WEBVTT 00:00:00.000 --> 00:00:02.180 Every time you speak into your smartphone, you 00:00:02.180 --> 00:00:04.139 know, and it magically transcribes your voice 00:00:04.139 --> 00:00:06.620 into a text message, it isn't actually hearing 00:00:06.620 --> 00:00:09.080 your words. Right. Not the way a human does anyway. 00:00:09.300 --> 00:00:11.140 Exactly. It's essentially blind to language. 00:00:11.439 --> 00:00:13.880 Instead, what that phone is doing is calculating 00:00:13.880 --> 00:00:17.000 this massive invisible web of probabilities in 00:00:17.000 --> 00:00:19.899 real time. It's just guessing, really. Guessing 00:00:19.899 --> 00:00:22.460 the most mathematically likely sequence of sounds 00:00:22.460 --> 00:00:25.920 based on this totally chaotic ocean of audio 00:00:25.920 --> 00:00:29.899 data. And today, we are taking you, the learner, 00:00:30.039 --> 00:00:33.020 inside the architecture of that randomness. Welcome 00:00:33.020 --> 00:00:35.880 to the Deep Dive. Glad to be here. So our mission 00:00:35.880 --> 00:00:39.539 today is to decode a rather dense, highly technical 00:00:39.539 --> 00:00:41.619 Wikipedia article we have. have in our sources. 00:00:42.039 --> 00:00:44.799 It breaks down this secret cheat code that data 00:00:44.799 --> 00:00:47.700 scientists and statisticians use to map out complex, 00:00:47.820 --> 00:00:49.500 chaotic systems. Yeah, it's all about something 00:00:49.500 --> 00:00:52.500 called probabilistic graphical models, or PGMs 00:00:52.500 --> 00:00:55.179 for short. PGMs, right. And the goal here is 00:00:55.179 --> 00:00:57.159 to give you a mental cheat sheet for how these 00:00:57.159 --> 00:00:59.960 actually work. Because it is an incredibly powerful 00:00:59.960 --> 00:01:02.280 framework. I mean, when you're trying to understand 00:01:02.280 --> 00:01:06.060 any real world situation where dozens or even 00:01:06.060 --> 00:01:08.519 hundreds of different random variables are interacting 00:01:08.519 --> 00:01:11.450 all at once, the the math just becomes paralyzing. 00:01:11.489 --> 00:01:13.870 Oh, I can imagine. Yeah, so a graphical model 00:01:13.870 --> 00:01:17.849 is the tool that cuts through that noise. To 00:01:17.849 --> 00:01:20.329 visualize what we're talking about today, I want 00:01:20.329 --> 00:01:23.750 you to imagine stepping into a vast dimly lit 00:01:23.750 --> 00:01:26.250 room. Okay, setting the scene, I like it. Right, 00:01:26.450 --> 00:01:28.790 and floating all around you is this glowing, 00:01:29.349 --> 00:01:31.730 highly complex, detective style string board. 00:01:31.829 --> 00:01:33.909 Oh wow. You've got these push bins representing 00:01:33.909 --> 00:01:36.030 different variables, and they are connected by 00:01:36.030 --> 00:01:38.329 these pulsing threads of light, just weaving 00:01:38.329 --> 00:01:40.969 this massive three -dimensional web. I love that 00:01:40.969 --> 00:01:43.569 visual. So we're basically standing inside the 00:01:43.569 --> 00:01:45.870 brain of a machine learning algorithm, just looking 00:01:45.870 --> 00:01:48.569 at how it maps the world. Exactly. That's exactly 00:01:48.569 --> 00:01:50.129 what it is. Alright, let's start with the fundamental 00:01:50.129 --> 00:01:52.489 building blocks then. We have our floating push 00:01:52.489 --> 00:01:54.969 pins and our strings of light. The source material 00:01:54.969 --> 00:01:57.969 defines a graphical model as a graph that expresses 00:01:57.969 --> 00:02:00.569 the, and I'm quoting here, the conditional dependence 00:02:00.569 --> 00:02:03.650 structure between random variables. Right. It 00:02:03.650 --> 00:02:07.650 also says it provides a compact, factorized representation 00:02:07.650 --> 00:02:11.229 of a multi -dimensional space, which, you know, 00:02:11.270 --> 00:02:13.289 is a lot of heavy terminology right out of the 00:02:13.289 --> 00:02:15.409 gate. It is a mouthful. Let's break that down 00:02:15.409 --> 00:02:18.289 because understanding that one sentence is really 00:02:18.289 --> 00:02:20.009 the key to everything else we're going to talk 00:02:20.009 --> 00:02:23.300 about. Please do. So. In statistics, a random 00:02:23.300 --> 00:02:25.539 variable is really just something that can take 00:02:25.539 --> 00:02:27.860 on different values. And we aren't completely 00:02:27.860 --> 00:02:29.419 sure what value it's going to take. It could 00:02:29.419 --> 00:02:31.639 be a coin. Yeah, or like the weather tomorrow, 00:02:32.060 --> 00:02:35.159 the price of a stock, or, to use the detective 00:02:35.159 --> 00:02:37.639 board analogy, the identity of a suspect in a 00:02:37.639 --> 00:02:41.180 crime. Got it. Now, if you have 100 of these 00:02:41.180 --> 00:02:44.199 variables, trying to calculate the probability 00:02:44.199 --> 00:02:46.919 of every single possible combination of events 00:02:46.919 --> 00:02:50.110 happening at the exact same time, computationally 00:02:50.110 --> 00:02:52.409 impossible just too many variables way too many 00:02:52.409 --> 00:02:54.650 even our most powerful supercomputers would choke 00:02:54.650 --> 00:02:57.819 on it The math just scales exponentially into 00:02:57.819 --> 00:03:01.379 infinity. So the graph, this PGM, is basically 00:03:01.379 --> 00:03:03.900 a way of cheating that math. It's a way of isolating 00:03:03.900 --> 00:03:06.419 what actually matters. What's fascinating here 00:03:06.419 --> 00:03:09.099 is that it bridges graph theory and probability. 00:03:09.879 --> 00:03:11.599 Instead of assuming everything in the universe 00:03:11.599 --> 00:03:14.539 affects everything else equally, the graph maps 00:03:14.539 --> 00:03:19.500 out specific relationships. It factors the massive, 00:03:19.740 --> 00:03:22.639 incomprehensible problem into bite -sized, solvable 00:03:22.639 --> 00:03:26.240 pieces. Oh, I see. the math along the strings 00:03:26.240 --> 00:03:29.580 that are actually connected. By doing that, it 00:03:29.580 --> 00:03:32.099 encodes a distribution over a multi -dimensional 00:03:32.099 --> 00:03:34.560 space without having to calculate the whole space 00:03:34.560 --> 00:03:37.240 at once. Okay, let's unpack this and ground it 00:03:37.240 --> 00:03:40.120 in that detective string board analogy. Say we 00:03:40.120 --> 00:03:44.099 have three suspects. Suspect A, Suspect B, and 00:03:44.099 --> 00:03:46.900 Suspect C. Right. The classic trio. Right. And 00:03:46.900 --> 00:03:49.060 you're the detective trying to figure out who 00:03:49.060 --> 00:03:51.900 committed a robbery. Initially, you have strings 00:03:51.900 --> 00:03:53.639 connecting all three of them because they all 00:03:53.639 --> 00:03:55.960 run in the same circles. Makes sense. But then 00:03:55.960 --> 00:03:58.340 you uncover airtight security footage proving 00:03:58.340 --> 00:04:00.780 Suspect A was in a completely different country 00:04:00.780 --> 00:04:02.659 at the time of the robbery. So now the value 00:04:02.659 --> 00:04:05.300 of variable A is known. It's an absolute certainty. 00:04:05.300 --> 00:04:07.699 Right. And because we know the absolute truth 00:04:07.699 --> 00:04:10.919 about A, Let's say we realized that suspect B 00:04:10.919 --> 00:04:13.840 and suspect C's possible motives had everything 00:04:13.840 --> 00:04:16.100 to do with A's involvement. Okay, following you. 00:04:16.379 --> 00:04:18.899 So without A, B, and C actually have nothing 00:04:18.899 --> 00:04:21.139 to do with each other. If I were visualizing 00:04:21.139 --> 00:04:23.699 this on a physical board, my instinct would be 00:04:23.699 --> 00:04:26.019 to just take scissors and cut the string between 00:04:26.019 --> 00:04:29.379 B and C forever. Is that conditional independence? 00:04:29.660 --> 00:04:31.759 Well, let's refine that slightly. Because the 00:04:31.759 --> 00:04:33.740 underlying relationship doesn't just cease to 00:04:33.740 --> 00:04:35.930 exist in the real world. Oh, it doesn't. No, 00:04:36.009 --> 00:04:38.129 think of the connection between them. Less like 00:04:38.129 --> 00:04:40.629 a physical string that gets destroyed and more 00:04:40.629 --> 00:04:44.949 like a road with a toll booth. Okay, a toll booth. 00:04:45.170 --> 00:04:47.149 I'm tracking. In conditional independence, the 00:04:47.149 --> 00:04:49.269 flow of new information is what gets blocked. 00:04:50.269 --> 00:04:52.850 If suspect A is the toll booth and we don't know 00:04:52.850 --> 00:04:55.709 anything about A yet, information about suspect 00:04:55.709 --> 00:04:58.129 B's whereabouts might give us clues about C. 00:04:58.290 --> 00:05:00.509 Because they're connected through A. Exactly. 00:05:00.790 --> 00:05:03.399 The traffic is flowing freely between them. But 00:05:03.399 --> 00:05:05.399 the moment we lock down the truth about suspect 00:05:05.399 --> 00:05:08.199 A, the moment we close the toll booth, no new 00:05:08.199 --> 00:05:11.500 information can travel from B to C. Wow. Knowing 00:05:11.500 --> 00:05:13.920 more about B tells us absolutely nothing new 00:05:13.920 --> 00:05:16.939 about C. The road is blocked. And that is the 00:05:16.939 --> 00:05:19.600 essence of conditional independence. That makes 00:05:19.600 --> 00:05:22.629 so much more sense. It's about the flow of information. 00:05:23.290 --> 00:05:25.509 So once we establish that we are drawing this 00:05:25.509 --> 00:05:27.689 map of connected variables, we have to look at 00:05:27.689 --> 00:05:29.589 the strings themselves. Right, the connections. 00:05:29.750 --> 00:05:32.649 And looking at the concepts in our sources, there's 00:05:32.649 --> 00:05:36.129 a major fork in the road here. The entire field 00:05:36.129 --> 00:05:39.949 basically splits into two main families of models 00:05:39.949 --> 00:05:42.769 based on one really simple visual detail. Yep. 00:05:43.230 --> 00:05:46.089 Do the strings have arrows or not? Exactly. Do 00:05:46.089 --> 00:05:48.449 the strings connecting these variables have arrows 00:05:48.449 --> 00:05:50.569 pointing in a specific direction, or are they 00:05:50.569 --> 00:05:52.649 just plane lines? These are the two heavyweights 00:05:52.649 --> 00:05:55.250 of the PGM world. Let's tackle the plane lines 00:05:55.250 --> 00:05:57.709 first. Okay. These are called undirected graphical 00:05:57.709 --> 00:06:02.110 models, or Markov random fields. In an undirected 00:06:02.110 --> 00:06:04.949 graph, the presence of an edge, just a simple 00:06:04.949 --> 00:06:07.389 line between two nodes, implies that there is 00:06:07.389 --> 00:06:09.759 a mutual relationship or correlation. between 00:06:09.759 --> 00:06:11.620 those variables. So just a general link, the 00:06:11.620 --> 00:06:13.800 source actually gives a conceptual example of 00:06:13.800 --> 00:06:16.680 four nodes, A, B, C, and D. Let's say node A 00:06:16.680 --> 00:06:19.519 connects to B, C, and D, but there are no lines 00:06:19.519 --> 00:06:21.980 connecting B, C, and D directly to each other. 00:06:22.040 --> 00:06:24.100 Right. What is that structure actually doing 00:06:24.100 --> 00:06:27.040 mathematically? It's establishing our tollbooth 00:06:27.040 --> 00:06:30.240 rule on a larger scale. It tells us that B, C, 00:06:30.240 --> 00:06:32.800 and D are conditionally independent given A. 00:06:32.959 --> 00:06:35.500 So A is the tollbooth for all of them. Exactly. 00:06:35.560 --> 00:06:38.459 If we lock down the value of A, The variables 00:06:38.459 --> 00:06:41.680 b, c, and d are totally isolated from one another. 00:06:42.319 --> 00:06:45.560 In a Markov random field, the math focuses on 00:06:45.560 --> 00:06:48.579 pairs of connected nodes. You look at a and b, 00:06:48.980 --> 00:06:51.259 and you evaluate how compatible their states 00:06:51.259 --> 00:06:54.129 are. Yeah, the text mentions a specific formula 00:06:54.129 --> 00:06:56.490 for this, where you calculate the joint probability 00:06:56.490 --> 00:06:59.509 by multiplying functions of those connected pairs, 00:06:59.589 --> 00:07:01.629 like function of A and B times function of A 00:07:01.629 --> 00:07:03.670 and C. Yeah, the non -negative functions. But, 00:07:03.689 --> 00:07:06.170 you know, reading math equations aloud is a surefire 00:07:06.170 --> 00:07:08.870 way to lose anyone listening. How do we visualize 00:07:08.870 --> 00:07:10.910 what that calculation is actually doing without 00:07:10.910 --> 00:07:12.970 getting totally bogged down in the notation? 00:07:13.129 --> 00:07:15.709 Think of it as a scoring system for a group project. 00:07:15.850 --> 00:07:17.970 Oh. You don't need to evaluate how the entire 00:07:17.970 --> 00:07:20.089 company works together all at once. You just 00:07:20.089 --> 00:07:21.509 look at the pairs of people who who actually 00:07:21.509 --> 00:07:23.370 sit next to each other. Oh, I like that. Yeah. 00:07:23.569 --> 00:07:25.889 So you give a compatibility score to A and B 00:07:25.889 --> 00:07:28.290 working together. Then you give a compatibility 00:07:28.290 --> 00:07:30.350 score to A and C working together. And then you 00:07:30.350 --> 00:07:32.589 just multiply them. Exactly. You multiply all 00:07:32.589 --> 00:07:35.829 those local pairwise scores together, and it 00:07:35.829 --> 00:07:38.110 gives you a global score for how well the whole 00:07:38.110 --> 00:07:41.089 system is functioning. It's a way of calculating 00:07:41.089 --> 00:07:44.290 the probability of the whole based purely on 00:07:44.290 --> 00:07:47.149 the harmony of the local neighborhoods. a local 00:07:47.149 --> 00:07:49.149 neighborhood scoring system. I really like that. 00:07:49.149 --> 00:07:51.529 It feels very symmetrical and neat. It is. But 00:07:51.529 --> 00:07:54.629 then we cross the aisle to the directed models. 00:07:54.670 --> 00:07:57.850 These are called directed acyclic graphs or Bayesian 00:07:57.850 --> 00:08:01.009 networks. And here, the strings are no longer 00:08:01.009 --> 00:08:04.009 just plain lines. They have arrows. And those 00:08:04.009 --> 00:08:06.370 arrows fundamentally change the interpretation 00:08:06.370 --> 00:08:08.250 of the map. Because they enforce a direction. 00:08:08.370 --> 00:08:10.389 Right. In a Bayesian network, we aren't just 00:08:10.389 --> 00:08:12.810 looking at mutual correlation anymore. We are 00:08:12.810 --> 00:08:15.730 looking at influence. The node where the arrow 00:08:15.689 --> 00:08:18.410 starts is called a parent and the node the arrow 00:08:18.410 --> 00:08:21.269 points to is the child. See, when I hear parent 00:08:21.269 --> 00:08:24.250 and child, my mind immediately goes to a family 00:08:24.250 --> 00:08:28.089 tree. It implies that this variable precedes 00:08:28.089 --> 00:08:30.410 or physically creates that one. Is that the right 00:08:30.410 --> 00:08:33.570 way to think about a directed graph? If we connect 00:08:33.570 --> 00:08:36.129 this to the bigger picture, it's a very useful 00:08:36.129 --> 00:08:38.919 starting point. But it's better to think of it 00:08:38.919 --> 00:08:41.419 strictly as an influence or a generative process. 00:08:41.620 --> 00:08:43.840 Okay, influence. The family tree analogy works 00:08:43.840 --> 00:08:46.100 because time and genetics flow in one direction. 00:08:46.120 --> 00:08:48.059 Right. Your grandfather influences your father 00:08:48.059 --> 00:08:50.679 and your father influences you. Right. In a Bayesian 00:08:50.679 --> 00:08:53.419 network, the probability of the entire system 00:08:53.419 --> 00:08:55.820 is calculated by looking at the probability of 00:08:55.820 --> 00:08:58.240 each child variable, given the state of its parents. 00:08:58.379 --> 00:09:01.259 So the math is literally probability of X, given 00:09:01.259 --> 00:09:03.879 the parents of X. Exactly. The conditionality 00:09:03.879 --> 00:09:05.840 is baked directly into the direction of the arrow. 00:09:06.059 --> 00:09:09.480 The text mentions naive Bayes classifiers, hidden 00:09:09.480 --> 00:09:12.120 Markov models, neural networks. They all use 00:09:12.120 --> 00:09:14.700 this directed flow. So if undirected graphs are 00:09:14.700 --> 00:09:17.179 good for things sitting at the same table, directed 00:09:17.179 --> 00:09:19.700 graphs are for a sequence of events. But the 00:09:19.700 --> 00:09:22.419 source specifically calls them directed acyclic 00:09:22.419 --> 00:09:25.799 graphs. Acyclic meaning what? No cycles. Right. 00:09:25.919 --> 00:09:28.460 No cycles. You can't loop back. That seems like 00:09:28.460 --> 00:09:30.919 a big rule. It is a strict rule for Bayesian 00:09:30.919 --> 00:09:33.980 networks. You can never start at a node. Follow 00:09:33.980 --> 00:09:36.259 the arrows and end up back where you started. 00:09:36.580 --> 00:09:38.799 You can't be your own ancestor. That makes sense. 00:09:39.039 --> 00:09:42.299 Time and influence must march forward. And because 00:09:42.299 --> 00:09:45.000 of this strict one -way flow, Bayesian networks 00:09:45.000 --> 00:09:48.019 allow for a very specific type of analysis called 00:09:48.279 --> 00:09:51.139 D separation. Wait. D separation. You totally 00:09:51.139 --> 00:09:53.419 lost me there. The text mentioned it briefly 00:09:53.419 --> 00:09:56.139 as a criterion to determine if sets of nodes 00:09:56.139 --> 00:09:59.019 are conditionally independent. But how does looking 00:09:59.019 --> 00:10:01.740 at the direction of arrows separate anything? 00:10:01.840 --> 00:10:03.940 It's a fantastic concept, really. And it goes 00:10:03.940 --> 00:10:06.200 back to our toll booth analogy, but this time 00:10:06.200 --> 00:10:08.340 with directional traffic. Okay. Directional toll 00:10:08.340 --> 00:10:10.860 booths. Right. D separation stands for directed 00:10:10.860 --> 00:10:13.259 separation. Let's imagine a simple chain of three 00:10:13.259 --> 00:10:16.019 events. Event A is your alarm clock failing to 00:10:16.019 --> 00:10:19.419 go off. Oh no. B is you being late for work. 00:10:19.820 --> 00:10:22.659 And event C is you getting fired. Wow. A very 00:10:22.659 --> 00:10:24.720 stressful directed graph today. Very stressful, 00:10:24.899 --> 00:10:27.820 yeah. Now the arrow points from alarm, which 00:10:27.820 --> 00:10:31.320 is A, to late, which is B, and from late to fired, 00:10:31.340 --> 00:10:34.879 which is C. Got it. A to B to C. If you don't 00:10:34.879 --> 00:10:36.919 know anything about whether someone was late, 00:10:37.379 --> 00:10:40.120 knowing their alarm failed gives you a pretty 00:10:40.120 --> 00:10:43.139 strong clue that they might get fired. The information 00:10:43.139 --> 00:10:46.000 flows all the way down the chain. Makes total 00:10:46.000 --> 00:10:50.110 sense. But... What if you already know for an 00:10:50.110 --> 00:10:52.730 absolute fact that the person was late for work? 00:10:53.250 --> 00:10:56.529 You have observed variable B. Okay, so I know 00:10:56.529 --> 00:10:59.809 B is true. In the rules of deseparation, observing 00:10:59.809 --> 00:11:02.549 the middle node in a chain blocks the flow of 00:11:02.549 --> 00:11:05.409 information. If I already know you were late, 00:11:05.570 --> 00:11:08.070 discovering that your alarm failed doesn't change 00:11:08.070 --> 00:11:10.549 my probability of you getting fired. Oh wow, 00:11:10.690 --> 00:11:12.629 because being late is the thing that gets you 00:11:12.629 --> 00:11:15.250 fired, not the alarm. Exactly. The late node 00:11:15.250 --> 00:11:17.710 already contains all the predictive power. The 00:11:17.710 --> 00:11:20.490 arrows dictate exactly how and when that information 00:11:20.490 --> 00:11:23.110 gets blocked. So by tracing the arrows, an algorithm 00:11:23.110 --> 00:11:25.269 can instantly know which variables to ignore 00:11:25.269 --> 00:11:27.049 and which ones to actually pay attention to, 00:11:27.070 --> 00:11:29.110 which must save massive amounts of competing 00:11:29.110 --> 00:11:31.230 power. Huge amounts, yeah. That is brilliant. 00:11:31.370 --> 00:11:33.250 But my detective brain is flashing a warning 00:11:33.250 --> 00:11:35.830 light right now. Uh -oh. What's the issue? Well, 00:11:35.889 --> 00:11:38.789 both of these heavyweights, the undirected mutual 00:11:38.789 --> 00:11:41.929 models and the directed acyclic models, they 00:11:41.929 --> 00:11:44.740 both assume the world is perfectly neat. Oh, 00:11:44.840 --> 00:11:46.639 I see where you're going. Right. Like, either 00:11:46.639 --> 00:11:48.500 everything is a two -way street or everything 00:11:48.500 --> 00:11:51.340 flows forward in one direction. But what happens 00:11:51.340 --> 00:11:53.779 when the real world is messy? What happens when 00:11:53.779 --> 00:11:56.779 things do loop back on themselves? That is exactly 00:11:56.779 --> 00:11:59.279 where we leave the neat, tidy world of the textbooks 00:11:59.279 --> 00:12:02.299 and enter the Wild West of graphical models. 00:12:02.379 --> 00:12:04.320 The Wild West? Yeah, because the researchers 00:12:04.320 --> 00:12:06.820 who build these systems, they know reality is 00:12:06.820 --> 00:12:08.940 messy. So they started breaking the foundational 00:12:08.940 --> 00:12:11.279 rules to accommodate that reality. And the text 00:12:11.279 --> 00:12:15.299 outlines several of these other types of models. 00:12:15.360 --> 00:12:18.440 The first one that jumps out is the cyclic -directed 00:12:18.440 --> 00:12:21.559 graphical model, which seems to directly violate 00:12:21.559 --> 00:12:23.600 the acyclic rule we just spent all that time 00:12:23.600 --> 00:12:26.399 establishing. It allows variables to depend on 00:12:26.399 --> 00:12:29.620 parents in a closed loop. It does. And from a 00:12:29.620 --> 00:12:32.059 purely mathematical standpoint, a cycle creates 00:12:32.059 --> 00:12:34.500 an enormous headache. Think about it if you're 00:12:34.500 --> 00:12:36.299 trying to calculate the probability of an event, 00:12:36.659 --> 00:12:39.019 but the event is constantly feeding back into 00:12:39.019 --> 00:12:42.120 its own cause. The standard rules of factorization 00:12:42.120 --> 00:12:45.360 we discussed completely break down. I mean, how 00:12:45.360 --> 00:12:48.059 do you calculate a score when the score keeps 00:12:48.059 --> 00:12:50.259 changing itself? It's like a snake eating its 00:12:50.259 --> 00:12:52.759 own tail. The example in the source is actually 00:12:52.759 --> 00:12:55.940 wild. It describes a situation where variable 00:12:55.940 --> 00:12:59.779 D depends on A, B, and C, but then C depends 00:12:59.779 --> 00:13:02.360 on B and D. Exactly. It creates this infinite 00:13:02.360 --> 00:13:04.840 feedback loop of influence. But think about real 00:13:04.840 --> 00:13:08.019 life for a second. Ecosystems are feedback loops. 00:13:08.840 --> 00:13:11.519 Predator and prey populations constantly influence 00:13:11.519 --> 00:13:14.200 each other in cycles. Oh, that's true. Economics 00:13:14.200 --> 00:13:17.240 is full of feedback loops. If inflation rises, 00:13:17.860 --> 00:13:20.139 central banks raise interest rates, which slows 00:13:20.139 --> 00:13:23.080 the economy, which then impacts inflation. You 00:13:23.080 --> 00:13:25.340 can't model that accurately with a strict one 00:13:25.340 --> 00:13:28.039 -way arrow. So the math just has to adapt. Right. 00:13:28.340 --> 00:13:30.799 Statisticians developed complex discovery algorithms 00:13:30.799 --> 00:13:34.200 specifically to handle these cyclic graphs, basically 00:13:34.200 --> 00:13:36.980 forcing the math to reconcile with reality. There's 00:13:36.980 --> 00:13:39.200 another hybrid in the source that I find absolutely 00:13:39.200 --> 00:13:41.500 fascinating, the chain graph. Oh yeah, that's 00:13:41.500 --> 00:13:43.500 a good one. It's described as a graph that allows 00:13:43.500 --> 00:13:46.379 both directed arrows and undirected lines in 00:13:46.379 --> 00:13:48.720 the same exact model. It's like the ultimate 00:13:48.720 --> 00:13:50.639 diplomatic treaty between the two heavyweights. 00:13:50.779 --> 00:13:53.480 Right, because it keeps the rule of no directed 00:13:53.480 --> 00:13:55.340 cycles, so you still can't follow the arrows 00:13:55.340 --> 00:13:58.700 back home. But along the way, you can have clusters 00:13:58.700 --> 00:14:01.360 of variables that just mutually interact. It 00:14:01.360 --> 00:14:03.299 gives you the best of both worlds, really. But 00:14:03.299 --> 00:14:05.360 how does that actually look in practice? Like, 00:14:05.360 --> 00:14:07.950 why would you need both in the same map. Okay. 00:14:08.049 --> 00:14:11.610 Imagine modeling a complex disease. At the top 00:14:11.610 --> 00:14:14.110 of the map, you have genetic predispositions. 00:14:14.690 --> 00:14:16.990 Those are directed arrows. Because you inherit 00:14:16.990 --> 00:14:19.649 them. Right. The influence flows strictly downward 00:14:19.649 --> 00:14:22.590 from your DNA to your cellular function. You 00:14:22.590 --> 00:14:24.929 can't change your DNA based on a cold you caught. 00:14:25.070 --> 00:14:27.289 Right. The arrow of biology only goes one way 00:14:27.289 --> 00:14:29.879 there. Exactly. But once you get down to the 00:14:29.879 --> 00:14:32.899 cellular level, you might have hundreds of different 00:14:32.899 --> 00:14:35.179 proteins and enzymes interacting with each other 00:14:35.179 --> 00:14:37.360 all at once. And they don't have a strict parent 00:14:37.360 --> 00:14:40.100 -child thing going on. No, one protein doesn't 00:14:40.100 --> 00:14:41.899 necessarily cause the other. They're just mutually 00:14:41.899 --> 00:14:45.120 reacting in a chemical soup. Those relationships 00:14:45.120 --> 00:14:47.940 are undirected. So a chain graph lets a data 00:14:47.940 --> 00:14:50.559 scientist model the strict downward flow of genetics 00:14:50.559 --> 00:14:54.200 while simultaneously capturing the mutual, undirected 00:14:54.200 --> 00:14:56.600 correlation of the immediate cellular environment. 00:14:56.840 --> 00:14:59.539 That is incredibly elegant. It really shows the 00:14:59.539 --> 00:15:01.659 adaptability of the math. I mean, they've mutated 00:15:01.659 --> 00:15:04.059 the basic graph concept to fit whatever messy 00:15:04.059 --> 00:15:06.820 data they have. The text mentions a few honorable 00:15:06.820 --> 00:15:08.980 engines too, like tree -augmented classifiers, 00:15:09.240 --> 00:15:11.820 TAN models, and targeted Bayesian network learning, 00:15:12.059 --> 00:15:15.240 or TBNL. Right, and ancestral graphs, which add 00:15:15.240 --> 00:15:18.039 bi -directed edges, and restrictive Boltzmann 00:15:18.039 --> 00:15:20.440 machines, which are these bipartite generative 00:15:20.440 --> 00:15:22.480 models. Lots of specialized tools. But I want 00:15:22.480 --> 00:15:24.559 to zero in on one specific concept it mentions 00:15:24.559 --> 00:15:27.360 before we move on. Factor graphs and something 00:15:27.360 --> 00:15:30.659 called belief propagation. Ah. belief propagation. 00:15:30.899 --> 00:15:33.120 The text notes that factor graphs are undirected 00:15:33.120 --> 00:15:35.500 by partite graphs used to implement this belief 00:15:35.500 --> 00:15:38.100 propagation. But, as we established earlier, 00:15:38.159 --> 00:15:40.679 we want to avoid just listing terms. We need 00:15:40.679 --> 00:15:43.000 to know how this works, what is a factor graph 00:15:43.000 --> 00:15:45.720 doing, and what exactly is a belief propagating. 00:15:46.080 --> 00:15:49.059 Okay, to understand belief propagation, we need 00:15:49.059 --> 00:15:53.179 a good analogy. Imagine you're at a massive crowded 00:15:53.179 --> 00:15:55.820 cocktail party in a sprawling mansion. Sounds 00:15:55.820 --> 00:15:58.500 fun. You're deep inside one of the interior rooms 00:15:58.500 --> 00:16:00.539 and you want to know if it's raining outside. 00:16:00.879 --> 00:16:03.059 You don't have a window and you don't have your 00:16:03.059 --> 00:16:06.360 phone. Okay, so I'm a node in a graphical model 00:16:06.360 --> 00:16:08.379 trying to figure out the state of a variable 00:16:08.379 --> 00:16:11.100 I can't observe directly. Look at Q speaking 00:16:11.100 --> 00:16:14.059 the language. Exactly. Now in a graphical model 00:16:14.059 --> 00:16:16.139 you can only communicate with the nodes you're 00:16:16.139 --> 00:16:18.519 directly connected to. So you ask the person 00:16:18.519 --> 00:16:20.840 standing next to you. Hey, do you think it's 00:16:20.840 --> 00:16:22.919 raining? And they don't know either. Right, they 00:16:22.919 --> 00:16:24.460 don't know, so they ask the person next to them 00:16:24.460 --> 00:16:26.879 who asks the person next to them. It's just a 00:16:26.879 --> 00:16:29.039 giant game of telephone. But it's smarter than 00:16:29.039 --> 00:16:32.340 telephone. Eventually, someone near the entryway 00:16:32.340 --> 00:16:35.379 notices a guest walking in with a wet umbrella. 00:16:35.779 --> 00:16:39.440 Ah, new data. Yes. That person updates their 00:16:39.440 --> 00:16:41.519 belief about the weather to almost certainly 00:16:41.519 --> 00:16:44.629 raining. They pass that updated belief back to 00:16:44.629 --> 00:16:46.490 the person next to them. Hey, I just saw a wet 00:16:46.490 --> 00:16:49.070 umbrella. I'm 90 % sure it's raining Okay, so 00:16:49.070 --> 00:16:52.509 the message travels back and as it travels That 00:16:52.509 --> 00:16:55.210 person combines that message with what they're 00:16:55.210 --> 00:16:58.309 hearing from someone else who maybe heard thunder 00:16:59.390 --> 00:17:02.490 They synthesize these messages, update their 00:17:02.490 --> 00:17:05.150 own belief, and pass the new synthesis along 00:17:05.150 --> 00:17:08.109 to you. And belief propagation is just that process 00:17:08.109 --> 00:17:10.630 happening mathematically. Yes, exactly. It's 00:17:10.630 --> 00:17:13.519 a message passing algorithm. all the variables 00:17:13.519 --> 00:17:16.140 in the network, continuously pass mathematical 00:17:16.140 --> 00:17:18.900 messages back and forth along the connected strengths. 00:17:18.960 --> 00:17:21.019 Just constantly updating. Right, updating their 00:17:21.019 --> 00:17:22.900 probabilities based on what their neighbors are 00:17:22.900 --> 00:17:25.339 telling them until the entire room reaches a 00:17:25.339 --> 00:17:27.420 consensus about the most likely state of the 00:17:27.420 --> 00:17:29.579 world outside. And where does the factor graph 00:17:29.579 --> 00:17:32.119 come in? A factor graph is just a specialized, 00:17:32.240 --> 00:17:34.839 highly efficient map designed specifically to 00:17:34.839 --> 00:17:37.160 organize how those messages are passed, making 00:17:37.160 --> 00:17:39.140 sure the computation doesn't just get totally 00:17:39.140 --> 00:17:41.980 overwhelmed by all the chatter. Okay, this brings 00:17:41.980 --> 00:17:45.180 us to the ultimate so -what of this entire deep 00:17:45.180 --> 00:17:47.700 dive. We've built these magnificent structures 00:17:47.700 --> 00:17:50.460 in our minds. We have tollbooths blocking information, 00:17:50.720 --> 00:17:53.359 we have local neighborhood scoring systems, we 00:17:53.359 --> 00:17:55.839 have diplomatic treaties between arrows and lines, 00:17:56.220 --> 00:17:58.779 and cocktail parties passing messages of belief. 00:17:59.059 --> 00:18:02.220 It's a busy world. It is, but outside of theoretical 00:18:02.220 --> 00:18:04.920 mathematics, what's this actually do? The source 00:18:04.920 --> 00:18:07.319 has a whole section dedicated to applications, 00:18:07.359 --> 00:18:10.450 and it claims this framework provides algorithms 00:18:10.450 --> 00:18:13.089 for quoting, again, discovering and analyzing 00:18:13.089 --> 00:18:15.930 structure in complex distributions. When you 00:18:15.930 --> 00:18:17.769 look at the list of applications, it really reads 00:18:17.769 --> 00:18:20.410 like a catalog of modern technological miracles. 00:18:21.049 --> 00:18:24.670 Causal inference, information extraction, decoding 00:18:24.670 --> 00:18:27.529 low -density parity check codes, modeling protein 00:18:27.529 --> 00:18:30.029 structures. Gene regulatory networks, too. Let's 00:18:30.029 --> 00:18:31.910 tie this all the way back to the hook from the 00:18:31.910 --> 00:18:34.299 beginning of our deep dive. Speech recognition. 00:18:34.819 --> 00:18:36.740 When I'm talking to my phone, how is it using 00:18:36.740 --> 00:18:39.279 these maps to understand me? So historically, 00:18:39.559 --> 00:18:41.680 speech recognition relied heavily on a specific 00:18:41.680 --> 00:18:44.299 type of directed graphical model called a hidden 00:18:44.299 --> 00:18:47.440 Markov model. When you speak, the phone receives 00:18:47.440 --> 00:18:51.039 a chaotic wave of acoustic noise. It has to guess 00:18:51.039 --> 00:18:54.059 the hidden state, which is the actual word you 00:18:54.059 --> 00:18:56.500 intended to say based on the observed state, 00:18:56.799 --> 00:18:59.279 which is the messy audio data. And it uses the 00:18:59.279 --> 00:19:01.380 arrows to figure out the context. It uses the 00:19:01.380 --> 00:19:03.960 conditional dependencies, yeah? The model knows 00:19:03.960 --> 00:19:06.000 the statistical rules of the English language. 00:19:06.720 --> 00:19:08.700 It knows that if the last hidden state it guessed 00:19:08.700 --> 00:19:11.700 was the word peanut, the probability that the 00:19:11.700 --> 00:19:13.779 next directed arrow points to the word butter 00:19:13.779 --> 00:19:17.299 is incredibly high. Right, peanut butter. Exactly. 00:19:17.940 --> 00:19:20.039 The probability that the next arrow points to 00:19:20.039 --> 00:19:23.140 elephant is near zero. It uses the structure 00:19:23.140 --> 00:19:25.539 of the graph to filter out the noise and lock 00:19:25.539 --> 00:19:27.920 in on the most probable sequence of words. That's 00:19:27.920 --> 00:19:30.180 amazing. The text also mentions computer vision. 00:19:30.279 --> 00:19:32.359 I imagine that uses a different map entirely. 00:19:32.700 --> 00:19:34.900 Usually computer vision relies on undirected 00:19:34.900 --> 00:19:37.420 models, like the Markov random fields we discussed 00:19:37.420 --> 00:19:39.859 earlier. Think about a digital photograph. Just 00:19:39.859 --> 00:19:42.019 a bunch of pixels. Right, it's just a grid of 00:19:42.019 --> 00:19:43.980 millions of pixels, each with a color value. 00:19:44.680 --> 00:19:47.019 If a computer is trying to identify a stop sign 00:19:47.019 --> 00:19:50.059 in a noisy image, it looks at the mutual correlation 00:19:50.059 --> 00:19:52.440 between neighboring pixels. Oh, the local neighborhood 00:19:52.440 --> 00:19:55.279 scoring system again? You got it. The algorithm 00:19:55.279 --> 00:19:57.910 knows that if one pretzel is red, The pixels 00:19:57.910 --> 00:20:00.289 immediately connected to it are statistically 00:20:00.289 --> 00:20:03.529 highly likely to also be red. So it groups them? 00:20:03.730 --> 00:20:06.549 Right. It uses those undirected connections to 00:20:06.549 --> 00:20:09.410 smooth out the noise, group the red pixels together, 00:20:09.910 --> 00:20:12.250 recognize that octagonal shape, and confidently 00:20:12.250 --> 00:20:16.549 conclude that is a stop sign. It finds the structure 00:20:16.549 --> 00:20:18.630 in the static. It is kind of blowing my mind 00:20:18.630 --> 00:20:21.309 that the underlying logic used to read my texts 00:20:21.309 --> 00:20:23.869 is the same underlying logic used by a self -driving 00:20:23.869 --> 00:20:26.500 car to not run a red light. It's universal. And 00:20:26.500 --> 00:20:29.940 the source also lists disease diagnosis. A medical 00:20:29.940 --> 00:20:32.180 software program using a Bayesian network can 00:20:32.180 --> 00:20:34.480 take dozens of disparate variables, a patient's 00:20:34.480 --> 00:20:37.339 age, a slight fever, a specific genetic marker, 00:20:37.819 --> 00:20:40.359 a recent travel history, and use those directed 00:20:40.359 --> 00:20:42.799 arrows to calculate the actual probability of 00:20:42.799 --> 00:20:45.779 a specific underlying disease. It is profound 00:20:45.779 --> 00:20:48.680 when you step back and look at it. This raises 00:20:48.680 --> 00:20:51.019 an important question about what it means to 00:20:51.019 --> 00:20:54.920 discover structure. How so? Well, what do humans 00:20:54.920 --> 00:20:57.700 speech? The pixels in a photograph and the symptoms 00:20:57.700 --> 00:21:00.640 of a rare disease have in common. To the naked 00:21:00.640 --> 00:21:03.759 eye, nothing. But to a probabilistic graphical 00:21:03.759 --> 00:21:06.359 model, they're all just complex distributions 00:21:06.359 --> 00:21:08.440 of random variables. They're all just pushpins 00:21:08.440 --> 00:21:11.319 on the board. Yes. It means that hidden inside 00:21:11.319 --> 00:21:14.039 the seemingly random chaos of speech patterns 00:21:14.039 --> 00:21:17.440 or genetic mutations or digital noise, there's 00:21:17.440 --> 00:21:21.019 an underlying structure of dependence. PGMs are 00:21:21.019 --> 00:21:23.279 just the flashlights we use to go into that dark 00:21:23.279 --> 00:21:25.519 room and find that structure. It's the invisible 00:21:25.519 --> 00:21:27.660 architecture running beneath the surface of modern 00:21:27.660 --> 00:21:30.160 science. It forces you to realize that very few 00:21:30.160 --> 00:21:32.740 things are truly entirely independent. The world 00:21:32.740 --> 00:21:35.039 is really just a web of conditional probabilities. 00:21:35.279 --> 00:21:36.980 Let's take a second to recap the journey we've 00:21:36.980 --> 00:21:39.119 been on because we have covered a massive amount 00:21:39.119 --> 00:21:41.319 of intellectual ground today. We really have. 00:21:41.380 --> 00:21:44.240 We started with a pile of random variables and 00:21:44.240 --> 00:21:46.240 learned that to cheat the impossible math of 00:21:46.240 --> 00:21:47.839 calculating everything at once, we have to build 00:21:47.839 --> 00:21:51.180 a map, a graph. to map the flow of information. 00:21:51.380 --> 00:21:53.980 Right. If the relationship is just a mutual correlation 00:21:53.980 --> 00:21:57.240 sitting at the same table, we use Planeline's 00:21:57.240 --> 00:21:59.900 undirected Markov models. And we evaluate them 00:21:59.900 --> 00:22:02.309 using those local neighborhood scores. But if 00:22:02.309 --> 00:22:04.690 there is a flow of influence, a parent -child 00:22:04.690 --> 00:22:07.589 dynamic driving events forward, we use directed 00:22:07.589 --> 00:22:10.569 arrows, Bayesian networks. We learned how observing 00:22:10.569 --> 00:22:13.490 a middle event can act like a toll booth, shutting 00:22:13.490 --> 00:22:16.430 down the flow of new information through deseparation. 00:22:17.569 --> 00:22:20.289 And when the world refuses to fit into neat boxes, 00:22:20.549 --> 00:22:23.630 we embrace the hybrids. We allow feedback loops 00:22:23.630 --> 00:22:26.589 with cyclic graphs. We forge diplomatic treaties 00:22:26.589 --> 00:22:29.569 with chain graphs. And we pass messages of belief 00:22:29.569 --> 00:22:32.519 through factor graphs. the math to the messiness 00:22:32.519 --> 00:22:35.079 of reality, and ultimately using it to decode 00:22:35.079 --> 00:22:38.640 the world around us, from our DNA to human speech. 00:22:38.829 --> 00:22:41.430 It is a dense, heavy topic, but the elegance 00:22:41.430 --> 00:22:43.789 of how it all connects is just incredible. Thank 00:22:43.789 --> 00:22:46.170 you for joining us on this exploration. We tackled 00:22:46.170 --> 00:22:48.789 a highly technical subject today, but hopefully 00:22:48.789 --> 00:22:50.990 you're walking away with a brand new lens through 00:22:50.990 --> 00:22:53.549 which to view the technology you use every single 00:22:53.549 --> 00:22:57.009 day. It certainly changes how you perceive randomness. 00:22:57.309 --> 00:22:59.490 You start seeing the underlying strings connecting 00:22:59.490 --> 00:23:01.369 everything. It really does. And actually, there 00:23:01.369 --> 00:23:03.910 is one final lingering thought I want to leave 00:23:03.910 --> 00:23:06.309 you with. Oh, I like these. The source notes 00:23:06.309 --> 00:23:09.480 over and over that graphical are used for discovering 00:23:09.480 --> 00:23:12.660 and analyzing structure in complex distributions. 00:23:13.339 --> 00:23:16.039 It makes you wonder, if you were to sit down 00:23:16.039 --> 00:23:19.019 right now and draw a probabilistic graphical 00:23:19.019 --> 00:23:22.200 model of your own life. Wow. OK. Yeah, mapping 00:23:22.200 --> 00:23:25.039 out all the seemingly random variables of your 00:23:25.039 --> 00:23:27.420 daily choices, your habits, your social environment, 00:23:27.519 --> 00:23:30.640 your stress levels, what hidden dependencies 00:23:30.640 --> 00:23:33.640 would you discover? What unexpected parent nodes 00:23:33.640 --> 00:23:35.940 are secretly acting as toll booths in your life, 00:23:36.339 --> 00:23:37.900 controlling your outcomes and pulling the strings 00:23:37.900 --> 00:23:40.099 without you even realizing it? That is a question 00:23:40.099 --> 00:23:42.059 worth spending some time on. Something for you 00:23:42.059 --> 00:23:44.099 to mull over until next time. Keep exploring.