WEBVTT 00:00:00.000 --> 00:00:02.660 Imagine someone walks up to you and hands you 00:00:02.660 --> 00:00:06.360 a tiny, a really tiny slip of paper. Like maybe 00:00:06.360 --> 00:00:09.000 the size of a fortune cookie insert? Yeah, exactly 00:00:09.000 --> 00:00:11.220 like that. Barely bigger than a fortune cookie 00:00:11.220 --> 00:00:14.460 insert. And they tell you with just absolute 00:00:14.460 --> 00:00:17.679 deadpan sincerity that written on this scrap 00:00:17.679 --> 00:00:21.239 is the fundamental mathematical algorithm behind 00:00:21.239 --> 00:00:23.879 every single decision you have ever made. Right. 00:00:24.179 --> 00:00:27.000 And not just that, but every decision you will 00:00:27.000 --> 00:00:29.160 ever make. You probably assume it was a joke. 00:00:29.289 --> 00:00:31.410 Right? I mean, I would. Oh, absolutely. It sounds 00:00:31.410 --> 00:00:34.490 like science fiction. It really does. But today, 00:00:34.890 --> 00:00:37.350 we're essentially holding that exact metaphorical 00:00:37.350 --> 00:00:40.210 slip of paper. Welcome to the deep dive, by the 00:00:40.210 --> 00:00:42.829 way. Today's deep dive is based on what might 00:00:42.829 --> 00:00:45.320 actually be our smallest source text ever. It 00:00:45.320 --> 00:00:47.799 is incredibly brief. Yeah, it's literally just 00:00:47.799 --> 00:00:50.560 a remarkably short Wikipedia stub about this 00:00:50.560 --> 00:00:52.740 concept in decision theory. It's called the complete 00:00:52.740 --> 00:00:54.799 class theorem. We're talking about, I don't know, 00:00:54.859 --> 00:00:58.060 a handful of really dense sentences. But the 00:00:58.060 --> 00:00:59.899 sheer density of the information, I mean, that 00:00:59.899 --> 00:01:01.659 is what makes it so compelling. We are dealing 00:01:01.659 --> 00:01:05.400 with an intensely concentrated nugget of mathematical 00:01:05.400 --> 00:01:08.480 statistics here. In academia, you sometimes find 00:01:08.480 --> 00:01:12.700 these tiny unassuming theorems that just quietly 00:01:12.700 --> 00:01:16.680 act as the load -bearer. pillars for entire fields, 00:01:17.359 --> 00:01:20.920 like behavioral economics or logic. And that 00:01:20.920 --> 00:01:23.500 is exactly what we have here. It really is fascinating. 00:01:23.620 --> 00:01:25.340 I have to admit, when I first looked at the source 00:01:25.340 --> 00:01:27.379 material, I was pretty skeptical. I was like, 00:01:27.519 --> 00:01:30.500 how much can we really extract from this? Right, 00:01:30.500 --> 00:01:32.620 it doesn't look like much. But then as you parse 00:01:32.620 --> 00:01:35.400 the actual phrasing, you realize this text is 00:01:35.400 --> 00:01:38.519 delivering this huge universal ultimatum about 00:01:38.519 --> 00:01:41.280 the actual mechanics of human choice. So our 00:01:41.280 --> 00:01:43.560 mission today for you, the listener, is to take 00:01:43.560 --> 00:01:46.560 this highly compressed statistical puzzle and 00:01:46.560 --> 00:01:49.219 carefully expand it, word by word. Expand it 00:01:49.219 --> 00:01:51.040 and really look at the implications. Exactly. 00:01:51.120 --> 00:01:52.760 We're going to examine the ultimate filter for 00:01:52.760 --> 00:01:54.920 making the optimal choice and see how it actually 00:01:54.920 --> 00:01:57.079 maps onto our everyday lives. I think that is 00:01:57.079 --> 00:01:59.000 the right approach. We really need to dismantle 00:01:59.000 --> 00:02:01.159 the machinery of this theorem piece by piece, 00:02:01.200 --> 00:02:03.420 because once you actually grasp the underlying 00:02:03.420 --> 00:02:05.640 mechanisms, the specific mathematical constraints 00:02:05.640 --> 00:02:08.780 the text is outlining, it just forces a total 00:02:08.780 --> 00:02:11.819 reevaluation of what you even consider rational 00:02:11.819 --> 00:02:14.080 thought. OK, let's unpack this before we get 00:02:14.080 --> 00:02:17.139 into the really heavy statistical claims. The 00:02:17.139 --> 00:02:19.810 text establishes a sort of baseline. Right. It 00:02:19.810 --> 00:02:22.710 has to set the stage. Yeah. It gives us this 00:02:22.710 --> 00:02:25.569 strict requirement for what makes a decision 00:02:25.569 --> 00:02:28.229 mathematically valid in the first place. And 00:02:28.229 --> 00:02:31.310 it introduces a specific term. It calls it an 00:02:31.310 --> 00:02:34.189 admissible decision rule. Yes. And the way the 00:02:34.189 --> 00:02:36.810 text defines admissibility is critical because 00:02:36.810 --> 00:02:39.909 it does so entirely by negation. Meaning it tells 00:02:39.909 --> 00:02:42.189 us what it isn't. Exactly. It does not tell you 00:02:42.189 --> 00:02:44.009 what a perfect decision looks like. It only tells 00:02:44.009 --> 00:02:46.949 you what an invalid decision looks like. It states 00:02:46.949 --> 00:02:49.550 that if you are using an inadmissible rule than, 00:02:49.789 --> 00:02:51.990 quote, there is a rule that is sometimes better 00:02:51.990 --> 00:02:55.430 and never worse. Wow. OK. So if we invert that 00:02:55.430 --> 00:02:58.150 logic, an admissible decision rule is one where 00:02:58.150 --> 00:03:01.250 absolutely no alternative exists that is sometimes 00:03:01.250 --> 00:03:03.889 better and never worse. Precisely. That phrase, 00:03:03.909 --> 00:03:06.569 sometimes better and never worse, that's the 00:03:06.569 --> 00:03:08.650 threshold for keeping a choice on the table at 00:03:08.650 --> 00:03:10.650 all. I want to ground this for a second. Let's 00:03:10.650 --> 00:03:12.789 look at a classic commuter dilemma. OK. Let's 00:03:12.789 --> 00:03:15.250 hear it. Say you're driving to work, and you 00:03:15.250 --> 00:03:18.469 have two routes. Route A is the highway. Based 00:03:18.469 --> 00:03:21.509 on all available data, it takes exactly 20 minutes 00:03:21.509 --> 00:03:24.129 every single day, regardless of conditions. And 00:03:24.129 --> 00:03:25.969 it's like a magical highway, but sure. Right, 00:03:26.129 --> 00:03:28.389 work with me here. Route B is the back roads. 00:03:28.669 --> 00:03:31.650 It usually takes 20 minutes, but if you get stuck 00:03:31.650 --> 00:03:33.169 behind a delivery truck or something, it takes 00:03:33.169 --> 00:03:35.569 30 minutes. We have to be careful there, though. 00:03:35.830 --> 00:03:38.729 For that analogy to really hold up mathematically, 00:03:39.270 --> 00:03:42.250 we have to assume strict ceteris paribus. Which 00:03:42.250 --> 00:03:45.199 means? all other things being equal. Right. Yes, 00:03:45.280 --> 00:03:48.659 exactly. Because if route B offers a stunning 00:03:48.659 --> 00:03:51.740 scenic view that you really enjoy, then the utility 00:03:51.740 --> 00:03:54.060 of that drive changes. It might not be strictly 00:03:54.060 --> 00:03:56.099 worse to take 30 minutes if you're enjoying the 00:03:56.099 --> 00:03:59.419 scenery. OK, fair point. But if we are isolating 00:03:59.419 --> 00:04:02.000 purely for time and fuel efficiency, if getting 00:04:02.000 --> 00:04:05.479 there fast is the only goal, then yes, your analogy 00:04:05.479 --> 00:04:08.080 works perfectly. Right. So assuming we only care 00:04:08.080 --> 00:04:10.180 about getting to the office as fast as possible 00:04:10.180 --> 00:04:14.159 in that tightly constrained scenario, route A 00:04:14.159 --> 00:04:16.639 is our sometimes better, never worse option. 00:04:16.660 --> 00:04:18.600 Because the highway is never going to take 30 00:04:18.600 --> 00:04:20.420 minutes, so it is never worse than the back roads. 00:04:21.040 --> 00:04:23.379 But it is sometimes better, because on the days 00:04:23.379 --> 00:04:25.339 there happens to be a truck on the back roads, 00:04:25.740 --> 00:04:27.959 the highway beats it by 10 minutes. Exactly. 00:04:28.600 --> 00:04:30.740 So if you choose the back roads, knowing this 00:04:30.740 --> 00:04:33.100 information, you are choosing a mathematically 00:04:33.100 --> 00:04:36.740 dominated strategy. You're picking a rule where 00:04:36.740 --> 00:04:40.319 a strictly superior alternative just plainly 00:04:40.319 --> 00:04:42.980 exists. So the theorem is basically saying that 00:04:42.980 --> 00:04:46.300 the first step of decision theory is just eliminating 00:04:46.300 --> 00:04:49.019 the obviously dumb choices. That's a great way 00:04:49.019 --> 00:04:51.639 to put it. It really is. That concept of dominance 00:04:51.639 --> 00:04:54.480 is the foundational sorting mechanism in decision 00:04:54.480 --> 00:04:57.819 theory. By eliminating those dominated, sometimes 00:04:57.819 --> 00:05:00.160 worse options, it leaves us with our admissible 00:05:00.160 --> 00:05:02.860 class of choices. It just clears the junk out. 00:05:02.980 --> 00:05:06.339 It sweeps the board clean of mathematically unjustifiable 00:05:06.339 --> 00:05:08.980 actions. But while that sounds obvious when we 00:05:08.980 --> 00:05:10.939 talk about a highway, it is merely the setup. 00:05:11.420 --> 00:05:13.939 The complete class theorem uses this baseline 00:05:13.939 --> 00:05:16.800 of admissibility to launch its actual, far more 00:05:16.800 --> 00:05:19.360 aggressive premise. Which brings us to the core 00:05:19.360 --> 00:05:22.540 of the text, because once we have isolated those 00:05:22.540 --> 00:05:24.839 admissible decisions, once we've thrown out the 00:05:24.839 --> 00:05:27.180 back roads and kept only the strategies that 00:05:27.180 --> 00:05:29.660 aren't mathematically self -defeating, the theorem 00:05:29.660 --> 00:05:33.480 asks a huge question. What do all of these surviving 00:05:33.480 --> 00:05:36.500 valid decisions have in common? And the answer 00:05:36.500 --> 00:05:39.100 the text provides is absolute. Yeah, it really 00:05:39.100 --> 00:05:42.490 is. It says they are all fundamentally Bayesian. 00:05:42.850 --> 00:05:45.589 What's fascinating here is the sheer uncompromising 00:05:45.589 --> 00:05:47.990 nature of that claim. The theorem mathematically 00:05:47.990 --> 00:05:50.589 proves that all admissible decision rules are 00:05:50.589 --> 00:05:53.009 equivalent to the Bayesian decision rule for, 00:05:53.170 --> 00:05:55.910 as the text specifically words it, some utility 00:05:55.910 --> 00:05:58.149 function and some prior distribution. Wow. There 00:05:58.149 --> 00:06:01.040 are no exceptions in the math. every single one. 00:06:01.220 --> 00:06:02.920 I want to really dig into those two specific 00:06:02.920 --> 00:06:05.220 terms, utility function and prior distribution, 00:06:05.639 --> 00:06:07.699 because they seem to form the engine of this 00:06:07.699 --> 00:06:09.540 entire framework. I mean, if every good choice 00:06:09.540 --> 00:06:11.639 we make is DASION, we have to understand what 00:06:11.639 --> 00:06:13.879 we're actually calculating in our heads. Well, 00:06:13.879 --> 00:06:16.180 let's start with the utility function. In statistics, 00:06:16.399 --> 00:06:19.079 your utility function is the mathematical representation 00:06:19.079 --> 00:06:23.600 of your goals, your risk tolerance, and the specific 00:06:23.600 --> 00:06:26.379 payoffs you are chasing. So it's like a quantifiable 00:06:26.379 --> 00:06:29.180 metric of what you value in any given scenario. 00:06:29.680 --> 00:06:31.720 Exactly. It is what you want out of the situation. 00:06:31.800 --> 00:06:34.060 OK. And the prior distribution. That is your 00:06:34.060 --> 00:06:36.560 pre -existing probability map. It encompasses 00:06:36.560 --> 00:06:40.060 the historical data, the baseline odds, and essentially 00:06:40.060 --> 00:06:42.139 the state of the world as you understand it, 00:06:42.259 --> 00:06:43.879 right up until the very moment you make your 00:06:43.879 --> 00:06:46.019 choice. OK. I want to take those two mechanisms 00:06:46.019 --> 00:06:48.600 and apply them to something way less abstract 00:06:48.600 --> 00:06:51.000 than theoretical statistics. Sure. Let's look 00:06:51.000 --> 00:06:54.110 at a highly relatable scenario. Say you are trying 00:06:54.110 --> 00:06:56.250 to decide between going to a local pizza place 00:06:56.250 --> 00:07:00.269 or a popular taco truck for dinner. And classic 00:07:00.269 --> 00:07:03.230 dilemma. Right. If we apply the theorem here, 00:07:04.050 --> 00:07:06.350 your utility function is essentially your craving, 00:07:06.750 --> 00:07:08.709 maybe combined with your caloric needs and your 00:07:08.709 --> 00:07:11.910 budget. OK. Let's say we quantify it. Your utility 00:07:11.910 --> 00:07:14.790 for eating a great size of pizza tonight is 100 00:07:14.790 --> 00:07:18.589 units of satisfaction. But the taco truck represents, 00:07:18.990 --> 00:07:22.509 say, 80 units. Oh. So pizza is the clear winner 00:07:22.509 --> 00:07:25.629 on utility. But you cannot just make the decision 00:07:25.629 --> 00:07:27.709 based on utility alone because the outcomes are 00:07:27.709 --> 00:07:29.610 not guaranteed. Because life happens. Right. 00:07:29.990 --> 00:07:32.550 And that is exactly where the prior distribution 00:07:32.550 --> 00:07:34.829 enters the calculation. Your prior distribution 00:07:34.829 --> 00:07:37.649 is your knowledge of the environment. So you 00:07:37.649 --> 00:07:39.649 know from past experience that the pizza place 00:07:39.649 --> 00:07:43.930 is clear across town. At 7 p .m. there is an 00:07:43.930 --> 00:07:46.699 80 percent chance of gridlock traffic. which 00:07:46.699 --> 00:07:49.060 obviously degrades your utility. Sitting in traffic 00:07:49.060 --> 00:07:51.620 is definitely negative utility. Yeah, huge penalty. 00:07:52.360 --> 00:07:54.319 But the taco truck is just down the street, meaning 00:07:54.319 --> 00:07:57.439 there's only a 5 % chance of a delay. Now, let's 00:07:57.439 --> 00:07:59.399 look at the actual mechanics of the Bayesian 00:07:59.399 --> 00:08:01.959 update here, rather than just stating the final 00:08:01.959 --> 00:08:05.000 result. To make an admissible choice, you are 00:08:05.000 --> 00:08:07.569 calculating the expected utility. Okay, how does 00:08:07.569 --> 00:08:10.250 that work? You multiply the raw utility of the 00:08:10.250 --> 00:08:12.389 pizza, those hundred units you mentioned, by 00:08:12.389 --> 00:08:14.750 the probability of actually securing it without 00:08:14.750 --> 00:08:17.250 that massive negative utility penalty of sitting 00:08:17.250 --> 00:08:20.540 in traffic for an hour. Got it. So, if the traffic 00:08:20.540 --> 00:08:23.779 penalty reduces the net utility of the pizza 00:08:23.779 --> 00:08:27.560 to, let's say, 40 units, but the reliable taco 00:08:27.560 --> 00:08:31.259 truck offers a solid expected utility of 75 units, 00:08:31.779 --> 00:08:34.320 then the mathematically admissible choice is 00:08:34.320 --> 00:08:36.620 the tacos. Even though I wanted the pizza more 00:08:36.620 --> 00:08:38.779 in a vacuum. Exactly, because we don't live in 00:08:38.779 --> 00:08:40.940 a vacuum. But what happens when the environment 00:08:40.940 --> 00:08:43.360 shifts? Like, let's say you're walking out the 00:08:43.360 --> 00:08:45.919 door fully intending to get tacos because it's 00:08:45.919 --> 00:08:48.600 the optimal admissible choice. But you check 00:08:48.600 --> 00:08:51.320 your phone and see a notification that a massive 00:08:51.320 --> 00:08:53.820 water main break just flooded the street where 00:08:53.820 --> 00:08:57.120 the taco truck usually parks. Ah. Well, that 00:08:57.120 --> 00:08:59.860 is the very essence of a Bayesian update. Your 00:08:59.860 --> 00:09:02.100 prior distribution has just been radically altered 00:09:02.100 --> 00:09:04.519 by new evidence. The probability of the taco 00:09:04.519 --> 00:09:06.820 truck being accessible just plummeted from 95 00:09:06.820 --> 00:09:09.519 percent down to, say, 10 percent. Which fundamentally 00:09:09.519 --> 00:09:12.159 rewrites the expected utility. Suddenly, the 00:09:12.159 --> 00:09:14.559 previously admissible taco choice becomes mathematically 00:09:14.559 --> 00:09:16.960 inferior. Yes. If you ignore that notification 00:09:16.960 --> 00:09:19.610 and just to the taco truck anyway, you are no 00:09:19.610 --> 00:09:21.889 longer operating optimally. You are choosing 00:09:21.889 --> 00:09:24.450 an inadmissible rule. Unless of course your utility 00:09:24.450 --> 00:09:27.149 function is highly idiosyncratic. What do you 00:09:27.149 --> 00:09:30.090 mean? Well, if you genuinely value the physical 00:09:30.090 --> 00:09:32.789 exercise of walking to a flooded street more 00:09:32.789 --> 00:09:35.730 than you value actually eating food, then the 00:09:35.730 --> 00:09:38.370 choice remains admissible for your specific utility 00:09:38.370 --> 00:09:40.889 function. OK, that's wild to think about. But 00:09:40.889 --> 00:09:44.029 it actually explains a lot of seemingly irrational 00:09:44.029 --> 00:09:46.789 human behavior. When people make choices that 00:09:46.789 --> 00:09:49.610 look totally foolish to outsiders, it is often 00:09:49.610 --> 00:09:52.110 because their internal utility function heavily 00:09:52.110 --> 00:09:54.490 weights variables that others simply cannot see. 00:09:54.690 --> 00:09:58.820 Like what? Like pride or spite. or just a desire 00:09:58.820 --> 00:10:01.440 for novelty. They are still making an admissible 00:10:01.440 --> 00:10:03.440 choice based on their math, we just don't have 00:10:03.440 --> 00:10:05.360 access to their numbers. So what does this all 00:10:05.360 --> 00:10:07.240 mean? We're looking at a framework here that 00:10:07.240 --> 00:10:09.440 leaves absolutely no room for mystical thinking. 00:10:10.179 --> 00:10:12.759 The text's conclusion is structured as this rigid 00:10:12.759 --> 00:10:15.879 dichotomy. It states, quote, Thus, for every 00:10:15.879 --> 00:10:18.440 decision rule, either the rule may be reformulated 00:10:18.440 --> 00:10:21.279 as a Bayesian procedure, or there is a rule that 00:10:21.279 --> 00:10:24.139 is sometimes better and never worse. There is 00:10:24.139 --> 00:10:26.870 no third option. Right. If we connect this to 00:10:26.870 --> 00:10:29.570 the bigger picture, you begin to realize why 00:10:29.570 --> 00:10:31.789 this theorem is so disruptive to how we view 00:10:31.789 --> 00:10:35.090 ourselves. Because humans deeply romanticize 00:10:35.090 --> 00:10:37.769 intuition. Oh, heavily. We love the narrative 00:10:37.769 --> 00:10:39.970 of the brilliant executive who ignores all the 00:10:39.970 --> 00:10:42.850 data and just goes with their gut. Or the master 00:10:42.850 --> 00:10:45.370 chess player who makes a move based on a feeling. 00:10:45.769 --> 00:10:49.509 We treat intuition as this magical third category 00:10:49.509 --> 00:10:52.450 of cognition that somehow transcends logic. But 00:10:52.450 --> 00:10:55.169 this little Wikipedia stub effectively outlaws 00:10:55.169 --> 00:10:57.710 that third category. It traps us in a binary. 00:10:57.990 --> 00:11:00.029 You are either performing a Bayesian calculation 00:11:00.029 --> 00:11:03.049 or you are making an error. Exactly. It completely 00:11:03.049 --> 00:11:05.429 demystifies intuition. If an executive makes 00:11:05.429 --> 00:11:07.970 a gut call that ultimately succeeds and that 00:11:07.970 --> 00:11:10.549 decision was truly optimal and admissible, the 00:11:10.549 --> 00:11:12.789 complete class theorem dictates that they did 00:11:12.789 --> 00:11:15.450 not bypass logic at all. They just didn't realize 00:11:15.450 --> 00:11:18.360 they were doing math. Right. Instead... Their 00:11:18.360 --> 00:11:21.539 brain executed a rapid subconscious Bayesian 00:11:21.539 --> 00:11:24.620 procedure. Their quote unquote gut feeling was 00:11:24.620 --> 00:11:27.320 actually just a highly compressed prior distribution. 00:11:27.960 --> 00:11:31.320 It's a massive internal database built over decades 00:11:31.320 --> 00:11:34.139 of industry experience, micro -observations, 00:11:34.600 --> 00:11:37.629 and pattern recognition. They ran the expected 00:11:37.629 --> 00:11:39.830 utility math instantly. They just didn't realize 00:11:39.830 --> 00:11:41.870 they were doing algebra. Their conscious mind 00:11:41.870 --> 00:11:44.149 experienced the output of the equation as a feeling. 00:11:44.509 --> 00:11:46.809 Precisely. Take a professional quarterback reading 00:11:46.809 --> 00:11:49.110 a complex defense at the line of scrimmage. They 00:11:49.110 --> 00:11:51.269 have, what, three seconds to make a choice? Yeah, 00:11:51.389 --> 00:11:53.940 barely that. They aren't guessing. They are accessing 00:11:53.940 --> 00:11:56.299 thousands of hours of film study, which is their 00:11:56.299 --> 00:11:58.899 prior distribution, and instantly calculating 00:11:58.899 --> 00:12:01.460 the probability of a blitz against the utility 00:12:01.460 --> 00:12:03.740 of a specific passing route. So when they throw 00:12:03.740 --> 00:12:05.600 an interception? When they throw an interception, 00:12:05.659 --> 00:12:08.139 it is not because their magic failed. It is because 00:12:08.139 --> 00:12:11.759 their prior distribution was incomplete. Or they 00:12:11.759 --> 00:12:13.960 failed to properly update their probabilities 00:12:13.960 --> 00:12:15.899 when a linebacker shifted at the last second. 00:12:16.000 --> 00:12:18.220 They just made an inadmissible choice. It is 00:12:18.220 --> 00:12:21.580 a profoundly humbling way to view human intellect. 00:12:21.690 --> 00:12:24.610 It suggests that our absolute greatest moments 00:12:24.610 --> 00:12:28.509 of genius are just instances where our internal 00:12:28.509 --> 00:12:31.490 mathematical models align perfectly with reality. 00:12:31.789 --> 00:12:34.440 It really is. But, you know, accepting that every 00:12:34.440 --> 00:12:37.399 valid choice is inherently Bayesian creates a 00:12:37.399 --> 00:12:39.940 massive problem for statisticians. How do you 00:12:39.940 --> 00:12:42.200 actually prove that mathematically when human 00:12:42.200 --> 00:12:44.940 variables are infinite? Right. That is the wall 00:12:44.940 --> 00:12:47.120 you hit. And this is where the source text introduces 00:12:47.120 --> 00:12:49.659 its only specific academic example. Yes, the 00:12:49.659 --> 00:12:52.539 text shifts from the abstract theorem to the 00:12:52.539 --> 00:12:54.659 practical constraints required to actually prove 00:12:54.659 --> 00:12:56.720 it. It brings up the work of Thomas Shelburne 00:12:56.720 --> 00:13:00.159 Ferguson from his 1994 text, Mathematical Statistics, 00:13:00.659 --> 00:13:02.730 a decision theoretic approach. Yeah, this stub 00:13:02.730 --> 00:13:04.850 highlights a very specific theorem from Ferguson. 00:13:05.330 --> 00:13:07.490 It reads, quote, if the sample space is closed 00:13:07.490 --> 00:13:10.070 and the parameter space is finite, then the class 00:13:10.070 --> 00:13:12.429 of Bayes rules is complete. We really need to 00:13:12.429 --> 00:13:13.909 elevate the discussion here, because these aren't 00:13:13.909 --> 00:13:16.990 just casual terms. No, they are incredibly strict 00:13:16.990 --> 00:13:21.110 boundaries. Ferguson is outlining the exact conditions 00:13:21.110 --> 00:13:24.139 required for the math to be ironclad. When he 00:13:24.139 --> 00:13:26.720 demands a closed sample space, he means that 00:13:26.720 --> 00:13:29.879 the set of all possible outcomes must be strictly 00:13:29.879 --> 00:13:32.720 bounded and known in advance. So no surprises. 00:13:32.940 --> 00:13:36.200 Exactly. You cannot have unexpected black swan 00:13:36.200 --> 00:13:38.860 events. Right. You are essentially demanding 00:13:38.860 --> 00:13:42.159 a ceiling on reality. You have to know every 00:13:42.159 --> 00:13:43.919 single state of nature that could possibly occur 00:13:43.919 --> 00:13:46.120 before you even make your decision. Yes. And 00:13:46.120 --> 00:13:48.340 the second constraint, demanding that the parameter 00:13:48.340 --> 00:13:50.860 space is finite, means you must be dealing with 00:13:50.860 --> 00:13:53.679 a limited, countable number of variables. This 00:13:53.679 --> 00:13:55.399 raises an important question about the limits 00:13:55.399 --> 00:13:58.019 of mathematical models. Why did Ferguson have 00:13:58.019 --> 00:14:00.740 to artificially bind the space to prove the theorem? 00:14:01.080 --> 00:14:02.600 Why couldn't he just prove it for everything? 00:14:02.779 --> 00:14:05.200 Because without a finite parameter space, you 00:14:05.200 --> 00:14:07.419 run straight into the problem of computational 00:14:07.419 --> 00:14:10.399 intractability. In an unbounded environment, 00:14:10.700 --> 00:14:13.120 the number of prior probabilities you have to 00:14:13.120 --> 00:14:15.919 constantly update expands exponentially. The 00:14:15.919 --> 00:14:18.440 math just explodes. Here's where it gets really 00:14:18.440 --> 00:14:21.820 interesting. because real life is almost never 00:14:21.820 --> 00:14:24.419 a closed sample space with finite parameters. 00:14:25.000 --> 00:14:27.419 Never. Operating under Ferguson's constraints 00:14:27.419 --> 00:14:29.360 is like trying to calculate the optimal route 00:14:29.360 --> 00:14:31.840 to work, but doing so under the assumption that 00:14:31.840 --> 00:14:34.019 the roads will always remain exactly as they 00:14:34.019 --> 00:14:37.399 are. Right. But in reality, the city planners 00:14:37.399 --> 00:14:40.500 are actively adding new roads, changing the speed 00:14:40.500 --> 00:14:43.159 limits, and physically moving your office building 00:14:43.159 --> 00:14:45.960 while you are currently driving. The parameter 00:14:45.960 --> 00:14:49.009 space is constantly, endlessly expanding. That 00:14:49.009 --> 00:14:52.049 is an excellent way to conceptualize it. In a 00:14:52.049 --> 00:14:54.970 closed finite system like a board game or maybe 00:14:54.970 --> 00:14:57.409 a highly controlled clinical trial, the complete 00:14:57.409 --> 00:15:00.129 class theorem holds perfectly. You can map every 00:15:00.129 --> 00:15:02.909 single variable. Surely in chess. Exactly. But 00:15:02.909 --> 00:15:05.769 as you transition into macroeconomics or global 00:15:05.769 --> 00:15:08.750 politics or honestly even just navigating a messy 00:15:08.750 --> 00:15:11.570 interpersonal conflict, you are introducing infinite 00:15:11.570 --> 00:15:14.529 parameters and open -ended sample spaces. The 00:15:14.529 --> 00:15:16.690 pure pristine application of the theorem becomes 00:15:16.690 --> 00:15:19.539 impossible to execute in real time. Even if the 00:15:19.539 --> 00:15:21.879 underlying logic remains completely sound, you 00:15:21.879 --> 00:15:23.980 just simply cannot compute an infinite number 00:15:23.980 --> 00:15:26.820 of variables. Which is exactly why advanced decision 00:15:26.820 --> 00:15:29.019 theory struggles to create perfectly predictive 00:15:29.019 --> 00:15:32.220 models for everyday human behavior. The theory 00:15:32.220 --> 00:15:34.679 is complete, but our capacity to measure the 00:15:34.679 --> 00:15:38.840 parameters is profoundly incomplete. Wow. So, 00:15:39.000 --> 00:15:40.779 to summarize the intellectual journey we have 00:15:40.779 --> 00:15:42.919 taken today, and remember, this is all based 00:15:42.919 --> 00:15:46.379 on just a few short sentences from a stub. We 00:15:46.379 --> 00:15:49.299 started with the concept of admissibility. learning 00:15:49.299 --> 00:15:51.980 that the very first step of rationality is simply 00:15:51.980 --> 00:15:55.039 eliminating choices where a strictly better alternative 00:15:55.039 --> 00:15:58.200 exists. Sweeping the board clean. Right. And 00:15:58.200 --> 00:16:01.200 from there, we discovered that every single optimal 00:16:01.200 --> 00:16:04.519 valid decision that survives that filter is fundamentally 00:16:04.519 --> 00:16:07.190 a Bayesian calculation. We are constantly weighing 00:16:07.190 --> 00:16:09.590 our utility functions against our prior knowledge. 00:16:09.789 --> 00:16:12.029 We learned that there is no magical third option. 00:16:12.210 --> 00:16:14.309 You are either updating your probabilities based 00:16:14.309 --> 00:16:16.830 on evidence to maximize your outcomes, or you 00:16:16.830 --> 00:16:19.049 are actively leaving a better option on the table. 00:16:19.350 --> 00:16:21.830 And we saw how statisticians like Sergusson had 00:16:21.830 --> 00:16:24.549 to artificially freeze reality into these closed, 00:16:24.850 --> 00:16:28.090 finite boxes just to mathematically prove the 00:16:28.090 --> 00:16:30.649 completeness of these rules, which really highlights 00:16:30.649 --> 00:16:34.409 the massive gap between pure math and the absolute 00:16:34.409 --> 00:16:36.590 chaos of the real world. It is a comprehensive 00:16:36.590 --> 00:16:38.669 framework for understanding choice. It really 00:16:38.669 --> 00:16:40.789 changes how you see things. It does. It really 00:16:40.789 --> 00:16:43.309 does. But before we finish this deep dive, there 00:16:43.309 --> 00:16:46.370 is one final easily overlooked detail in the 00:16:46.370 --> 00:16:47.970 source text that I want you, the listener, to 00:16:47.970 --> 00:16:51.830 consider. Ah, yes. It is buried in a tiny parenthetical 00:16:51.830 --> 00:16:57.190 aside in the very first paragraph. And it just... 00:16:56.809 --> 00:16:59.450 beautifully reframes this entire struggle between 00:16:59.450 --> 00:17:02.149 perfect math and imperfect reality. You're talking 00:17:02.149 --> 00:17:04.490 about the caveat about the sequence. Yes. The 00:17:04.490 --> 00:17:06.990 text states that an admissible rule is equivalent 00:17:06.990 --> 00:17:09.670 to a Bayesian rule for a specific utility in 00:17:09.670 --> 00:17:12.069 prior distribution and then it quietly adds in 00:17:12.069 --> 00:17:14.569 parentheses or for the limit of a sequence of 00:17:14.569 --> 00:17:16.910 prior distributions. That one phrase mathematically 00:17:16.910 --> 00:17:19.799 acknowledges the passage of time. Exactly. By 00:17:19.799 --> 00:17:22.240 introducing the limit of a sequence, the theorem 00:17:22.240 --> 00:17:24.319 acknowledges that our prior knowledge is not 00:17:24.319 --> 00:17:27.420 just some static snapshot. It is a sequence. 00:17:27.799 --> 00:17:31.019 It's a continuous dynamic flow of data that is 00:17:31.019 --> 00:17:33.900 constantly updating with every single micro interaction 00:17:33.900 --> 00:17:36.720 we have with the world. And in calculus, a limit 00:17:36.720 --> 00:17:38.819 represents a value that a sequence approaches 00:17:38.819 --> 00:17:41.859 as the inputs approach infinity. It is a horizon 00:17:41.859 --> 00:17:44.140 line. You can get closer and closer, but you 00:17:44.140 --> 00:17:46.160 never actually touch it. So I really want you 00:17:46.160 --> 00:17:48.900 to ponder what that parenthetical actually implies 00:17:48.900 --> 00:17:52.500 for human rationality. If making the ultimate 00:17:52.500 --> 00:17:55.500 mathematically perfect choice requires us to 00:17:55.500 --> 00:17:57.400 chase the theoretical limit of an infinitely 00:17:57.400 --> 00:18:00.579 updating sequence of prior assumptions, at what 00:18:00.579 --> 00:18:03.059 point does our biology force us to fail? We are 00:18:03.059 --> 00:18:06.000 bounded by time and cognitive capacity. We literally 00:18:06.000 --> 00:18:08.539 cannot compute to infinity. Precisely. At what 00:18:08.539 --> 00:18:10.619 point does our human inability to process an 00:18:10.619 --> 00:18:13.440 infinite sequence force us into making mathematically 00:18:13.440 --> 00:18:16.160 inadmissible choices simply because we just ran 00:18:16.160 --> 00:18:17.940 out of time to run the Bayesian update? We have 00:18:17.940 --> 00:18:20.119 to act eventually. Right, we have to choose a 00:18:20.119 --> 00:18:22.819 career or buy a house or pick a partner now. 00:18:23.319 --> 00:18:25.619 Even though the sequence of data is still flowing, 00:18:26.140 --> 00:18:29.079 we are forced to be imperfect. Not because the 00:18:29.079 --> 00:18:31.339 theorem is flawed, but because the clock runs 00:18:31.339 --> 00:18:33.599 out before we can reach the limit. It suggests 00:18:33.599 --> 00:18:36.720 that true mathematical optimality isn't a destination 00:18:36.720 --> 00:18:39.740 we can ever actually reach. It is merely an asymptote 00:18:39.740 --> 00:18:42.269 we continually strive toward in the dark. And 00:18:42.269 --> 00:18:45.089 to think, we've found that profound realization 00:18:45.089 --> 00:18:47.910 hidden in a tiny Wikipedia scub, just a handful 00:18:47.910 --> 00:18:49.970 of sentences outlining the ultimate algorithm 00:18:49.970 --> 00:18:52.650 of choice. Keep that horizon line in mind the 00:18:52.650 --> 00:18:54.190 next time you are trying to make a difficult 00:18:54.190 --> 00:18:56.650 decision. You are just a biological, Bayesian 00:18:56.650 --> 00:18:59.049 engine doing your absolute best before the time 00:18:59.049 --> 00:18:59.529 runs out.