WEBVTT 00:00:00.000 --> 00:00:02.859 You know, usually when we think about changing 00:00:02.859 --> 00:00:05.919 our minds, we view it as this kind of moment 00:00:05.919 --> 00:00:09.060 of defeat. Right, like a structural failure or 00:00:09.060 --> 00:00:11.019 something. Yeah, exactly. You build a belief 00:00:11.019 --> 00:00:13.919 system, a crack appears in the foundation, and 00:00:13.919 --> 00:00:15.960 you just assume the whole house is going to come 00:00:15.960 --> 00:00:18.399 down. And it feels incredibly destabilizing. 00:00:18.579 --> 00:00:20.739 I mean, human beings are fundamentally wired 00:00:20.739 --> 00:00:25.420 to treat our convictions as absolute destinations 00:00:25.420 --> 00:00:28.359 rather than provisional rest stops. But what 00:00:28.359 --> 00:00:30.859 if changing your mind wasn't a collapse at all? 00:00:31.199 --> 00:00:33.899 What if it was actually like a mathematical upgrade? 00:00:34.090 --> 00:00:36.570 Oh, I like that framing. Think of a really good 00:00:36.570 --> 00:00:39.609 detective working a complex case. A savvy detective 00:00:39.609 --> 00:00:42.229 doesn't just look at a new clue and a total vacuum, 00:00:42.390 --> 00:00:45.189 right? No, of course not. They take that brand 00:00:45.189 --> 00:00:47.310 new fingerprint or that fresh alibi and they 00:00:47.310 --> 00:00:48.649 weigh it against everything they already knew 00:00:48.649 --> 00:00:51.130 about the case. Exactly. They update their work 00:00:51.130 --> 00:00:53.850 in Peary. And today we're taking a deep dive 00:00:53.850 --> 00:00:56.149 into the very math of doing exactly that. Yes, 00:00:56.149 --> 00:00:58.850 we are. We've got this massive comprehensive 00:00:58.850 --> 00:01:01.409 Wikipedia article sitting in front of us. It 00:01:01.409 --> 00:01:04.469 covers the history. the dense mathematics, and 00:01:04.469 --> 00:01:07.069 the real -world applications of a concept called 00:01:07.069 --> 00:01:12.040 Bayesian inference. one of the most powerful 00:01:12.040 --> 00:01:14.640 mental models you can acquire. And our mission 00:01:14.640 --> 00:01:17.739 for you today is to basically demystify it. We 00:01:17.739 --> 00:01:20.379 want to translate this heavy statistical framework 00:01:20.379 --> 00:01:22.659 into a mental shortcut that you can actually 00:01:22.659 --> 00:01:25.000 use. Right, because the goal is to walk away 00:01:25.000 --> 00:01:27.579 from this deep dive as a sharper, more adaptable 00:01:27.579 --> 00:01:30.680 thinker, especially in a world where we are constantly 00:01:30.680 --> 00:01:33.680 bombarded with, you know, conflicting information. 00:01:33.780 --> 00:01:35.900 So to set the stage here, Bayesian inference 00:01:35.900 --> 00:01:39.099 isn't just some dry statistical formula to be 00:01:39.099 --> 00:01:41.340 memorized in a universal lecture and then forgotten. 00:01:41.739 --> 00:01:44.319 No, not at all. It is a foundational framework 00:01:44.319 --> 00:01:46.799 for human reasoning. Its roots actually trace 00:01:46.799 --> 00:01:49.420 all the way back to a mathematician and minister 00:01:49.420 --> 00:01:52.739 named Thomas Bayes in the 1700s. The 1700s, yeah. 00:01:52.900 --> 00:01:55.140 Yeah, and it was later rigorously developed by 00:01:55.140 --> 00:01:57.599 Pierre -Simon Laplace, and what these two managed 00:01:57.599 --> 00:02:00.239 to do was build this unprecedented bridge. The 00:02:00.239 --> 00:02:02.299 bridge between what and what? Well, they connected 00:02:02.299 --> 00:02:05.420 pure objective mathematics with human subjective 00:02:05.420 --> 00:02:08.460 belief. They essentially wrote the formula for 00:02:08.460 --> 00:02:11.039 how to rationally update what you believe when 00:02:11.039 --> 00:02:13.759 new information finally arrives. To understand 00:02:13.759 --> 00:02:16.400 how that mathematical bridge is actually built 00:02:16.400 --> 00:02:18.560 and, you know, how it's currently being used 00:02:18.560 --> 00:02:21.780 to do wild things like map the cosmos or train 00:02:21.780 --> 00:02:24.740 AI, we first need to strip the complexity all 00:02:24.740 --> 00:02:27.199 the way down. We definitely do. So we need to 00:02:27.199 --> 00:02:29.280 look at a highly relatable thought experiment 00:02:29.280 --> 00:02:31.719 from our source material. It involves cookies. 00:02:32.340 --> 00:02:34.599 It is remarkable how often high -level mathematics 00:02:34.599 --> 00:02:37.580 relies on baked goods to make a point. It really 00:02:37.580 --> 00:02:39.719 grounds the abstract ideas. So I want you to 00:02:39.719 --> 00:02:41.840 picture two identical bowls of cookies sitting 00:02:41.840 --> 00:02:44.340 on a table in front of you. Let's call it Fred's 00:02:44.340 --> 00:02:46.240 Cookie Bowls. OK, I'm picturing it. Two bowls. 00:02:46.719 --> 00:02:48.780 Right. Bowl number one has 10 chocolate chip 00:02:48.780 --> 00:02:52.560 cookies and 30 plain cookies. So 40 total, but 00:02:52.560 --> 00:02:55.319 mostly plain. Got it. Bowl number two has 20 00:02:55.319 --> 00:02:58.360 chocolate chip and 20 plain. A perfect 50 -50 00:02:58.360 --> 00:03:01.280 split. The setup is clear. Two bowls, different 00:03:01.280 --> 00:03:04.919 ratios. Now, our friend Fred blindfolds you, 00:03:05.039 --> 00:03:06.939 shuffles the two bowls around on the table so 00:03:06.939 --> 00:03:09.180 you totally lose track of them, and asks you 00:03:09.180 --> 00:03:12.439 to pick a bowl at random. Okay. Then, from that 00:03:12.439 --> 00:03:15.800 random bowl, you reach in and pick a cookie at 00:03:15.800 --> 00:03:18.120 random. You take off the blindfold and look at 00:03:18.120 --> 00:03:21.379 your hand. It is a plain cookie. Right, a plain 00:03:21.379 --> 00:03:24.319 cookie. Question BayesianInference asks is this, 00:03:25.080 --> 00:03:27.840 what is the probability that you just pulled 00:03:27.840 --> 00:03:30.840 that plain cookie from bowl number one. And this 00:03:30.840 --> 00:03:33.439 is where human intuition usually jumps in, far 00:03:33.439 --> 00:03:36.280 ahead of the actual math. Oh, totally. Our intuition 00:03:36.280 --> 00:03:38.360 immediately screams that it's definitely more 00:03:38.360 --> 00:03:40.979 than a 50 % chance, because bowl number one has 00:03:40.979 --> 00:03:42.860 way more plain cookies in it. Right. You're holding 00:03:42.860 --> 00:03:45.460 a plain cookie so it feels obvious. But Bayesian 00:03:45.460 --> 00:03:48.520 inference gives us the precise language and the 00:03:48.520 --> 00:03:50.659 precise mathematical engine to actually prove 00:03:50.659 --> 00:03:53.819 it. So let's lay out the formal vocabulary. First, 00:03:53.919 --> 00:03:55.879 we have what is called the prior. The prior, 00:03:56.020 --> 00:03:57.719 yes. Before you even looked at the cookie in 00:03:57.719 --> 00:04:00.240 your hand, what was your chance of picking bowl 00:04:00.240 --> 00:04:03.300 number one? It was 50 -50. You picked a bowl 00:04:03.300 --> 00:04:06.400 entirely at random while blindfolded. Exactly. 00:04:06.740 --> 00:04:09.319 The prior probability is your baseline estimate 00:04:09.319 --> 00:04:12.280 of a hypothesis before any new evidence is observed. 00:04:12.580 --> 00:04:14.699 It's just your starting point. Then we introduce 00:04:14.699 --> 00:04:17.360 the likelihood. This measures the compatibility 00:04:17.360 --> 00:04:20.019 of the new evidence with the hypothesis. Right, 00:04:20.139 --> 00:04:21.879 so the likelihood of pulling a plain cookie from 00:04:21.879 --> 00:04:25.560 bowl 1 is 30 out of 40, which is 75%. And the 00:04:25.560 --> 00:04:27.040 likelihood of pulling a plain cookie from bowl 00:04:27.040 --> 00:04:31.300 2 is 20 out of 40, or 50%. Finally, we take those 00:04:31.300 --> 00:04:34.180 numbers and calculate the posterior. The posterior. 00:04:34.439 --> 00:04:37.360 That is your updated mathematically sound belief 00:04:37.360 --> 00:04:39.759 after incorporating the new evidence. Yeah. When 00:04:39.759 --> 00:04:42.759 you plug your 50 % prior and those likelihoods 00:04:42.759 --> 00:04:45.399 into Bayes' theorem, the posterior calculates 00:04:45.399 --> 00:04:48.759 out to exactly 60%. There's a 60 % chance you 00:04:48.759 --> 00:04:50.930 pick from bowl number one. Synthesizing that 00:04:50.930 --> 00:04:53.610 into the core formula basically looks like this. 00:04:54.050 --> 00:04:56.250 The posterior is proportional to the likelihood 00:04:56.250 --> 00:04:59.089 multiplied by the prior. Or, formulated another 00:04:59.089 --> 00:05:01.990 way, posterior equals likelihood times prior 00:05:01.990 --> 00:05:04.970 divided by the total evidence. Exactly. And the 00:05:04.970 --> 00:05:06.829 reason this matters for you listening to this 00:05:06.829 --> 00:05:09.930 right now is because it mathematically prevents 00:05:09.930 --> 00:05:12.970 the human tendency to overreact to new data. 00:05:13.100 --> 00:05:15.660 Oh, that makes sense. It forces us to anchor 00:05:15.660 --> 00:05:18.240 our new observations to our existing knowledge 00:05:18.240 --> 00:05:21.120 base. So because you saw a plain cookie, your 00:05:21.120 --> 00:05:24.079 brain might want to jump to being like 80 or 00:05:24.079 --> 00:05:26.819 90 percent sure it was bowl one. Right. But the 00:05:26.819 --> 00:05:29.660 math restrains you to 60 percent because it forces 00:05:29.660 --> 00:05:32.000 you to remember that your initial chance of picking 00:05:32.000 --> 00:05:34.980 bowl one was only a coin toss. It's like a built 00:05:34.980 --> 00:05:37.279 -in shock absorber for your opinions. It's a 00:05:37.279 --> 00:05:39.350 great way to put it. But that brings up a really 00:05:39.350 --> 00:05:41.990 fascinating friction point. Because the poster 00:05:41.990 --> 00:05:44.670 your belief, your final conclusion, relies so 00:05:44.670 --> 00:05:47.850 heavily on your initial prior belief. What happens 00:05:47.850 --> 00:05:50.730 if your starting belief is just incredibly stubborn? 00:05:51.230 --> 00:05:53.810 Ah, yeah. That leads us straight into a vital 00:05:53.810 --> 00:05:56.269 mathematical principle known as Cromwell's Rule. 00:05:56.490 --> 00:05:58.709 Right, Cromwell's Rule. It states that if your 00:05:58.709 --> 00:06:01.610 prior probability for a model, your initial foundational 00:06:01.610 --> 00:06:05.910 belief, is exactly zero or exactly one, then 00:06:05.910 --> 00:06:07.870 absolutely no amount of new evidence can ever 00:06:07.870 --> 00:06:11.230 change your mind. And the reason is purely mechanical. 00:06:11.949 --> 00:06:15.120 Mathematically, Multiplying any number by zero 00:06:15.120 --> 00:06:18.019 always yields zero. Right, it's just basic multiplication. 00:06:18.220 --> 00:06:22.240 Exactly. If you assign a prior of zero to a hypothesis, 00:06:22.579 --> 00:06:24.699 meaning you believe it is utterly impossible, 00:06:25.319 --> 00:06:28.240 it doesn't matter if the new evidence has a massive 00:06:28.240 --> 00:06:31.699 undeniable likelihood. The posterior will remain 00:06:31.699 --> 00:06:34.199 zero. You are multiplying the near evidence by 00:06:34.199 --> 00:06:36.500 zero. Let me play devil's advocate here, though, 00:06:36.519 --> 00:06:38.259 because I think a lot of people might push back 00:06:38.259 --> 00:06:40.899 on that being a bad thing. We often admire people 00:06:40.899 --> 00:06:43.259 with strong, unwavering convictions, right? We 00:06:43.259 --> 00:06:46.259 do, yeah. Being 100 % sure of your principles 00:06:46.259 --> 00:06:48.800 is frequently framed as a strong leadership trait. 00:06:49.519 --> 00:06:52.060 Socially, unwavering conviction might be rewarded. 00:06:52.939 --> 00:06:55.480 But mathematically, it's a dead end. It is a 00:06:55.480 --> 00:06:58.420 complete cognitive trap. Think of a heavy lock 00:06:58.420 --> 00:07:01.430 door. If you approach that door and you're 100 00:07:01.430 --> 00:07:03.970 % certain that it is locked, you won't even bother 00:07:03.970 --> 00:07:06.089 to try the handle. No, why would you? Right. 00:07:06.470 --> 00:07:07.970 Even if someone is standing right next to you, 00:07:08.110 --> 00:07:10.310 holding a key, explicitly telling you, hey, I 00:07:10.310 --> 00:07:13.209 just unlocked this, your absolute certainty prevents 00:07:13.209 --> 00:07:15.209 you from verifying the new evidence. You just 00:07:15.209 --> 00:07:17.430 turn around and walk away. That perfectly illustrates 00:07:17.430 --> 00:07:20.189 the danger. If we pull back to look at the larger 00:07:20.189 --> 00:07:22.529 implications, Cromwell's rule mathematically 00:07:22.529 --> 00:07:26.189 reveals why hard fundamentalist convictions are 00:07:26.189 --> 00:07:29.509 entirely insensitive to counter -evidence. Wow. 00:07:29.930 --> 00:07:32.269 Absolute certainty is the enemy of discovery. 00:07:32.550 --> 00:07:35.410 Yes. If you want to be a rational, adaptable 00:07:35.410 --> 00:07:38.839 thinker, a turbasion, you must always leave a 00:07:38.839 --> 00:07:41.480 tiny fraction of a percentage open to being wrong. 00:07:41.639 --> 00:07:46.420 Even just a sliver. Even if your prior is 0 .0001%, 00:07:46.420 --> 00:07:49.199 that sliver of doubt is required. If you don't 00:07:49.199 --> 00:07:51.660 leave that sliver, the entire mathematical engine 00:07:51.660 --> 00:07:53.839 of learning breaks down. You literally cannot 00:07:53.839 --> 00:07:56.720 update a belief if you start at exactly 100 % 00:07:56.720 --> 00:07:59.300 or exactly 0%. It's mechanically impossible to 00:07:59.300 --> 00:08:01.399 learn anything new if you already know everything. 00:08:01.399 --> 00:08:04.160 Precisely. So, we've seen how a single piece 00:08:04.160 --> 00:08:07.160 of evidence, like one cookie, or one locked door, 00:08:07.579 --> 00:08:10.620 updates a single belief. But real life is rarely 00:08:10.620 --> 00:08:13.519 that clean, right? Very rarely. How does this 00:08:13.519 --> 00:08:16.300 scale up? What does Bayesian inference look like 00:08:16.300 --> 00:08:18.759 when we are accumulating evidence over a long 00:08:18.759 --> 00:08:22.319 period of time in deeply complex scenarios? To 00:08:22.319 --> 00:08:24.920 see that scaling effect, we can examine the medieval 00:08:24.920 --> 00:08:27.620 archaeology example from our source text. It 00:08:27.620 --> 00:08:30.540 beautifully demonstrates how Bayes' theorem handles 00:08:30.540 --> 00:08:33.860 sequential compounding data. Oh, I loved this 00:08:33.860 --> 00:08:36.730 part. Imagine an archaeologist is digging at 00:08:36.730 --> 00:08:39.549 a newly discovered site. They know from the general 00:08:39.549 --> 00:08:41.450 region that the site is from the medieval period, 00:08:41.809 --> 00:08:43.669 somewhere between the 11th and 16th centuries. 00:08:44.049 --> 00:08:46.730 But they have no idea exactly when, in that 500 00:08:46.730 --> 00:08:49.110 -year span, the site was actually inhabited. 00:08:49.450 --> 00:08:52.230 Right. So as they begin to excavate, they find 00:08:52.230 --> 00:08:55.360 50 fragments of broken pottery. Some of the pottery 00:08:55.360 --> 00:08:58.820 is glazed and some is decorated. And the archaeologist 00:08:58.820 --> 00:09:00.940 has historical data to compare this against. 00:09:01.460 --> 00:09:03.519 They know that if a site is from the early medieval 00:09:03.519 --> 00:09:06.799 period, say the 11th century, only about 1 % 00:09:06.799 --> 00:09:08.779 of the pottery would typically be glazed and 00:09:08.779 --> 00:09:11.559 50 % would be decorated. Okay, so mostly decorated, 00:09:11.720 --> 00:09:14.600 barely any glaze. Exactly, but if it's from the 00:09:14.600 --> 00:09:18.019 late medieval period, the 16th century, the manufacturing 00:09:18.019 --> 00:09:22.139 techniques changed. By then, 81 % would be glazed 00:09:22.139 --> 00:09:25.669 and only 5 % decorated. So the evidence is trickling 00:09:25.669 --> 00:09:29.190 in one single fragment at a time. How does the 00:09:29.190 --> 00:09:31.909 Bayesian math actually process a slow trickle 00:09:31.909 --> 00:09:34.710 of clues? It starts with setting what statisticians 00:09:34.710 --> 00:09:37.889 call a uniform prior. Meaning they don't have 00:09:37.889 --> 00:09:40.620 a favorite century yet. Right. Because the archaeologist 00:09:40.620 --> 00:09:43.799 has no initial reason to favor one specific century 00:09:43.799 --> 00:09:46.740 over another, they assign a 20 % probability 00:09:46.740 --> 00:09:49.840 to each of the five centuries, an equal unbiased 00:09:49.840 --> 00:09:52.620 spread. OK, fair enough. Then they dig up the 00:09:52.620 --> 00:09:54.259 first pottery fragment. Let's say it's highly 00:09:54.259 --> 00:09:57.220 glazed. They apply Bayes' theorem and the probability 00:09:57.220 --> 00:09:59.460 shift. The likelihood of the 16th century goes 00:09:59.460 --> 00:10:02.019 up a bit, and the likelihood of the 11th century 00:10:02.019 --> 00:10:04.759 drops. Because 16th century pots were way more 00:10:04.759 --> 00:10:07.820 likely to be glazed. Yes. But here is the critical 00:10:07.820 --> 00:10:11.049 mechanism. That new updated probability, the 00:10:11.049 --> 00:10:13.750 posterior, now becomes the prior for the second 00:10:13.750 --> 00:10:16.350 fragment. Today's posterior is tomorrow's prior. 00:10:16.789 --> 00:10:18.690 You just carry the updated math forward to the 00:10:18.690 --> 00:10:20.990 next clue. That's the compounding power of it. 00:10:21.330 --> 00:10:24.690 This process repeats 50 times. Every fragment 00:10:24.690 --> 00:10:27.129 updates the belief. And by the time the 50th 00:10:27.129 --> 00:10:28.970 fragment is pulled from the dirt and analyzed, 00:10:29.549 --> 00:10:32.009 the math has converged to a highly specific, 00:10:32.250 --> 00:10:34.899 dominant conclusion. It has. The calculations 00:10:34.899 --> 00:10:38.340 reveal a 63 % chance the site is from the 14th 00:10:38.340 --> 00:10:41.639 century. A 36 % chance it's from the 15th century, 00:10:41.840 --> 00:10:44.419 and practically zero chance it's from the 11th 00:10:44.419 --> 00:10:47.000 or 12th. And there is a mathematical guarantee 00:10:47.000 --> 00:10:50.179 backing this process up, known as the Bernstein 00:10:50.179 --> 00:10:53.419 von Mises theorem. Yes, quite a mouthful. Without 00:10:53.419 --> 00:10:56.019 getting bogged down in the dense academic terminology, 00:10:56.500 --> 00:10:58.940 what this theorem essentially proves is that 00:10:58.940 --> 00:11:01.779 if you collect enough independent data, your 00:11:01.779 --> 00:11:03.840 final conclusion will eventually converge on 00:11:03.840 --> 00:11:06.299 the truth, regardless of what your initial prior 00:11:06.299 --> 00:11:09.259 was. The sheer volume of incoming data will eventually 00:11:09.259 --> 00:11:11.740 wash away the initial bias. That is so cool. 00:11:11.879 --> 00:11:14.159 It is. If our archaeologists had started out 00:11:14.159 --> 00:11:16.460 completely stubbornly convinced that the site 00:11:16.460 --> 00:11:18.679 was from the 11th century, assigning it a 90 00:11:18.679 --> 00:11:21.879 % prior, those 50 fragments of highly glazed 00:11:21.879 --> 00:11:23.759 pottery would have relentlessly dragged the math 00:11:23.759 --> 00:11:26.279 back to the 14th century anyway. The data always 00:11:26.279 --> 00:11:28.259 wins in the end, as long as you keep updating. 00:11:29.379 --> 00:11:31.059 But hold on. Let me push back on the premise 00:11:31.059 --> 00:11:33.799 here. Sure. If the initial prior can just be 00:11:33.799 --> 00:11:36.259 whatever the researcher wants it to be, a uniform 00:11:36.259 --> 00:11:40.509 prior, or just a subjective expert opinion. How 00:11:40.509 --> 00:11:44.629 can Bayesian inference be considered hard objective 00:11:44.629 --> 00:11:47.409 science? That is the million dollar question. 00:11:47.590 --> 00:11:49.730 Aren't we basically letting personal bias slip 00:11:49.730 --> 00:11:51.450 through the back door of the scientific method? 00:11:51.799 --> 00:11:55.480 That exact critique completely fractured the 00:11:55.480 --> 00:11:57.879 statistical world in the 20th century. After 00:11:57.879 --> 00:12:01.460 the 1920s, a rival school of thought called frequentist 00:12:01.460 --> 00:12:04.700 statistics actually dominated the field. Frequentist 00:12:04.700 --> 00:12:07.299 statistics. Right. Frequentists rely entirely 00:12:07.299 --> 00:12:09.980 on objective sampling data. They detest the concept 00:12:09.980 --> 00:12:12.220 of a prior because they view it as inherently 00:12:12.220 --> 00:12:15.019 unscientific and subjective. They only want to 00:12:15.019 --> 00:12:17.059 look at the frequency of events as they appear 00:12:17.059 --> 00:12:19.299 in the data itself. To put an analogy on it, 00:12:19.379 --> 00:12:21.220 a frequentist is like someone trying to predict 00:12:21.230 --> 00:12:23.529 if it will rain tomorrow by pulling up 50 years 00:12:23.529 --> 00:12:26.450 of almanacs, seeing that it rained on 12 % of 00:12:26.450 --> 00:12:29.029 this specific calendar date historically, and 00:12:29.029 --> 00:12:31.549 concluding the chance of rain is exactly 12%. 00:12:31.549 --> 00:12:33.769 Because they only look at the historical frequency. 00:12:33.990 --> 00:12:35.789 Right. A Bayesian is the person who looks at 00:12:35.789 --> 00:12:38.929 that same almanac, but then walks outside, sees 00:12:38.929 --> 00:12:41.409 massive black thunder clouds rolling over the 00:12:41.409 --> 00:12:45.370 horizon, and says, I'm updating that 12 % prior 00:12:45.370 --> 00:12:48.529 to a 90 % posterior because of the new evidence 00:12:48.529 --> 00:12:51.080 right in front of me. They incorporate the present 00:12:51.080 --> 00:12:53.820 state into the historical data. Exactly. That 00:12:53.820 --> 00:12:56.440 is a highly effective way to separate the two. 00:12:57.000 --> 00:12:59.259 And the Bayesian approach, relying on inverse 00:12:59.259 --> 00:13:02.120 probability to infer backwards from effects to 00:13:02.120 --> 00:13:05.659 causes, using a subjective belief, it absolutely 00:13:05.659 --> 00:13:08.799 infuriated many traditional scientists. Really? 00:13:08.899 --> 00:13:11.580 Like who? Well, the famous philosopher of science, 00:13:11.779 --> 00:13:15.039 Karl Popper, explicitly rejected Bayesian rationalism. 00:13:15.360 --> 00:13:17.679 Popper argued that using Bayes' rule creates 00:13:17.679 --> 00:13:20.559 a vicious circle. How so? He said it presupposes 00:13:20.559 --> 00:13:23.159 what it attempts to justify because you are baking 00:13:23.159 --> 00:13:26.299 your own beliefs into the starting formula. So 00:13:26.299 --> 00:13:28.159 Popper is essentially saying you can't use your 00:13:28.159 --> 00:13:30.360 own opinion as part of the math to prove your 00:13:30.360 --> 00:13:32.659 own opinion. Exactly. Which sounds like a completely 00:13:32.659 --> 00:13:35.080 fair critique. Okay. How did the Beijing camp 00:13:35.080 --> 00:13:37.700 defend against that? The defense relies on radical 00:13:37.700 --> 00:13:41.049 transparency. Bayesians argue that everyone has 00:13:41.049 --> 00:13:43.090 biases and assumptions when they look at data. 00:13:43.309 --> 00:13:45.649 That's true. The frequentist approach often hides 00:13:45.649 --> 00:13:48.070 those assumptions behind complex sampling models 00:13:48.070 --> 00:13:51.169 and arbitrary confidence intervals. The Bayesian 00:13:51.169 --> 00:13:54.110 approach, conversely, forces you to explicitly 00:13:54.110 --> 00:13:57.470 state your bias out loud as a mathematical prior. 00:13:57.690 --> 00:13:59.570 Oh, I see. You have to write down exactly what 00:13:59.570 --> 00:14:01.309 your assumptions are before you start calculating. 00:14:01.590 --> 00:14:04.029 Precisely. Furthermore, a statistician named 00:14:04.029 --> 00:14:07.110 Abraham Wald later proved mathematically that 00:14:07.110 --> 00:14:09.950 every admissible statistical decision procedure 00:14:09.950 --> 00:14:13.009 is, at its core, either a Bayesian procedure 00:14:13.009 --> 00:14:16.169 or a limit of one. Wait, really? Yeah, it turns 00:14:16.169 --> 00:14:18.950 out to be the unavoidable foundational math of 00:14:18.950 --> 00:14:21.409 making decisions under uncertainty. So instead 00:14:21.409 --> 00:14:23.730 of pretending we don't have biases, we put a 00:14:23.730 --> 00:14:25.470 hard number on them, put them out in the open, 00:14:25.789 --> 00:14:27.870 and let the incoming data correct us over time. 00:14:27.960 --> 00:14:30.799 Exactly. And despite those massive philosophical 00:14:30.799 --> 00:14:33.379 debates, the Bayesian approach came roaring back 00:14:33.379 --> 00:14:36.679 to dominance in the 1980s. It experienced a massive 00:14:36.679 --> 00:14:39.679 resurgence, largely thanks to the rise of powerful 00:14:39.679 --> 00:14:43.000 computers. Right. Yes. Bayesian math can get 00:14:43.000 --> 00:14:45.960 computationally overwhelming very quickly. But 00:14:45.960 --> 00:14:48.500 with the invention of Markov chain Monte Carlo 00:14:48.500 --> 00:14:52.159 methods, The real -world applications exploded. 00:14:52.620 --> 00:14:55.039 We see that term Markov chain Monte Carlo or 00:14:55.039 --> 00:14:57.720 MCMC all over the source material. But how do 00:14:57.720 --> 00:15:00.559 we actually compute this when the math gets insane? 00:15:01.000 --> 00:15:03.820 How does MCMC actually work? Think of it like 00:15:03.820 --> 00:15:06.899 a blindfolded hiker trying to find the highest 00:15:06.899 --> 00:15:09.759 peak in a massive sprawling mountain range. Okay, 00:15:09.960 --> 00:15:12.039 blindfolded hiker. The hiker can't see the whole 00:15:12.039 --> 00:15:14.700 map to calculate where the peak is, but they 00:15:14.700 --> 00:15:17.179 can take a step, feel if the ground is sloping 00:15:17.179 --> 00:15:19.940 up or down, and make a decision to stay or move 00:15:19.940 --> 00:15:23.100 higher. Just feeling their way up. Right. Markov 00:15:23.100 --> 00:15:25.480 chain Monte Carlo algorithms do something similar 00:15:25.480 --> 00:15:28.039 in a landscape of complex probabilities. They 00:15:28.039 --> 00:15:30.480 wander around, taking steps, testing the math 00:15:30.480 --> 00:15:32.710 locally, and eventually they map a the shape 00:15:32.710 --> 00:15:34.830 of the mountain finding the highest probability 00:15:34.830 --> 00:15:37.610 without needing to calculate every single inch 00:15:37.610 --> 00:15:40.269 of the infinite terrain. That computational power 00:15:40.269 --> 00:15:42.750 brings us to where this subjective math is actually 00:15:42.750 --> 00:15:44.990 operating in your daily life right now. Yeah. 00:15:45.169 --> 00:15:47.629 Let's trace this from your inbox to the literal 00:15:47.629 --> 00:15:49.690 edge of the universe. Let's do it. If you've 00:15:49.690 --> 00:15:52.110 ever checked your email spam folder, you are 00:15:52.110 --> 00:15:55.429 looking at Bayesian inference at work. Yes, specifically 00:15:55.429 --> 00:15:58.840 naive Bayes classifiers. The software starts 00:15:58.840 --> 00:16:00.940 with a prior belief about what a normal email 00:16:00.940 --> 00:16:03.519 looks like. Then it looks at the new evidence, 00:16:03.940 --> 00:16:06.100 the specific words in the incoming email. If 00:16:06.100 --> 00:16:08.860 it sees the word lottery or Prince, the algorithm 00:16:08.860 --> 00:16:10.860 notes that the likelihood of those words appearing 00:16:10.860 --> 00:16:14.080 in a legitimate email is very low, but incredibly 00:16:14.080 --> 00:16:17.340 high for spam. The posterior probability updates 00:16:17.340 --> 00:16:20.559 and the emails aggressively filtered out. The 00:16:20.559 --> 00:16:23.059 algorithm learns over time based on past data. 00:16:23.399 --> 00:16:25.899 But how does a computer which only understands 00:16:25.899 --> 00:16:30.039 ones and zeros possess a subjective bias? It's 00:16:30.039 --> 00:16:32.860 because the programmers initially feed it a massive 00:16:32.860 --> 00:16:35.620 data set of human -labeled spam to establish 00:16:35.620 --> 00:16:38.289 its prior. Right. Humans give it the bias. From 00:16:38.289 --> 00:16:41.629 tech, we jump to cosmology. Scientists use Bayesian 00:16:41.629 --> 00:16:43.970 inference to literally map the universe. They 00:16:43.970 --> 00:16:46.529 do. Our sources point out that researchers use 00:16:46.529 --> 00:16:49.750 Bayesian model comparison on the cosmic microwave 00:16:49.750 --> 00:16:52.950 background data, the Planck's 2018 data, to fit 00:16:52.950 --> 00:16:54.970 the parameters of the standard model of the Big 00:16:54.970 --> 00:16:57.909 Bang, the Lambda CDM model. Cosmology is the 00:16:57.909 --> 00:17:00.070 perfect application for Bayes' theorem because 00:17:00.070 --> 00:17:03.029 of a fundamental physical limitation. We only 00:17:03.029 --> 00:17:05.549 have one universe. Yeah, we can't exactly run 00:17:05.549 --> 00:17:08.609 a control group on the big bang. Exactly. A frequentist 00:17:08.609 --> 00:17:11.109 would prefer to sample multiple universes to 00:17:11.109 --> 00:17:13.829 find an average frequency of how big bangs usually 00:17:13.829 --> 00:17:16.869 play out. But we can't do that. We only have 00:17:16.869 --> 00:17:20.170 one set of cosmic microwave background data. 00:17:20.230 --> 00:17:23.470 So what do they do? Cosmologists set prior parameters 00:17:23.470 --> 00:17:26.569 based on existing physics, run the data through 00:17:26.569 --> 00:17:28.890 those Markov chain Monte Carlo simulations we 00:17:28.890 --> 00:17:31.349 just talked about, and update their beliefs about 00:17:31.349 --> 00:17:34.029 the fundamental structure of reality. So it filters 00:17:34.029 --> 00:17:36.349 our spam and it measures the Big Bang. But here 00:17:36.349 --> 00:17:38.809 is where this beautiful mathematical framework 00:17:38.809 --> 00:17:42.619 hits a very messy, very human limit. The courtroom. 00:17:42.880 --> 00:17:45.339 Oh, this is a fascinating example in the UK. 00:17:45.519 --> 00:17:48.490 There was a legal case known as RV Adams The 00:17:48.490 --> 00:17:51.130 defense actually brought in an expert to explain 00:17:51.130 --> 00:17:53.529 Bayes' theorem to the jury, hoping to help them 00:17:53.529 --> 00:17:55.690 combine different pieces of conflicting evidence 00:17:55.690 --> 00:17:57.609 mathematically. And the court's reaction to that 00:17:57.609 --> 00:18:00.269 was aggressively hostile. Really hostile. The 00:18:00.269 --> 00:18:02.589 Court of Appeal ultimately ruled that introducing 00:18:02.589 --> 00:18:05.910 Bayes' theorem plunges the jury into inappropriate 00:18:05.910 --> 00:18:08.930 and unnecessary realms of theory and complexity, 00:18:09.710 --> 00:18:11.849 deflecting them from their proper task. It really 00:18:11.849 --> 00:18:14.769 makes you wonder. Are human brains just not wired 00:18:14.769 --> 00:18:18.210 for this type of rigorous explicit accumulation 00:18:18.210 --> 00:18:21.829 of evidence? It's tough. We naturally crave cohesive 00:18:21.829 --> 00:18:25.009 stories and emotional narratives, not probability 00:18:25.009 --> 00:18:28.309 matrices. There is a profound friction between 00:18:28.309 --> 00:18:31.769 mathematical logic and human nature. The source 00:18:31.769 --> 00:18:34.289 text highlights a fascinating counter -argument 00:18:34.289 --> 00:18:36.849 regarding this friction by a theorist named Gardner 00:18:36.849 --> 00:18:39.950 -Medwin. was his argument. He argues that juries 00:18:39.950 --> 00:18:41.809 shouldn't just be calculating the probability 00:18:41.809 --> 00:18:43.750 of guilt, they should be looking at the probability 00:18:43.750 --> 00:18:45.970 of the evidence given that the defendant is innocent. 00:18:46.150 --> 00:18:47.849 Okay, let's break down what that actually means 00:18:47.849 --> 00:18:50.410 for a regular person sitting on a jury. If you 00:18:50.410 --> 00:18:53.170 are going to use Bayes' theorem to compute a 00:18:53.170 --> 00:18:55.910 final posterior probability of someone's guilt, 00:18:56.569 --> 00:18:58.950 the math demands that you start with a prior 00:18:58.950 --> 00:19:02.089 probability of guilt. But what is the prior probability 00:19:02.089 --> 00:19:04.589 that a random person sitting on trial committed 00:19:04.589 --> 00:19:07.490 a crime before you even look at the DNA or the 00:19:07.490 --> 00:19:10.670 witnesses? How do you even set that number? If 00:19:10.670 --> 00:19:12.650 100 ,000 people live in the city where the crime 00:19:12.650 --> 00:19:15.900 happened, is the prior one in 100 ,000? Or do 00:19:15.900 --> 00:19:17.779 you base it on the defendant's past criminal 00:19:17.779 --> 00:19:20.279 record? Yeah. Suddenly you are asking a jury. 00:19:20.880 --> 00:19:23.640 to assign a mathematical number to a person's 00:19:23.640 --> 00:19:26.079 underlying suspiciousness before looking at the 00:19:26.079 --> 00:19:28.259 facts of the case. Which is exactly the problem. 00:19:28.400 --> 00:19:30.819 The math works perfectly, but the act of setting 00:19:30.819 --> 00:19:34.079 a prior incidence of guilt feels fundamentally 00:19:34.079 --> 00:19:37.099 opposed to the legal concept of innocent until 00:19:37.099 --> 00:19:39.839 proven guilty. It totally clashes. It requires 00:19:39.839 --> 00:19:42.480 knowing a baseline probability of criminality 00:19:42.480 --> 00:19:44.900 that is highly controversial to establish in 00:19:44.900 --> 00:19:47.400 a specific individual trial. It encapsulates 00:19:47.400 --> 00:19:50.329 why this is such a powerful yet demanding mental 00:19:50.329 --> 00:19:53.769 model. So to wrap up our deep dive today, what 00:19:53.769 --> 00:19:55.690 are the actionable takeaways? I think the main 00:19:55.690 --> 00:19:58.690 one is transparency. Yeah. Thinking like a Bajan 00:19:58.690 --> 00:20:00.890 means explicitly acknowledging that you have 00:20:00.890 --> 00:20:03.910 priors, you have initial biases, you have starting 00:20:03.910 --> 00:20:06.950 assumptions, and that is perfectly OK as long 00:20:06.950 --> 00:20:09.289 as you make them transparent. Exactly. It means 00:20:09.289 --> 00:20:11.650 you have to remain genuinely open to new evidence 00:20:11.650 --> 00:20:14.150 and let that evidence iteratively update your 00:20:14.150 --> 00:20:19.160 beliefs over time. And above all, avoid 100 % 00:20:19.160 --> 00:20:21.980 certainty. Leave the door unlocked, even if it's 00:20:21.980 --> 00:20:24.019 just a fraction of a percent. Before we sign 00:20:24.019 --> 00:20:26.519 off, I want to leave you with one final, truly 00:20:26.519 --> 00:20:28.680 mind -bending concept from the source material. 00:20:29.220 --> 00:20:31.819 It's a theory called Solomonov's inductive inference. 00:20:32.019 --> 00:20:34.309 Okay, take us down the rabbit hole. Ray Salomonash 00:20:34.309 --> 00:20:37.289 mathematically combined Bayesian statistics with 00:20:37.289 --> 00:20:40.309 Occam's razor, which is the philosophical principle 00:20:40.309 --> 00:20:42.569 that the simplest explanation is usually the 00:20:42.569 --> 00:20:45.230 correct one. Right. He used these two concepts 00:20:45.230 --> 00:20:47.869 to theorize a framework for universal prediction. 00:20:48.569 --> 00:20:50.890 The underlying idea is that if our environment, 00:20:51.250 --> 00:20:53.970 our physical reality follows any unknown but 00:20:53.970 --> 00:20:56.849 computable probability distribution, then Bayesian 00:20:56.849 --> 00:20:59.470 updating could theoretically be used to predict 00:20:59.470 --> 00:21:02.430 the yet unseen parts of the universe in an optimal 00:21:02.430 --> 00:21:05.640 perfect - fashion. Wait so if the universe operates 00:21:05.640 --> 00:21:09.039 on some kind of underlying computable code Bayes 00:21:09.039 --> 00:21:11.799 theorem is the literal key to predicting the 00:21:11.799 --> 00:21:14.980 future. Given enough incoming data and the correct 00:21:14.980 --> 00:21:18.900 universal prior to start with you could hypothetically 00:21:18.900 --> 00:21:21.640 predict the unseen sequence of reality perfectly. 00:21:22.359 --> 00:21:24.680 Every random event would just be another data 00:21:24.680 --> 00:21:27.400 point updating the master equation. Look around 00:21:27.400 --> 00:21:29.720 the room you're in right now. It really asks 00:21:29.720 --> 00:21:32.480 you to ponder whether everything in your life, 00:21:33.180 --> 00:21:35.200 the traffic on your commute, the decisions of 00:21:35.200 --> 00:21:37.700 the people around you, the seemingly random chaos 00:21:37.700 --> 00:21:40.440 of your day, is just a computable sequence. It's 00:21:40.440 --> 00:21:42.119 a wild thought. It might all just be waiting 00:21:42.119 --> 00:21:44.680 for the right prior and enough data to be perfectly 00:21:44.680 --> 00:21:47.259 mathematically predicted. It brings us right 00:21:47.259 --> 00:21:49.299 back to our savvy detective at the start of the 00:21:49.299 --> 00:21:51.700 show. It really does. Because in the end, we're 00:21:51.700 --> 00:21:53.579 all just detectives trying to weigh the clues 00:21:53.579 --> 00:21:56.440 of today against the mysteries of tomorrow. making 00:21:56.440 --> 00:21:58.519 sure we never ever lock the door on the truth.