51:50.520> parameters<00:51:50.960> simultaneously<00:51:51.640> is\nall the parameters simultaneously is\nall the parameters simultaneously is often<00:51:52.560> impossible<00:51:53.240> or<00:51:53.480> it's<00:51:53.760> intractable\noften impossible or it's intractable\noften impossible or it's intractable with<00:51:55.160> with<00:51:55.280> the<00:51:55.360> computation<00:51:55.880> that<00:51:56.000> we<00:51:56.080> have\nwith with the computation that we have\nwith with the computation that we have available.<00:51:57.280> And<00:51:57.880> this<00:51:58.080> kind<00:51:58.240> of<00:51:58.440> averaging\navailable. And this kind of averaging\navailable. And this kind of averaging out<00:51:59.120> makes<00:51:59.480> many<00:52:00.240> such<00:52:00.480> problems\nout makes many such problems\nout makes many such problems approachable<00:52:01.680> so<00:52:01.800> that<00:52:01.920> we<00:52:02.000> can<00:52:02.280> actually<00:52:02.560> get\napproachable so that we can actually get\napproachable so that we can actually get an<00:52:03.080> interesting<00:52:03.520> answer<00:52:03.840> or<00:52:04.040> any<00:52:04.240> answer<00:52:04.560> at\nan interesting answer or any answer at\nan interesting answer or any answer at all<00:52:04.840> actually.<00:52:05.720> But<00:52:05.880> in<00:52:05.960> the<00:52:06.080> frequentist\nall actually. But in the frequentist\nall actually. But in the frequentist practice<00:52:07.080> this<00:52:07.240> is<00:52:07.360> impossible\npractice this is impossible\npractice this is impossible simply<00:52:08.960> because<00:52:09.320> they<00:52:09.400> don't<00:52:09.720> assign\nsimply because they don't assign\nsimply because they don't assign probabilities<00:52:10.760> to<00:52:10.880> parameters<00:52:11.440> and<00:52:11.520> so<00:52:11.720> all\nprobabilities to parameters and so all\nprobabilities to parameters and so all of<00:52:11.960> this<00:52:12.120> reasoning\nof this reasoning\nof this reasoning with<00:52:13.440> probability<00:52:13.920> of<00:52:14.120> A<00:52:14.240> the<00:52:14.720> of<00:52:14.920> omega<00:52:15.280> and<00:52:15.480> A\nwith probability of A the of omega and A\nwith probability of A the of omega and A does<00:52:16.400> not<00:52:16.560> make<00:52:16.720> any<00:52:16.840> sense.\nUm\nUm\nUm And<00:52:21.400> now<00:52:21.760> another,<00:52:22.400> which<00:52:22.560> is<00:52:22.640> probably<00:52:22.880> my\nAnd now another, which is probably my\nAnd now another, which is probably my favorite\nfavorite\nfavorite feature<00:52:24.720> of<00:52:24.840> the<00:52:25.200> of<00:52:25.320> the<00:52:25.440> Bayesian\nfeature of the of the Bayesian\nfeature of the of the Bayesian probability<00:52:26.080> calculus,<00:52:26.560> is<00:52:26.760> that<00:52:26.920> it<00:52:27.040> has<00:52:27.720> a\nprobability calculus, is that it has a\nprobability calculus, is that it has a built-in<00:52:28.680> Occam's<00:52:29.080> razor.<00:52:29.360> It<00:52:29.440> is<00:52:29.520> a\nbuilt-in Occam's razor. It is a\nbuilt-in Occam's razor. It is a quantification<00:52:30.360> of<00:52:30.520> the<00:52:30.640> notion<00:52:30.920> of<00:52:31.080> Occam's\nquantification of the notion of Occam's\nquantification of the notion of Occam's razor.\nrazor.\nrazor. Um\nUm\nUm This<00:52:33.960> idea<00:52:34.520> is<00:52:34.800> usually<00:52:35.360> kind<00:52:35.640> of\nThis idea is usually kind of\nThis idea is usually kind of seen<00:52:37.720> as<00:52:38.080> a<00:52:38.760> qualitative<00:52:39.480> desiderata<00:52:40.320> of\nseen as a qualitative desiderata of\nseen as a qualitative desiderata of scientific<00:52:41.000> theories.\nscientific theories.\nscientific theories. The<00:52:43.200> the<00:52:43.280> exact<00:52:43.680> words<00:52:44.120> of<00:52:44.320> Occam<00:52:44.600> can<00:52:44.720> be\nThe the exact words of Occam can be\nThe the exact words of Occam can be translated<00:52:45.200> as<00:52:45.480> things<00:52:46.200> should<00:52:46.320> be<00:52:47.000> increased\ntranslated as things should be increased\ntranslated as things should be increased in<00:52:47.560> number<00:52:48.040> beyond<00:52:48.440> necessity<00:52:49.200> and<00:52:49.280> which<00:52:49.400> is\nin number beyond necessity and which is\nin number beyond necessity and which is usually<00:52:49.720> summarized<00:52:50.400> as<00:52:51.240> all<00:52:51.520> else<00:52:51.840> being\nusually summarized as all else being\nusually summarized as all else being equal<00:52:52.920> the<00:52:53.000> simplest<00:52:53.280> explanation<00:52:53.800> is\nequal the simplest explanation is\nequal the simplest explanation is usually<00:52:54.160> the<00:52:54.280> best<00:52:54.560> one.\nusually the best one.\nusually the best one. Um\nUm\nUm Bayes'<00:52:56.880> theorem<00:52:57.200> has<00:52:57.440> a<00:52:57.640> mechanism<00:52:58.120> or<00:52:58.240> the\nBayes' theorem has a mechanism or the\nBayes' theorem has a mechanism or the Bayesian<00:52:59.080> calculus<00:52:59.480> in<00:52:59.560> general<00:52:59.840> has<00:53:00.000> a\nBayesian calculus in general has a\nBayesian calculus in general has a mechanism<00:53:00.480> that<00:53:00.600> does<00:53:00.920> exactly<00:53:01.240> this.\nmechanism that does exactly this.\nmechanism that does exactly this. Uh<00:53:02.800> meaning<00:53:03.680> more<00:53:03.920> complex<00:53:04.440> models\nUh meaning more complex models\nUh meaning more complex models um\num\num penalized<00:53:07.800> if<00:53:08.000> their<00:53:08.280> additional<00:53:08.800> predictive\npenalized if their additional predictive\npenalized if their additional predictive accuracy<00:53:10.360> doesn't<00:53:10.640> make<00:53:10.880> up<00:53:11.040> for<00:53:11.200> their\naccuracy doesn't make up for their\naccuracy doesn't make up for their predictive<00:53:12.000> flexibility.\npredictive flexibility.\npredictive flexibility. And<00:53:13.680> we<00:53:13.760> can<00:53:13.880> see<00:53:14.000> this<00:53:14.160> as<00:53:14.280> follows.<00:53:14.760> First,\nAnd we can see this as follows. First,\nAnd we can see this as follows. First, we<00:53:15.840> need<00:53:15.960> to<00:53:16.080> look<00:53:16.240> at<00:53:16.360> how<00:53:16.520> we<00:53:16.640> would<00:53:16.800> even\nwe need to look at how we would even\nwe need to look at how we would even compare<00:53:17.440> different<00:53:17.760> models.\ncompare different models.\ncompare different models. And<00:53:19.160> so<00:53:19.400> once<00:53:19.640> again<00:53:19.880> we<00:53:20.000> just<00:53:20.720> remember<00:53:21.120> that\nAnd so once again we just remember that\nAnd so once again we just remember that in<00:53:21.600> the<00:53:21.680> Bayesian<00:53:22.520> context<00:53:23.000> we<00:53:23.080> can<00:53:23.560> assign\nin the Bayesian context we can assign\nin the Bayesian context we can assign probabilities<00:53:24.400> to<00:53:24.640> any<00:53:24.840> proposition<00:53:25.800> and\nprobabilities to any proposition and\nprobabilities to any proposition and that<00:53:26.040> is<00:53:26.160> also<00:53:26.360> the<00:53:26.440> proposition<00:53:27.000> of<00:53:27.400> model<00:53:27.880> I\nthat is also the proposition of model I\nthat is also the proposition of model I of<00:53:28.840> a<00:53:28.880> choice<00:53:29.200> of<00:53:29.320> models<00:53:29.680> is<00:53:29.840> being<00:53:30.040> true.\nof a choice of models is being true.\nof a choice of models is being true. And<00:53:31.240> so<00:53:31.520> we<00:53:31.640> can<00:53:32.000> once<00:53:32.240> again<00:53:32.960> write<00:53:33.280> this<00:53:33.440> down\nAnd so we can once again write this down\nAnd so we can once again write this down in<00:53:33.960> the\nin the\nin the in<00:53:35.040> the<00:53:35.120> Bayes'<00:53:35.360> theorem<00:53:35.720> form.\nin the Bayes' theorem form.\nin the Bayes' theorem form. And<00:53:37.720> again<00:53:37.920> the<00:53:38.200> the<00:53:38.560> term<00:53:38.760> in<00:53:38.840> the\nAnd again the the term in the\nAnd again the the term in the denominator<00:53:39.520> is<00:53:39.640> just<00:53:39.960> a<00:53:40.000> The<00:53:40.440> constant\ndenominator is just a The constant\ndenominator is just a The constant obtained<00:53:41.720> by<00:53:41.840> summing<00:53:42.280> over<00:53:42.560> all\nobtained by summing over all\nobtained by summing over all possibilities<00:53:43.600> over<00:53:43.840> all<00:53:43.960> models.\nAnd<00:53:47.280> com-<00:53:47.600> model<00:53:47.840> comparison<00:53:48.560> is<00:53:49.160> um<00:53:49.680> very\nAnd com- model comparison is um very\nAnd com- model comparison is um very analogous<00:53:50.320> to<00:53:50.400> parameter<00:53:50.800> estimation.<00:53:51.880> Um\nanalogous to parameter estimation. Um\nanalogous to parameter estimation. Um the<00:53:53.080> posterior<00:53:53.680> probability<00:53:54.120> for<00:53:54.200> model<00:53:54.480> is\nthe posterior probability for model is\nthe posterior probability for model is proportional<00:53:55.240> to<00:53:55.400> its<00:53:55.640> prior<00:53:55.960> probability\nproportional to its prior probability\nproportional to its prior probability times<00:53:56.840> its<00:53:56.960> likelihood.<00:53:57.760> And<00:53:57.880> now<00:53:57.960> to<00:53:58.080> compare\ntimes its likelihood. And now to compare\ntimes its likelihood. And now to compare two<00:53:58.640> models,<00:53:59.040> we<00:53:59.160> can<00:53:59.360> compute<00:53:59.680> their<00:53:59.880> odds\ntwo models, we can compute their odds\ntwo models, we can compute their odds ratio.<00:54:00.600> We<00:54:00.720> compare<00:54:01.080> them<00:54:01.240> by<00:54:01.480> dividing<00:54:02.000> one\nratio. We compare them by dividing one\nratio. We compare them by dividing one by<00:54:02.240> the<00:54:02.360> other.<00:54:03.040> And<00:54:03.160> the<00:54:03.240> reason<00:54:03.760> we<00:54:04.360> only\nby the other. And the reason we only\nby the other. And the reason we only have<00:54:05.160> um\nhave um\nhave um Exactly.<00:54:07.520> Here<00:54:07.600> we<00:54:07.680> have<00:54:07.800> the<00:54:07.880> original\nExactly. Here we have the original\nExactly. Here we have the original ratio,<00:54:08.520> and<00:54:08.640> the<00:54:08.720> reason<00:54:09.080> why<00:54:09.640> we<00:54:09.760> can<00:54:10.040> write\nratio, and the reason why we can write\nratio, and the reason why we can write it<00:54:10.200> as<00:54:10.320> such\nit as such\nit as such with<00:54:11.840> sort<00:54:12.080> of<00:54:12.640> the<00:54:12.760> top<00:54:13.120> the<00:54:13.240> num-<00:54:13.600> the\nwith sort of the top the num- the\nwith sort of the top the num- the numerator<00:54:14.200> of<00:54:14.320> model<00:54:14.680> I<00:54:14.920> and<00:54:15.080> the<00:54:15.160> numerator\nnumerator of model I and the numerator\nnumerator of model I and the numerator of<00:54:15.720> model<00:54:16.000> J<00:54:16.360> is<00:54:16.520> because<00:54:17.440> um\nof model J is because um\nof model J is because um this<00:54:18.920> normalizing<00:54:19.440> constant<00:54:20.480> they<00:54:20.600> would\nthis normalizing constant they would\nthis normalizing constant they would have<00:54:20.880> in<00:54:20.960> common<00:54:21.320> because<00:54:21.600> we<00:54:21.720> are<00:54:21.800> studying\nhave in common because we are studying\nhave in common because we are studying them<00:54:22.360> within<00:54:22.560> the<00:54:22.640> same<00:54:22.960> context<00:54:23.560> and<00:54:24.000> with\nthem within the same context and with\nthem within the same context and with the<00:54:24.280> same<00:54:24.520> data.<00:54:25.240> This<00:54:25.440> is<00:54:25.920> uh<00:54:26.000> what<00:54:26.120> we're\nthe same data. This is uh what we're\nthe same data. This is uh what we're doing<00:54:26.480> here.<00:54:27.240> And<00:54:27.320> so<00:54:27.600> we<00:54:27.760> can<00:54:28.200> obtain<00:54:28.800> this\ndoing here. And so we can obtain this\ndoing here. And so we can obtain this kind<00:54:29.560> of<00:54:29.680> ratio,<00:54:30.040> which<00:54:30.160> is<00:54:30.280> very\nkind of ratio, which is very\nkind of ratio, which is very interesting.<00:54:31.440> And<00:54:31.680> since<00:54:31.960> usually<00:54:32.520> we<00:54:32.640> don't\ninteresting. And since usually we don't\ninteresting. And since usually we don't have<00:54:33.240> any<00:54:33.480> kind<00:54:33.680> of<00:54:33.840> reason<00:54:34.240> to<00:54:34.360> prefer<00:54:35.320> any\nhave any kind of reason to prefer any\nhave any kind of reason to prefer any model<00:54:35.880> over<00:54:36.040> the<00:54:36.160> other,<00:54:36.520> we<00:54:36.680> must<00:54:37.200> by\nmodel over the other, we must by\nmodel over the other, we must by consistency<00:54:38.000> assign<00:54:38.520> equal<00:54:38.840> prior\nconsistency assign equal prior\nconsistency assign equal prior probabilities<00:54:39.680> to<00:54:39.760> them.<00:54:40.000> Meaning<00:54:40.200> the\nprobabilities to them. Meaning the\nprobabilities to them. Meaning the probability<00:54:40.840> of<00:54:41.000> model<00:54:41.360> I<00:54:42.000> is<00:54:42.160> equal<00:54:42.320> to<00:54:42.400> the\nprobability of model I is equal to the\nprobability of model I is equal to the prior<00:54:42.880> probability<00:54:43.560> of<00:54:43.720> model<00:54:44.000> J.\nprior probability of model J.\nprior probability of model J. And<00:54:45.320> so<00:54:45.440> this<00:54:45.720> odds<00:54:46.040> ratio,<00:54:46.720> it's<00:54:46.880> called,<00:54:47.800> um\nAnd so this odds ratio, it's called, um\nAnd so this odds ratio, it's called, um becomes<00:54:48.680> the<00:54:48.760> ratio<00:54:49.240> of<00:54:49.400> the<00:54:49.960> global\nbecomes the ratio of the global\nbecomes the ratio of the global likelihoods<00:54:50.880> of<00:54:50.960> the<00:54:51.040> model.\nlikelihoods of the model.\nlikelihoods of the model. Uh<00:54:53.240> like<00:54:53.480> so.<00:54:54.000> And<00:54:54.200> this<00:54:54.440> little<00:54:54.600> expression\nUh like so. And this little expression\nUh like so. And this little expression is<00:54:55.160> usually<00:54:55.400> called<00:54:55.600> the<00:54:55.680> Bayes<00:54:55.960> factor,\nis usually called the Bayes factor,\nis usually called the Bayes factor, which<00:54:56.520> is<00:54:56.680> why<00:54:56.880> I<00:54:56.960> denote<00:54:57.240> it<00:54:57.360> as<00:54:57.720> B<00:54:58.200> IJ.\nAnd<00:55:01.840> um\nAnd um\nAnd um these<00:55:02.680> likelihoods<00:55:03.040> are<00:55:03.120> calculated<00:55:03.640> from\nthese likelihoods are calculated from\nthese likelihoods are calculated from the<00:55:03.840> product<00:55:04.200> rules<00:55:04.560> as<00:55:04.680> seen<00:55:04.880> before.\nthe product rules as seen before.\nthe product rules as seen before. Um\nUm\nUm if<00:55:07.800> say<00:55:08.080> for<00:55:08.240> example<00:55:09.240> uh<00:55:09.680> model<00:55:10.280> I<00:55:10.680> has<00:55:10.920> only\nif say for example uh model I has only\nif say for example uh model I has only one<00:55:11.320> parameter,\none parameter,\none parameter, theta<00:55:13.080> one,<00:55:13.840> then<00:55:13.920> its<00:55:14.080> likelihood<00:55:14.640> will<00:55:15.280> be\ntheta one, then its likelihood will be\ntheta one, then its likelihood will be just<00:55:15.920> this,<00:55:16.280> right?<00:55:16.720> You<00:55:17.280> um\nyou<00:55:20.000> just\nyou just\nyou just then<00:55:22.200> aver-<00:55:22.520> average<00:55:22.920> out<00:55:23.640> the<00:55:24.320> the<00:55:25.000> the\nthen aver- average out the the the\nthen aver- average out the the the parameter,<00:55:25.800> and<00:55:25.920> you<00:55:25.960> just<00:55:26.200> obtain<00:55:26.600> the\nparameter, and you just obtain the\nparameter, and you just obtain the complete\ncomplete\ncomplete for<00:55:28.520> the<00:55:28.640> likelihood<00:55:29.120> of<00:55:29.200> the<00:55:29.320> model.<00:55:30.160> This<00:55:30.240> is\nfor the likelihood of the model. This is\nfor the likelihood of the model. This is essentially<00:55:30.760> the<00:55:30.840> observation<00:55:31.320> that<00:55:31.440> we<00:55:31.560> have\nessentially the observation that we have\nessentially the observation that we have seen<00:55:32.440> before.<00:55:33.200> However,<00:55:34.000> if<00:55:34.200> the<00:55:34.320> alternative\nseen before. However, if the alternative\nseen before. However, if the alternative model<00:55:35.400> J<00:55:35.800> has<00:55:36.040> three<00:55:36.240> parameters,<00:55:37.400> then<00:55:37.600> its\nmodel J has three parameters, then its\nmodel J has three parameters, then its likelihood<00:55:38.200> is<00:55:38.920> this<00:55:39.200> massive<00:55:39.520> expression.\nlikelihood is this massive expression.\nlikelihood is this massive expression. Um\nUm\nUm and<00:55:42.840> because<00:55:43.160> of<00:55:43.280> both<00:55:43.960> prior<00:55:44.200> probabilities,\nand because of both prior probabilities,\nand because of both prior probabilities, meaning<00:55:45.320> this<00:55:45.560> probability<00:55:46.480> here,<00:55:47.400> uh<00:55:47.680> of\nmeaning this probability here, uh of\nmeaning this probability here, uh of parameter<00:55:48.560> one<00:55:49.120> from<00:55:49.320> the<00:55:49.400> first<00:55:49.680> model<00:55:50.080> and\nparameter one from the first model and\nparameter one from the first model and of<00:55:50.560> the\nof the\nof the three<00:55:51.520> parameters<00:55:52.000> of<00:55:52.080> the<00:55:52.160> second<00:55:52.440> model,\nthree parameters of the second model,\nthree parameters of the second model, they<00:55:53.200> both<00:55:53.680> must<00:55:54.000> individually<00:55:54.720> add<00:55:54.960> up<00:55:55.120> to\nthey both must individually add up to\nthey both must individually add up to one<00:55:55.440> because<00:55:55.680> they're<00:55:55.760> probabilities.\none because they're probabilities.\none because they're probabilities. Uh<00:55:57.200> it<00:55:57.400> means<00:55:57.640> that<00:55:57.760> in<00:55:57.880> the<00:55:57.960> second<00:55:58.320> case,<00:55:58.720> the\nUh it means that in the second case, the\nUh it means that in the second case, the prior<00:55:59.160> probability,<00:56:00.280> this<00:56:00.440> one<00:56:00.560> right<00:56:00.720> here,\nprior probability, this one right here,\nprior probability, this one right here, is<00:56:02.120> much<00:56:02.360> more<00:56:02.560> {quote}<00:56:03.000> diluted,<00:56:03.720> meaning<00:56:04.160> it\nis much more {quote} diluted, meaning it\nis much more {quote} diluted, meaning it is<00:56:05.080> um\nis um\nis um distributed<00:56:07.240> among<00:56:08.280> more<00:56:08.520> parameters<00:56:09.080> and\ndistributed among more parameters and\ndistributed among more parameters and more<00:56:09.360> values,<00:56:10.360> meaning<00:56:10.920> each<00:56:11.200> value<00:56:11.800> for<00:56:12.000> each\nmore values, meaning each value for each\nmore values, meaning each value for each parameter<00:56:12.640> is<00:56:12.840> much<00:56:13.240> much<00:56:13.480> much<00:56:13.720> less<00:56:14.080> likely\nparameter is much much much less likely\nparameter is much much much less likely to<00:56:14.680> begin<00:56:15.000> with.\nUm<00:56:18.680> and<00:56:18.880> if<00:56:19.120> this<00:56:19.320> means<00:56:19.480> that<00:56:19.640> if<00:56:19.800> we<00:56:19.960> add<00:56:20.120> a\nUm and if this means that if we add a\nUm and if this means that if we add a parameter<00:56:20.680> that<00:56:20.840> doesn't<00:56:21.160> drastically\nparameter that doesn't drastically\nparameter that doesn't drastically improve<00:56:22.080> the<00:56:22.200> likelihood,<00:56:23.320> severe<00:56:23.680> drop<00:56:23.960> in\nimprove the likelihood, severe drop in\nimprove the likelihood, severe drop in prior<00:56:24.320> density<00:56:25.560> caused<00:56:25.920> by<00:56:26.000> that<00:56:26.160> new\nprior density caused by that new\nprior density caused by that new dimension<00:56:26.760> will<00:56:26.920> drag<00:56:27.320> the<00:56:27.440> entire<00:56:27.840> integral\ndimension will drag the entire integral\ndimension will drag the entire integral down.\ndown.\ndown. Basically,<00:56:29.480> the<00:56:29.600> model<00:56:30.040> is\nBasically, the model is\nBasically, the model is the\nthe\nthe the<00:56:32.160> probability<00:56:32.640> calculus<00:56:33.160> is\nthe probability calculus is\nthe probability calculus is mathematically<00:56:34.080> penalizing<00:56:35.440> uh<00:56:35.680> models<00:56:36.360> for\nmathematically penalizing uh models for\nmathematically penalizing uh models for exploring<00:56:37.280> useless<00:56:37.640> dimensions.\nexploring useless dimensions.\nexploring useless dimensions. Uh<00:56:39.080> and<00:56:39.240> once<00:56:39.440> again,<00:56:39.720> the<00:56:39.840> frequentist\nUh and once again, the frequentist\nUh and once again, the frequentist methodology<00:56:40.880> cannot<00:56:41.480> do<00:56:41.720> any<00:56:41.960> of<00:56:42.040> this\nmethodology cannot do any of this\nmethodology cannot do any of this because<00:56:42.680> they<00:56:42.760> don't<00:56:42.960> assign<00:56:43.200> probabilities\nbecause they don't assign probabilities\nbecause they don't assign probabilities to<00:56:43.800> parameters\nto parameters\nto parameters and<00:56:45.680> uh<00:56:45.760> also<00:56:46.480> therefore<00:56:46.720> can't<00:56:46.920> use<00:56:47.080> this.<00:56:47.880> Uh\nand uh also therefore can't use this. Uh\nand uh also therefore can't use this. Uh and<00:56:48.040> they<00:56:48.120> also<00:56:48.320> don't<00:56:48.520> use<00:56:48.680> priors,<00:56:49.080> of\nand they also don't use priors, of\nand they also don't use priors, of course.<00:56:50.080> Um<00:56:50.640> instead,<00:56:51.320> in<00:56:51.480> order<00:56:51.680> to<00:56:52.520> um\ncourse. Um instead, in order to um\ncourse. Um instead, in order to um somehow\nsomehow\nsomehow avoid<00:56:54.320> the<00:56:54.440> overfitting<00:56:55.000> of<00:56:55.120> their<00:56:55.240> models\navoid the overfitting of their models\navoid the overfitting of their models when<00:56:55.720> they<00:56:55.800> do<00:56:55.960> maximum<00:56:56.400> likelihood\nwhen they do maximum likelihood\nwhen they do maximum likelihood estimation,<00:56:58.000> uh<00:56:58.320> they<00:56:58.480> have<00:56:58.640> to<00:56:58.720> invent<00:56:59.080> all\nestimation, uh they have to invent all\nestimation, uh they have to invent all sorts<00:56:59.440> of<00:56:59.680> ad<00:56:59.840> hoc<00:57:00.360> rules<00:57:00.840> that<00:57:01.200> don't<00:57:01.560> follow\nsorts of ad hoc rules that don't follow\nsorts of ad hoc rules that don't follow from<00:57:01.960> the<00:57:02.040> rules<00:57:02.200> of<00:57:02.320> probability<00:57:02.920> and\nfrom the rules of probability and\nfrom the rules of probability and therefore<00:57:03.440> are<00:57:03.560> suboptimal<00:57:04.160> or<00:57:04.240> even<00:57:04.440> wrong.\ntherefore are suboptimal or even wrong.\ntherefore are suboptimal or even wrong. And<00:57:05.600> some<00:57:05.760> of<00:57:05.880> these<00:57:06.120> ad<00:57:06.280> hoc<00:57:06.400> fixes<00:57:06.760> are<00:57:06.880> very\nAnd some of these ad hoc fixes are very\nAnd some of these ad hoc fixes are very famous.<00:57:07.560> You're<00:57:07.840> all<00:57:07.960> familiar<00:57:08.280> with<00:57:08.400> them.\nfamous. You're all familiar with them.\nfamous. You're all familiar with them. We<00:57:08.640> have<00:57:08.880> validation<00:57:09.520> data<00:57:09.800> sets,\nWe have validation data sets,\nWe have validation data sets, cross-validation,\ncross-validation,\ncross-validation, regularization,<00:57:13.320> uh<00:57:13.440> some<00:57:13.560> kind<00:57:13.680> of\nregularization, uh some kind of\nregularization, uh some kind of information<00:57:14.200> criterion.\ninformation criterion.\ninformation criterion. And<00:57:16.680> especially<00:57:17.080> in<00:57:17.240> the<00:57:17.440> neural<00:57:17.680> network<00:57:18.480> uh\nAnd especially in the neural network uh\nAnd especially in the neural network uh side,<00:57:18.840> we<00:57:18.920> have<00:57:19.120> early<00:57:19.320> stopping<00:57:19.800> and<00:57:19.920> dropout\nside, we have early stopping and dropout\nside, we have early stopping and dropout and<00:57:20.520> all<00:57:20.640> of<00:57:20.720> these<00:57:20.920> other<00:57:21.280> strange<00:57:21.680> and\nand all of these other strange and\nand all of these other strange and arbitrary<00:57:22.200> rules<00:57:22.480> that<00:57:22.760> somehow<00:57:23.280> try<00:57:23.560> to\narbitrary rules that somehow try to\narbitrary rules that somehow try to avoid\navoid\navoid uh<00:57:25.240> you<00:57:25.360> know,<00:57:25.560> just<00:57:25.840> the<00:57:25.920> model<00:57:26.200> finding\nuh you know, just the model finding\nuh you know, just the model finding the<00:57:27.360> greediest<00:57:27.960> way<00:57:28.120> to<00:57:28.240> fit<00:57:28.520> all<00:57:28.680> the<00:57:28.760> points\nthe greediest way to fit all the points\nthe greediest way to fit all the points in<00:57:29.120> the<00:57:29.160> data<00:57:29.400> perfectly.\nin the data perfectly.\nin the data perfectly. And<00:57:32.560> uh<00:57:32.640> so<00:57:32.960> we<00:57:33.160> come<00:57:33.600> to<00:57:33.920> basically<00:57:34.280> the\nAnd uh so we come to basically the\nAnd uh so we come to basically the conclusion<00:57:34.800> of<00:57:34.880> the<00:57:34.920> presentation,<00:57:35.520> which<00:57:35.720> is\nconclusion of the presentation, which is\nconclusion of the presentation, which is the<00:57:37.160> Bayesian<00:57:37.760> probability<00:57:38.240> calculus\nthe Bayesian probability calculus\nthe Bayesian probability calculus teaches<00:57:39.000> us<00:57:39.160> that<00:57:39.280> the<00:57:39.400> best<00:57:39.680> practice<00:57:40.200> or\nteaches us that the best practice or\nteaches us that the best practice or good<00:57:41.120> practice<00:57:41.560> is<00:57:41.760> simply<00:57:42.680> that\ngood practice is simply that\ngood practice is simply that since<00:57:43.880> the<00:57:44.240> rules<00:57:44.560> of<00:57:45.080> manipulating\nsince the rules of manipulating\nsince the rules of manipulating probability<00:57:46.560> as<00:57:46.720> a<00:57:46.760> degree<00:57:47.080> of<00:57:47.200> implication\nprobability as a degree of implication\nprobability as a degree of implication are<00:57:49.000> unique,<00:57:49.960> it<00:57:50.120> means<00:57:50.400> that<00:57:50.600> any<00:57:50.800> deviation\nare unique, it means that any deviation\nare unique, it means that any deviation from<00:57:51.520> them<00:57:51.800> is<00:57:52.040> bound<00:57:52.320> to<00:57:52.440> lead<00:57:52.640> to<00:57:52.760> some\nfrom them is bound to lead to some\nfrom them is bound to lead to some suboptimal<00:57:54.400> paradoxical<00:57:55.120> or<00:57:55.240> even<00:57:55.840> really\nsuboptimal paradoxical or even really\nsuboptimal paradoxical or even really really<00:57:56.400> incorrect<00:57:56.840> results.\nreally incorrect results.\nreally incorrect results. And<00:57:58.200> that<00:57:58.320> is<00:57:58.520> why<00:57:59.320> it's<00:57:59.560> very<00:57:59.800> important<00:58:00.200> in\nAnd that is why it's very important in\nAnd that is why it's very important in the<00:58:00.360> literature<00:58:00.600> there's<00:58:00.720> great<00:58:01.000> focus<00:58:01.360> on\nthe literature there's great focus on\nthe literature there's great focus on somehow<00:58:01.920> not<00:58:02.240> deviating<00:58:02.640> from<00:58:02.760> these<00:58:02.960> rules.\nsomehow not deviating from these rules.\nsomehow not deviating from these rules. And<00:58:03.800> in<00:58:03.920> fact,<00:58:04.240> there's<00:58:04.440> so<00:58:04.680> often<00:58:04.960> when\nAnd in fact, there's so often when\nAnd in fact, there's so often when someone<00:58:05.840> tries<00:58:06.280> to<00:58:06.440> invent<00:58:06.800> a<00:58:06.840> new<00:58:07.080> trick<00:58:08.080> that\nsomeone tries to invent a new trick that\nsomeone tries to invent a new trick that somehow<00:58:08.640> doesn't<00:58:08.960> follow<00:58:09.160> from<00:58:09.280> the<00:58:09.360> rules,\nsomehow doesn't follow from the rules,\nsomehow doesn't follow from the rules, and<00:58:09.680> it<00:58:09.760> doesn't<00:58:10.040> take<00:58:10.240> long<00:58:10.480> for<00:58:10.600> someone<00:58:10.840> to\nand it doesn't take long for someone to\nand it doesn't take long for someone to find,<00:58:11.240> \"Okay,<00:58:11.480> this<00:58:11.640> actually<00:58:11.880> leads<00:58:12.000> to<00:58:12.120> some\nfind, \"Okay, this actually leads to some\nfind, \"Okay, this actually leads to some paradoxes<00:58:12.760> down<00:58:12.920> the<00:58:13.000> road.\"\nparadoxes down the road.\"\nparadoxes down the road.\" Um\nUm\nUm And<00:58:15.680> additionally,<00:58:16.480> one<00:58:16.720> needs<00:58:16.880> to<00:58:16.960> be<00:58:17.120> very\nAnd additionally, one needs to be very\nAnd additionally, one needs to be very very<00:58:17.640> careful<00:58:17.920> with<00:58:18.080> handling<00:58:18.400> infinite\nvery careful with handling infinite\nvery careful with handling infinite sets.<00:58:19.200> This<00:58:19.400> is<00:58:19.560> what<00:58:19.720> I<00:58:19.760> mentioned<00:58:20.120> at<00:58:20.200> the\nsets. This is what I mentioned at the\nsets. This is what I mentioned at the beginning.\nbeginning.\nbeginning. And<00:58:21.640> carelessness<00:58:22.120> in<00:58:22.200> this<00:58:22.360> context<00:58:22.840> can\nAnd carelessness in this context can\nAnd carelessness in this context can lead<00:58:23.080> to<00:58:23.200> paradoxes<00:58:23.680> that<00:58:23.800> has<00:58:24.200> that<00:58:24.320> have\nlead to paradoxes that has that have\nlead to paradoxes that has that have nothing<00:58:25.080> to<00:58:25.240> do\nnothing to do\nnothing to do with<00:58:26.200> the<00:58:26.280> theory<00:58:26.600> of<00:58:26.720> probability,<00:58:27.280> but\nwith the theory of probability, but\nwith the theory of probability, but instead<00:58:27.800> follow<00:58:28.120> from<00:58:28.360> the<00:58:28.680> unsound\ninstead follow from the unsound\ninstead follow from the unsound mathematical<00:58:29.520> practices.<00:58:30.600> Some<00:58:30.760> of<00:58:30.800> the\nmathematical practices. Some of the\nmathematical practices. Some of the famous<00:58:31.200> paradoxes<00:58:31.680> are<00:58:31.840> the\nfamous paradoxes are the\nfamous paradoxes are the Bertrand's<00:58:33.040> paradox\nBertrand's paradox\nBertrand's paradox and<00:58:34.560> the<00:58:34.680> Borel-Cantelli<00:58:35.760> Borel-Cantelli\nand the Borel-Cantelli Borel-Cantelli\nand the Borel-Cantelli Borel-Cantelli paradox.<00:58:37.560> And<00:58:37.840> you<00:58:37.920> can<00:58:38.080> avoid<00:58:38.520> them<00:58:38.760> by\nparadox. And you can avoid them by\nparadox. And you can avoid them by simply<00:58:39.320> using<00:58:39.640> finite<00:58:40.120> sets<00:58:40.880> or<00:58:41.080> infinite\nsimply using finite sets or infinite\nsimply using finite sets or infinite sets<00:58:41.640> that<00:58:41.760> are<00:58:41.840> produced<00:58:42.360> as<00:58:42.640> the\nsets that are produced as the\nsets that are produced as the well-behaved<00:58:43.240> limit<00:58:43.600> of<00:58:43.800> a<00:58:43.840> sequence<00:58:44.440> of\nwell-behaved limit of a sequence of\nwell-behaved limit of a sequence of finite<00:58:44.880> sets.<00:58:45.680> So,\nfinite sets. So,\nfinite sets. So, if<00:58:46.960> you<00:58:47.120> have<00:58:47.320> once<00:58:47.480> again<00:58:47.640> this<00:58:47.800> question<00:58:48.200> of\nif you have once again this question of\nif you have once again this question of like,<00:58:48.600> \"Okay,<00:58:48.840> you<00:58:48.960> have<00:58:49.160> somehow<00:58:49.800> the<00:58:49.920> real\nlike, \"Okay, you have somehow the real\nlike, \"Okay, you have somehow the real numbers<00:58:51.000> in<00:58:51.120> some<00:58:51.280> kind<00:58:51.480> of<00:58:51.560> range,<00:58:51.840> what<00:58:52.000> is\nnumbers in some kind of range, what is\nnumbers in some kind of range, what is the<00:58:52.200> probability<00:58:52.720> that<00:58:52.920> between<00:58:53.760> that<00:58:54.000> you\nthe probability that between that you\nthe probability that between that you cannot<00:58:54.560> pick<00:58:54.760> out<00:58:54.960> a<00:58:55.000> rational<00:58:55.400> number<00:58:55.680> from\ncannot pick out a rational number from\ncannot pick out a rational number from this<00:58:56.080> from<00:58:56.280> the<00:58:56.360> space?\"\nthis from the space?\"\nthis from the space?\" The<00:58:57.560> the<00:58:57.680> frequentists<00:58:58.200> or<00:58:58.280> measure\nThe the frequentists or measure\nThe the frequentists or measure theoretic<00:58:59.400> probability<00:58:59.880> calculus<00:59:00.200> will<00:59:00.320> tell\ntheoretic probability calculus will tell\ntheoretic probability calculus will tell you<00:59:00.560> the<00:59:00.640> probability<00:59:01.120> is<00:59:01.240> zero.\nyou the probability is zero.\nyou the probability is zero. But<00:59:02.240> that<00:59:02.440> is<00:59:02.640> of<00:59:02.760> course<00:59:02.960> absurd<00:59:03.360> because<00:59:03.680> it\nBut that is of course absurd because it\nBut that is of course absurd because it is<00:59:03.960> possible<00:59:04.440> to<00:59:04.560> draw<00:59:05.280> a<00:59:05.360> rational<00:59:05.720> number.\nis possible to draw a rational number.\nis possible to draw a rational number. And<00:59:06.240> so,<00:59:06.920> this<00:59:07.080> is<00:59:07.680> an<00:59:07.800> example<00:59:08.400> of<00:59:08.680> a<00:59:09.200> of<00:59:09.360> a\nAnd so, this is an example of a of a\nAnd so, this is an example of a of a result<00:59:09.840> that\nresult that\nresult that leads<00:59:10.880> to<00:59:10.960> paradoxes,<00:59:12.000> and<00:59:12.120> it<00:59:12.200> is<00:59:12.320> actually\nleads to paradoxes, and it is actually\nleads to paradoxes, and it is actually just<00:59:12.840> a<00:59:12.880> consequence<00:59:13.480> of<00:59:13.600> an<00:59:13.760> ill-posed\njust a consequence of an ill-posed\njust a consequence of an ill-posed question<00:59:14.680> and<00:59:14.840> has<00:59:15.000> nothing<00:59:15.320> to<00:59:15.440> do\nquestion and has nothing to do\nquestion and has nothing to do with<00:59:17.120> the<00:59:17.200> probability<00:59:17.640> calculus<00:59:18.080> per<00:59:18.280> se.\nwith the probability calculus per se.\nwith the probability calculus per se. Um<00:59:19.960> yes.\nUm yes.\nUm yes. And<00:59:21.520> there<00:59:21.640> we<00:59:21.720> go.<00:59:22.120> That's<00:59:22.320> it.\nAnd there we go. That's it.\nAnd there we go. That's it. If<00:59:23.840> you<00:59:23.920> have<00:59:24.000> any<00:59:24.120> questions,<00:59:24.600> feel<00:59:24.760> free<00:59:24.880> to\nIf you have any questions, feel free to\nIf you have any questions, feel free to to\nI<00:59:27.240> hope<00:59:27.360> it<00:59:27.480> was<00:59:27.600> interesting.\nOh,<00:59:31.440> you<00:59:31.520> couldn't<00:59:31.760> see<00:59:31.840> the<00:59:31.960> slides?", + "fetched_at": "2026-06-21T20:32:04Z", + "source": "yt-dlp-vtt" +} \ No newline at end of file diff --git a/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/video.log b/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/video.log new file mode 100644 index 00000000..8f2692f2 --- /dev/null +++ b/conductor/tracks/video_analysis_probability_logic_20260621/artifacts/video.log @@ -0,0 +1,10 @@ +# yt-dlp log +# url: https://youtu.be/0yF9TvMeAzM +# output: C:\projects\manual_slop\conductor\tracks\video_analysis_probability_logic_20260621\artifacts\video.mp4 +# returncode: 0 + +stdout: + + +stderr: + diff --git a/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase1_acquire.py b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase1_acquire.py new file mode 100644 index 00000000..4d3d82f8 --- /dev/null +++ b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase1_acquire.py @@ -0,0 +1,132 @@ +"""Generic Phase 1 Acquire driver for video_analysis_campaign children. + +Reads the child spec from CLI args: slug + URL + needs_yt_dlp_verify. +Calls extract_transcript (with yt-dlp VTT fallback) + download_video. +""" +from __future__ import annotations + +import argparse +import json +import re +import subprocess +import sys +import time +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[4] +sys.path.insert(0, str(ROOT)) + +from scripts.video_analysis.download_video import download_video +from scripts.video_analysis.extract_transcript import _fetch_raw_transcript + + +def _parse_vtt_segments(vtt_path: Path) -> list[dict]: + text = vtt_path.read_text(encoding="utf-8") + segments: list[dict] = [] + pattern = re.compile(r"(\d{2}):(\d{2}):(\d{2})\.(\d{3})\s+-->", re.MULTILINE) + blocks = re.split(r"\n\n+", text) + for block in blocks: + match = pattern.search(block) + if not match: + continue + h, m, s, ms = match.groups() + start = int(h) * 3600 + int(m) * 60 + int(s) + int(ms) / 1000.0 + lines = [ln.strip() for ln in block.split("\n") if ln.strip() and not pattern.match(ln) and "-->" not in ln] + text_content = " ".join(lines) + if text_content: + segments.append({"start": start, "duration": 0.0, "text": text_content}) + return segments + + +def phase1_acquire(slug: str, url: str, artifacts_dir: Path) -> dict: + print(f"Phase 1 Acquire for {slug}: {url}") + print(f"Artifacts: {artifacts_dir}") + artifacts_dir.mkdir(parents=True, exist_ok=True) + + m = re.search(r"(?:youtu\.be/|v=)([A-Za-z0-9_-]{11})", url) + if not m: + return {"status": "error", "error": f"Could not parse video_id from {url}"} + video_id = m.group(1) + + print("Step 1: extract_transcript (try youtube-transcript-api)") + transcript_path = artifacts_dir / "transcript.json" + last_exc = None + for attempt in range(3): + try: + segments = _fetch_raw_transcript(video_id) + data = { + "video_id": video_id, + "segments": segments, + "plain": "\n".join(s["text"] for s in segments), + "fetched_at": time.strftime("%Y-%m-%dT%H:%M:%SZ", time.gmtime()), + "source": "youtube-transcript-api", + } + transcript_path.write_text(json.dumps(data, indent=2, ensure_ascii=False), encoding="utf-8") + print(f" OK: wrote {transcript_path} ({len(segments)} segments)") + last_exc = None + break + except Exception as e: + last_exc = e + print(f" attempt {attempt+1} failed: {type(e).__name__}: {str(e)[:200]}") + if attempt < 2: + time.sleep(2 ** attempt) + + if last_exc is not None: + print("Step 1b: yt-dlp subtitle fallback") + completed = subprocess.run( + ["yt-dlp", "--write-auto-subs", "--sub-langs", "en", "--sub-format", "vtt", + "--skip-download", "--output", str(artifacts_dir / video_id), url], + capture_output=True, text=True, + ) + candidates = list(artifacts_dir.glob(f"{video_id}*.vtt")) + if not candidates: + print(f" yt-dlp subtitle fetch failed: {completed.stderr[:300]}") + print(f" No transcript available. Continuing with download only.") + transcript_path.write_text(json.dumps({ + "video_id": video_id, + "segments": [], + "plain": "", + "fetched_at": time.strftime("%Y-%m-%dT%H:%M:%SZ", time.gmtime()), + "source": "unavailable", + "error": str(last_exc)[:500], + "note": "youtube-transcript-api and yt-dlp VTT both failed. Frame OCR will be the primary signal.", + }, indent=2, ensure_ascii=False), encoding="utf-8") + else: + vtt_path = candidates[0] + segments = _parse_vtt_segments(vtt_path) + data = { + "video_id": video_id, + "segments": segments, + "plain": "\n".join(s["text"] for s in segments), + "fetched_at": time.strftime("%Y-%m-%dT%H:%M:%SZ", time.gmtime()), + "source": "yt-dlp-vtt", + } + transcript_path.write_text(json.dumps(data, indent=2, ensure_ascii=False), encoding="utf-8") + print(f" OK: parsed {len(segments)} segments from {vtt_path.name}") + + print("Step 2: download_video") + video_path = artifacts_dir / "video.mp4" + result = download_video(url, video_path) + if result.is_err(): + return {"status": "error", "error": f"download_video: {result.err.class_name}: {result.err.detail[:200]}"} + print(f" OK: wrote {video_path} ({video_path.stat().st_size} bytes)") + return {"status": "ok", "video_path": str(video_path), "transcript_path": str(transcript_path)} + + +def main() -> int: + parser = argparse.ArgumentParser() + parser.add_argument("slug") + parser.add_argument("url") + parser.add_argument("--artifacts-dir", required=False) + args = parser.parse_args() + if args.artifacts_dir: + artifacts_dir = Path(args.artifacts_dir) + else: + artifacts_dir = ROOT / "conductor" / "tracks" / f"video_analysis_{args.slug}_20260621" / "artifacts" + result = phase1_acquire(args.slug, args.url, artifacts_dir) + print(json.dumps(result, indent=2)) + return 0 if result["status"] == "ok" else 1 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase2_keyframes.py b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase2_keyframes.py new file mode 100644 index 00000000..a26ec471 --- /dev/null +++ b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase2_keyframes.py @@ -0,0 +1,37 @@ +"""Generic Phase 2 Keyframes driver.""" +from __future__ import annotations + +import argparse +import sys +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[4] +sys.path.insert(0, str(ROOT)) + +from scripts.video_analysis.extract_keyframes import extract_keyframes + + +def main() -> int: + parser = argparse.ArgumentParser() + parser.add_argument("slug") + parser.add_argument("--video", required=False) + parser.add_argument("--output-dir", required=False) + parser.add_argument("--threshold", type=float, default=0.4) + args = parser.parse_args() + + track_dir = ROOT / "conductor" / "tracks" / f"video_analysis_{args.slug}_20260621" / "artifacts" + video = Path(args.video) if args.video else track_dir / "video.mp4" + output = Path(args.output_dir) if args.output_dir else track_dir / "frames" + + print(f"Phase 2 Keyframes for {video}") + output.mkdir(parents=True, exist_ok=True) + result = extract_keyframes(video, output, threshold=args.threshold) + if result.is_err(): + print(f" ERR: {result.err.class_name}: {result.err.detail[:300]}") + return 1 + print(f" OK: kept {result.value['kept']} frames") + return 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase3_ocr.py b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase3_ocr.py new file mode 100644 index 00000000..724733a8 --- /dev/null +++ b/scripts/tier2/artifacts/video_analysis_campaign_20260621/phase3_ocr.py @@ -0,0 +1,39 @@ +"""Generic Phase 3 OCR driver.""" +from __future__ import annotations + +import argparse +import sys +import time +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[4] +sys.path.insert(0, str(ROOT)) + +from scripts.video_analysis.ocr_frames import ocr_frames + + +def main() -> int: + parser = argparse.ArgumentParser() + parser.add_argument("slug") + parser.add_argument("--frames-dir", required=False) + parser.add_argument("--output", required=False) + parser.add_argument("--backend", default="winsdk") + args = parser.parse_args() + + track_dir = ROOT / "conductor" / "tracks" / f"video_analysis_{args.slug}_20260621" / "artifacts" + frames = Path(args.frames_dir) if args.frames_dir else track_dir / "frames" + output = Path(args.output) if args.output else track_dir / "ocr.md" + + print(f"Phase 3 OCR for {frames} ({args.backend})") + t0 = time.time() + result = ocr_frames(frames, output, backend=args.backend) + elapsed = time.time() - t0 + if result.is_err(): + print(f" ERR: {result.err.class_name}: {result.err.detail[:300]}") + return 1 + print(f" OK: OCR'd {result.value['frames_ocrd']} frames in {elapsed:.1f}s") + return 0 + + +if __name__ == "__main__": + sys.exit(main())