WEBVTT 00:00:00.000 --> 00:00:04.339 You know, whether you are asking Siri to set 00:00:04.339 --> 00:00:07.419 a timer, or investing your life savings in the 00:00:07.419 --> 00:00:10.140 stock market, or even having your DNA analyzed, 00:00:10.519 --> 00:00:12.960 you are interacting with a ghost. A mathematical 00:00:12.960 --> 00:00:15.199 ghost, to be precise. Right, yeah, not a literal 00:00:15.199 --> 00:00:18.570 ghost, but there's this... this invisible architecture 00:00:18.570 --> 00:00:20.789 operating right beneath the surface of our digital 00:00:20.789 --> 00:00:23.670 lives. And it's constantly making these highly 00:00:23.670 --> 00:00:26.550 educated guesses about things it can't directly 00:00:26.550 --> 00:00:28.949 see. It is entirely behind the scenes. Exactly. 00:00:29.050 --> 00:00:31.469 So today we're opening the hood on that invisible 00:00:31.469 --> 00:00:34.009 architecture. We're doing a deep dive into the 00:00:34.009 --> 00:00:36.549 hidden Markov model. This is the mathematical 00:00:36.549 --> 00:00:38.950 engine that allows a machine to deduce the invisible 00:00:38.950 --> 00:00:41.829 forces shaping our world purely by looking at, 00:00:41.829 --> 00:00:44.679 well, the shadows they leave behind. We're using 00:00:44.679 --> 00:00:47.399 a comprehensive breakdown from Wikipedia as our 00:00:47.399 --> 00:00:50.340 source map today. It is a fantastic topic. The 00:00:50.340 --> 00:00:53.000 mission today is really to demystify how these 00:00:53.000 --> 00:00:55.640 algorithms actually work in practice. Yeah, so, 00:00:55.719 --> 00:00:57.759 okay, let's unpack this. At its core, a hidden 00:00:57.759 --> 00:01:01.140 Markov model or an HMM is basically a system 00:01:01.140 --> 00:01:03.439 split into two distinct layers, right? That's 00:01:03.439 --> 00:01:05.969 right, two layers. You have one layer. which 00:01:05.969 --> 00:01:08.209 is just a sequence of events we can actually 00:01:08.209 --> 00:01:10.290 observe in the real world. And then you have 00:01:10.290 --> 00:01:12.790 the second layer, which is a sequence of hidden 00:01:12.790 --> 00:01:15.409 states. We can't see them, but they are directly 00:01:15.409 --> 00:01:18.569 causing those observable events. And the primary 00:01:18.569 --> 00:01:21.370 objective of this model is to reverse engineer 00:01:21.370 --> 00:01:24.510 reality. It tries to learn about that hidden 00:01:24.510 --> 00:01:27.090 layer strictly by observing the visible layer. 00:01:27.109 --> 00:01:29.989 Like playing detective. Exactly like that. And 00:01:29.989 --> 00:01:33.069 the engine that makes this possible, the core 00:01:33.069 --> 00:01:35.859 rule, is what we call the Markov. property. Which 00:01:35.859 --> 00:01:37.540 is kind of a weird rule when you think about 00:01:37.540 --> 00:01:40.280 it. It is. It's a strict mathematical rule stating 00:01:40.280 --> 00:01:42.500 that the current hidden state is influenced only 00:01:42.500 --> 00:01:45.260 by the state immediately preceding it. It operates 00:01:45.260 --> 00:01:48.439 with zero long -term memory. Zero. None at all. 00:01:48.480 --> 00:01:51.060 Right. What happened two steps ago or ten steps 00:01:51.060 --> 00:01:53.239 ago doesn't mathematically matter to the current 00:01:53.239 --> 00:01:55.620 state. Which, I mean, if you think about how 00:01:55.620 --> 00:01:57.840 human memory works or even real -world physics, 00:01:57.900 --> 00:02:00.299 that sounds totally counterintuitive at first. 00:02:00.620 --> 00:02:03.340 Like, why force the system to have amnesia? Because 00:02:03.340 --> 00:02:06.030 without that amnesia, the calculations would 00:02:06.030 --> 00:02:09.430 literally break modern computers. By assuming 00:02:09.430 --> 00:02:12.289 that only the immediate past matters, the Markov 00:02:12.289 --> 00:02:15.629 property takes this chaotic, infinitely complex 00:02:15.629 --> 00:02:19.169 universe and simplifies it into calculable probabilities. 00:02:19.349 --> 00:02:21.610 Oh, I see. So it's a shortcut. A necessary one. 00:02:21.759 --> 00:02:24.759 If a system had to remember the entire history 00:02:24.759 --> 00:02:26.860 of everything that led up to a single moment, 00:02:27.099 --> 00:02:29.699 just to guess what happens next, the math becomes 00:02:29.699 --> 00:02:32.960 intractable. The Markov property trims the fat 00:02:32.960 --> 00:02:38.180 so the algorithm can actually run. I mean, the 00:02:38.180 --> 00:02:40.319 source material gives us two fantastic scenarios. 00:02:40.879 --> 00:02:42.659 Let's start with the Alice and Bob weather game. 00:02:42.800 --> 00:02:45.159 A classic example. Yeah, it's great. So imagine 00:02:45.159 --> 00:02:47.639 two friends, Alice and Bob, who live far apart 00:02:47.639 --> 00:02:50.800 and talk on the phone every day. Bob is a creature 00:02:50.800 --> 00:02:52.960 of habit. He really only does three things. He 00:02:52.960 --> 00:02:55.099 walks in the park, he goes shopping, or he cleans 00:02:55.099 --> 00:02:57.500 his apartment. And his choice is determined exclusively 00:02:57.500 --> 00:02:59.639 by the weather where he lives. Right, exclusively. 00:02:59.800 --> 00:03:01.819 The catch is that Alice has no idea what the 00:03:01.819 --> 00:03:04.900 weather is actually like where Bob lives. So 00:03:04.900 --> 00:03:06.860 the weather is the hidden state. And the only 00:03:06.860 --> 00:03:09.460 things Alice has access to are the observations. 00:03:10.139 --> 00:03:12.139 So Bob telling her, you know, hey, I cleaned 00:03:12.139 --> 00:03:15.419 my apartment today or I went for a walk. Exactly. 00:03:16.439 --> 00:03:19.400 But Alice isn't totally in the dark here. She 00:03:19.400 --> 00:03:21.860 knows the general rules of his city. Like, she 00:03:21.860 --> 00:03:24.280 knows it rains a lot there. She knows that if 00:03:24.280 --> 00:03:26.900 it's rainy today, there is maybe a 30 % chance 00:03:26.900 --> 00:03:29.900 tomorrow will be sunny. And she knows his habits. 00:03:29.979 --> 00:03:31.780 Right. She knows that if it's raining, there's 00:03:31.780 --> 00:03:34.340 a 50 % chance he'll stay in and clean. But if 00:03:34.340 --> 00:03:36.800 it's sunny, there's a 60 % chance he'll go for 00:03:36.800 --> 00:03:39.340 a walk. So using just those observations and 00:03:39.340 --> 00:03:41.740 those probabilities, Alice has to reconstruct 00:03:41.740 --> 00:03:44.389 the hidden weather patterns day by day. The text 00:03:44.389 --> 00:03:46.810 provides another, perhaps more mechanical example 00:03:46.810 --> 00:03:49.669 to illustrate this architecture. Imagine a genie 00:03:49.669 --> 00:03:52.349 in a hidden room. Okay, a genie. In this room, 00:03:52.349 --> 00:03:54.650 there are several urns, and each urn contains 00:03:54.650 --> 00:03:57.909 a specific known mix of uniquely labeled balls. 00:03:58.569 --> 00:04:01.409 The genie randomly picks an urn, draws a ball, 00:04:01.590 --> 00:04:03.750 and drops it onto a conveyor belt that leads 00:04:03.750 --> 00:04:06.169 out of the room. So you, standing outside the 00:04:06.169 --> 00:04:08.150 room, you just see the sequence of colored balls 00:04:08.150 --> 00:04:10.129 rolling down the conveyor belt. You never see 00:04:10.129 --> 00:04:13.780 the urns. Never. You only see the output. Watching 00:04:13.780 --> 00:04:16.079 this model feels a lot like sitting in Plato's 00:04:16.079 --> 00:04:18.600 cave. You know, like you're watching the shadows 00:04:18.600 --> 00:04:21.279 on a cave wall to try and guess the shape of 00:04:21.279 --> 00:04:24.639 the physical object casting them. You only ever 00:04:24.639 --> 00:04:27.500 see the byproduct of the reality, never the reality 00:04:27.500 --> 00:04:30.240 itself. Plato's cave is exactly the right way 00:04:30.240 --> 00:04:32.819 to think about it. The sequence of balls rolling 00:04:32.819 --> 00:04:35.500 out of the room is the shadow on the wall. and 00:04:35.500 --> 00:04:38.220 the hidden math is trying to reverse -engineer 00:04:38.220 --> 00:04:41.839 the true object casting it. To do that, the architecture 00:04:41.839 --> 00:04:44.759 of a hidden Markov model relies on two specific 00:04:44.759 --> 00:04:48.120 sets of rules. First you have transition probabilities. 00:04:48.259 --> 00:04:50.079 Transition probabilities, okay. These are the 00:04:50.079 --> 00:04:52.439 rules for moving from one hidden urn to the next 00:04:52.439 --> 00:04:54.459 or, you know, one weather state to the next. 00:04:54.839 --> 00:04:56.939 If you have a specific number of hidden states, 00:04:56.980 --> 00:05:00.639 say, n states, you have an n squared matrix of 00:05:00.639 --> 00:05:03.199 transition probabilities mapping every possible 00:05:03.199 --> 00:05:06.160 move. From any one state to any other. Correct. 00:05:06.329 --> 00:05:09.050 Then you have the emission probabilities. These 00:05:09.050 --> 00:05:11.670 dictate how likely a hidden state is to produce 00:05:11.670 --> 00:05:15.310 a specific observation, like Bob's 50 % chance 00:05:15.310 --> 00:05:18.129 of cleaning in the rain or the 80 % chance that 00:05:18.129 --> 00:05:20.810 urn number three produces a red ball. Wait, I 00:05:20.810 --> 00:05:22.550 have to push back here on behalf of the listener 00:05:22.550 --> 00:05:25.329 for a second. Go ahead. If the genie is choosing 00:05:25.329 --> 00:05:28.310 urns based only on the single previous urn... 00:05:28.379 --> 00:05:31.579 or if the weather today only depends on yesterday, 00:05:32.360 --> 00:05:34.600 because of that strict Markov property we talked 00:05:34.600 --> 00:05:37.699 about, doesn't that make the system incredibly 00:05:37.699 --> 00:05:39.759 short -sighted? It does seem that way. I mean, 00:05:39.920 --> 00:05:43.019 how can this accurately mild complex reality 00:05:43.019 --> 00:05:46.279 if it has zero long term memory? It feels like 00:05:46.279 --> 00:05:48.220 trying to predict a novel by only looking at 00:05:48.220 --> 00:05:50.439 the previous word. What's fascinating here is 00:05:50.439 --> 00:05:52.800 that while it feels limiting, this is where the 00:05:52.800 --> 00:05:55.040 math gets incredibly elegant. Even though the 00:05:55.040 --> 00:05:57.639 model only looks one step back, these strict 00:05:57.639 --> 00:06:00.800 interlocking probabilistic rules compound. Be 00:06:00.800 --> 00:06:03.439 compound. Well, it is not just looking at one 00:06:03.439 --> 00:06:06.279 transition in a vacuum. It is analyzing a long 00:06:06.279 --> 00:06:09.199 chain of them. When you chain enough of these 00:06:09.199 --> 00:06:12.060 short -term dependencies together through transition 00:06:12.060 --> 00:06:14.920 and emission matrices, they become incredibly 00:06:14.920 --> 00:06:17.980 powerful at mapping highly complex, seemingly 00:06:17.980 --> 00:06:21.319 long -term patterns in the data. Oh, wow. A single 00:06:21.319 --> 00:06:23.220 word doesn't tell you the plot of the novel, 00:06:23.800 --> 00:06:26.240 but observing the transition probabilities of 00:06:26.240 --> 00:06:29.180 10 ,000 words in a row absolutely maps out the 00:06:29.180 --> 00:06:31.199 grammar and tone of the book. Okay, that makes 00:06:31.199 --> 00:06:33.220 sense. So we've got the rules of the game. Yeah. 00:06:33.360 --> 00:06:35.319 We know how the matrices are set up. Now how 00:06:35.319 --> 00:06:37.399 do we actually win the game? Right. How do we 00:06:37.399 --> 00:06:39.899 solve the puzzle? Exactly. How do we use this 00:06:39.899 --> 00:06:43.000 model to solve the puzzle of the hidden truth? 00:06:43.540 --> 00:06:46.339 The source lays out three main inference tasks 00:06:46.339 --> 00:06:49.600 for these latent or hidden variables. There are 00:06:49.600 --> 00:06:51.980 three major questions we can ask the model to 00:06:51.980 --> 00:06:54.139 solve. The first is called filtering. Filtering, 00:06:54.220 --> 00:06:57.019 right. This is when we want to find the distribution 00:06:57.019 --> 00:06:59.699 over the hidden states at the very end of a sequence. 00:07:00.540 --> 00:07:02.579 We use something called the forward algorithm 00:07:02.579 --> 00:07:06.060 to figure out what state the process is in right 00:07:06.060 --> 00:07:09.040 at this exact moment, based on accumulating the 00:07:09.040 --> 00:07:10.920 probabilities of all the observations leading 00:07:10.920 --> 00:07:13.420 up to it. OK, so that's the present moment. Then 00:07:13.420 --> 00:07:15.759 there's the second task, which is smoothing. 00:07:16.160 --> 00:07:18.759 Yes, smoothing. This is when you're looking backward. 00:07:19.560 --> 00:07:22.019 You want to find the distribution of a hidden 00:07:22.019 --> 00:07:24.420 state somewhere in the middle of a past sequence. 00:07:25.180 --> 00:07:28.579 To do this, the model uses the forward -backward 00:07:28.579 --> 00:07:30.980 algorithm. Which is very clever mathematically. 00:07:31.220 --> 00:07:33.139 Yeah, it calculates the probabilities from the 00:07:33.139 --> 00:07:35.040 beginning of the sequence up to that middle point 00:07:35.040 --> 00:07:37.279 and also from the end of the sequence backward 00:07:37.279 --> 00:07:39.899 to that middle point to basically pinch the exact 00:07:39.899 --> 00:07:43.110 probability. And finally, the third task is finding 00:07:43.110 --> 00:07:45.670 the most likely explanation. This doesn't just 00:07:45.670 --> 00:07:48.449 look at one point in time. It asks, what is the 00:07:48.449 --> 00:07:50.990 joint probability of the entire sequence of hidden 00:07:50.990 --> 00:07:53.449 states that generated our observations? The whole 00:07:53.449 --> 00:07:55.670 thing. The whole sequence. And for this, we use 00:07:55.670 --> 00:07:58.189 the famous Viterbi algorithm. OK, here's where 00:07:58.189 --> 00:08:00.430 it gets really interesting. Let's put this into 00:08:00.430 --> 00:08:03.740 real -world terms for you. Think of these three 00:08:03.740 --> 00:08:06.699 tools like tracking a friend on a cross -country 00:08:06.699 --> 00:08:09.680 drive. I like this analogy Yeah, so filtering 00:08:09.680 --> 00:08:12.079 is like using their current speed and direction 00:08:12.079 --> 00:08:16.180 to guess exactly where they are right now smoothing 00:08:16.399 --> 00:08:18.839 is trying to figure out which specific gas station 00:08:18.839 --> 00:08:21.220 they stopped at three hours ago based on their 00:08:21.220 --> 00:08:23.620 overall trajectory. Right. But the most likely 00:08:23.620 --> 00:08:26.459 explanation, the Viterbi algorithm, is like looking 00:08:26.459 --> 00:08:29.800 at a pile of crumpled gas receipts and reconstructing 00:08:29.800 --> 00:08:32.879 the entire cross -country road trip map from 00:08:32.879 --> 00:08:35.980 start to finish. The road trip analogy perfectly 00:08:35.980 --> 00:08:38.419 captures the scale of the Viterbi algorithm, 00:08:38.419 --> 00:08:40.960 and it brings up a crucial mechanical distinction. 00:08:41.240 --> 00:08:43.720 Which is? You might wonder... Why can't we just 00:08:43.720 --> 00:08:46.379 use the filtering tool over and over again? Why 00:08:46.379 --> 00:08:48.539 not just find the most likely state for step 00:08:48.539 --> 00:08:51.279 one, then step two, then step three, and stitch 00:08:51.279 --> 00:08:53.500 them together to make our map? Right. I was actually 00:08:53.500 --> 00:08:55.779 just thinking that. If I know the most likely 00:08:55.779 --> 00:08:58.100 weather for Monday, Tuesday, and Wednesday individually. 00:08:58.440 --> 00:09:00.539 Shouldn't that be the most likely sequence for 00:09:00.539 --> 00:09:03.559 the week? Not necessarily. The most likely individual 00:09:03.559 --> 00:09:06.419 states don't always form the most likely continuous 00:09:06.419 --> 00:09:09.080 sequence. Wait, really? Why not? Because the 00:09:09.080 --> 00:09:11.940 transitions between certain states might be mathematically 00:09:11.940 --> 00:09:15.620 impossible or highly improbable. If you try to 00:09:15.620 --> 00:09:18.740 guess a 10 word sentence, the number of possible 00:09:18.740 --> 00:09:21.820 grammatical combinations is astronomical. If 00:09:21.820 --> 00:09:24.519 a machine brute forced it by checking every single 00:09:24.519 --> 00:09:26.440 path through the matrix, it would take years. 00:09:26.600 --> 00:09:29.340 Oh, because of all the branches. Exactly. The 00:09:29.340 --> 00:09:31.899 brilliance of the Viterbi algorithm is dynamic 00:09:31.899 --> 00:09:34.740 programming. Instead of mapping every route, 00:09:35.059 --> 00:09:38.120 it only remembers the single best path to reach 00:09:38.120 --> 00:09:41.200 the current word, instantly discarding all the 00:09:41.200 --> 00:09:44.259 suboptimal routes. It cuts a process that should 00:09:44.110 --> 00:09:47.090 to take centuries down to milliseconds. So it's 00:09:47.090 --> 00:09:49.330 ruthlessly efficient. It prunes the dead ends 00:09:49.330 --> 00:09:51.549 immediately instead of following them to the 00:09:51.549 --> 00:09:53.990 finish line. Exactly. The source gives a great 00:09:53.990 --> 00:09:56.309 example regarding part of speech tagging and 00:09:56.309 --> 00:09:58.909 language processing. If you are trying to understand 00:09:58.909 --> 00:10:01.169 a sentence, you don't just want the probability 00:10:01.169 --> 00:10:03.230 of what part of speech a single word is. Right. 00:10:03.230 --> 00:10:05.909 You need context. To actually make sense of the 00:10:05.909 --> 00:10:08.230 grammar, you need the entire sequence of parts 00:10:08.230 --> 00:10:10.669 of speech to align logically. You need the whole 00:10:10.669 --> 00:10:13.960 road trip. Which means you require the viterbi 00:10:13.960 --> 00:10:16.659 algorithm to find that most likely explanation 00:10:16.659 --> 00:10:19.559 for the whole sentence Ensuring that a noun actually 00:10:19.559 --> 00:10:21.899 follows an adjective in a way that makes linguistic 00:10:21.899 --> 00:10:25.220 sense that makes total sense We use the terby 00:10:25.220 --> 00:10:27.259 when the context of the whole sequence is what 00:10:27.259 --> 00:10:31.269 actually matters but All of this puzzle solving 00:10:31.269 --> 00:10:33.330 assumes we already know the rules, right? He 00:10:33.330 --> 00:10:35.110 does assume that. Like, it assumes that we know 00:10:35.110 --> 00:10:37.929 Alice's probabilities for Bob's weather, or the 00:10:37.929 --> 00:10:40.289 exact mix of colored balls and the genies' urns. 00:10:41.049 --> 00:10:43.529 How does a machine learn the rules of the hidden 00:10:43.529 --> 00:10:47.080 world from scratch if no one inputs them? And 00:10:47.080 --> 00:10:49.259 where does this actually touch your daily life? 00:10:49.440 --> 00:10:51.919 Well, the history here is quite rich. The core 00:10:51.919 --> 00:10:54.360 math was developed in the late 1960s by Leonard 00:10:54.360 --> 00:10:56.919 E. Baum and his colleagues, but it really took 00:10:56.919 --> 00:10:59.539 off in the mid -1970s when it was applied to 00:10:59.539 --> 00:11:01.500 speech recognition. Speech recognition, okay. 00:11:01.720 --> 00:11:05.779 And then it exploded in the 1980s with bioinformatics, 00:11:06.139 --> 00:11:10.179 specifically analyzing DNA sequences. Today, 00:11:10.379 --> 00:11:12.480 the applications are endless. We are talking 00:11:12.480 --> 00:11:15.820 computational finance, protein folding, handwriting 00:11:15.820 --> 00:11:18.940 recognition, even predicting solar irradiance 00:11:18.940 --> 00:11:21.539 variability. And Siri. The source specifically 00:11:21.539 --> 00:11:23.879 mentions Siri's speech recognition. Yes. Siri 00:11:23.879 --> 00:11:26.500 is a perfect everyday example. But let's dig 00:11:26.500 --> 00:11:28.220 into the mechanics of that. We're saying the 00:11:28.220 --> 00:11:30.259 algorithm learns the transition and emission 00:11:30.259 --> 00:11:34.000 probabilities, but how can it possibly learn 00:11:34.000 --> 00:11:36.799 the rules of the hidden states if those states 00:11:36.799 --> 00:11:40.409 are by definition, hidden. It is a paradox, isn't 00:11:40.409 --> 00:11:42.629 it? And I really need to logic this out. If it's 00:11:42.629 --> 00:11:44.470 guessing the rules based on the data, but it 00:11:44.470 --> 00:11:46.230 needs the rules to understand the data, isn't 00:11:46.230 --> 00:11:48.409 it just spinning its wheels? How does tweaking 00:11:48.409 --> 00:11:50.710 the probabilities actually lock it into the correct 00:11:50.710 --> 00:11:53.429 pattern instead of just, you know, a mathematically 00:11:53.429 --> 00:11:55.730 convenient hallucination? It is the ultimate 00:11:55.730 --> 00:11:57.809 chicken and egg problem. If we connect this to 00:11:57.809 --> 00:12:00.870 the bigger picture to solve it, we rely on algorithms 00:12:00.870 --> 00:12:03.289 like Baumwelsch, which is a type of expectation 00:12:03.289 --> 00:12:07.600 maximization algorithm, or EM. For more complex 00:12:07.600 --> 00:12:10.759 time series predictions, systems might use Markov 00:12:10.759 --> 00:12:14.539 chain Monte Carlo or MCMC sampling. Okay, lots 00:12:14.539 --> 00:12:17.740 of acronyms. True, but the core logic of expectation 00:12:17.740 --> 00:12:19.860 maximization directly answers your question. 00:12:20.519 --> 00:12:23.860 It uses iterative local maximum likelihood estimates. 00:12:24.080 --> 00:12:25.860 Okay, let's translate that for everyone. Think 00:12:25.860 --> 00:12:28.419 of it like trying to tune a blurry radio station 00:12:28.419 --> 00:12:30.889 in the dark. Great way to visualize it. You don't 00:12:30.889 --> 00:12:32.649 know the exact frequency. Those are the hidden 00:12:32.649 --> 00:12:35.070 rules. But you can hear the static, which are 00:12:35.070 --> 00:12:37.950 the errors in your output. Expectation maximization 00:12:37.950 --> 00:12:40.009 is basically the process of turning the dial 00:12:40.009 --> 00:12:42.629 just a millimeter to the left. Did the audio 00:12:42.629 --> 00:12:45.309 get clearer? Yes. Keep going. Did it get worse? 00:12:45.529 --> 00:12:48.330 Turn back. It iterates. Right. It iterates this 00:12:48.330 --> 00:12:51.029 microtuning until the music comes through crystal 00:12:51.029 --> 00:12:54.350 clear. That is precisely how it functions. The 00:12:54.350 --> 00:12:57.009 algorithm splits the work into two distinct steps. 00:12:57.549 --> 00:13:00.980 First is the expectation step. Given its current 00:13:00.980 --> 00:13:03.840 fuzzy guess of the rules, it asks, what is the 00:13:03.840 --> 00:13:06.940 most likely sequence of hidden states? It makes 00:13:06.940 --> 00:13:08.960 a rough map. Okay, step one is the rough map. 00:13:09.159 --> 00:13:11.960 Then comes the maximization step. Given that 00:13:11.960 --> 00:13:14.360 rough map, it asks, how should we mathematically 00:13:14.360 --> 00:13:17.019 update our transition and emission rules to make 00:13:17.019 --> 00:13:19.820 this specific map even more likely to occur? 00:13:20.000 --> 00:13:22.220 Oh, I see. It updates the rules, which changes 00:13:22.220 --> 00:13:24.039 the map, which updates the rules again. It loops 00:13:24.039 --> 00:13:26.539 this process over and over until the probabilities 00:13:26.539 --> 00:13:29.059 stabilize, the static fades, and the pattern 00:13:29.059 --> 00:13:31.159 locks in. So it literally pulls itself up by 00:13:31.159 --> 00:13:33.360 its own mathematical bootstraps. It does. And 00:13:33.360 --> 00:13:34.759 that brings it right back to you, the listener. 00:13:35.080 --> 00:13:37.580 Every single time your phone's voice assistant 00:13:37.580 --> 00:13:40.639 understands your spoken words, translating the 00:13:40.639 --> 00:13:42.860 messy audio waves of your voice into a hidden 00:13:42.860 --> 00:13:46.100 sequence of grammatical text, it is solving this 00:13:46.100 --> 00:13:49.860 exact probabilistic puzzle. It's using expectation 00:13:49.860 --> 00:13:52.820 maximization to tune the radio dial of your speech. 00:13:53.399 --> 00:13:55.519 It is quite literally mapping the audio shadows 00:13:55.519 --> 00:13:57.740 back to the linguistic urns in fractions of a 00:13:57.740 --> 00:14:01.299 second. But wait, there is a fatal flaw in everything 00:14:01.299 --> 00:14:04.490 we've talked about so far. A flaw? Yeah. Reality 00:14:04.490 --> 00:14:07.009 doesn't happen in neat discrete steps like drawing 00:14:07.009 --> 00:14:09.610 a single ball from an urn or flipping a coin 00:14:09.610 --> 00:14:12.529 between rainy and sunny. The real world is continuous. 00:14:12.909 --> 00:14:15.269 You know, it's messy. The stock market doesn't 00:14:15.269 --> 00:14:18.570 just go up or down in rigid boxes. It fluctuates 00:14:18.570 --> 00:14:20.909 wildly across a continuous spectrum. That is 00:14:20.909 --> 00:14:23.029 very true. So what happens to our Markov model? 00:14:23.230 --> 00:14:25.509 when the data doesn't fit into clean little boxes. 00:14:25.830 --> 00:14:28.350 It evolves. The source outlines several major 00:14:28.350 --> 00:14:31.450 extensions to handle exactly this kind of complexity. 00:14:31.710 --> 00:14:33.850 When you were dealing with continuous state spaces 00:14:33.850 --> 00:14:36.629 like tracking the continuous real -time trajectory 00:14:36.629 --> 00:14:38.789 of a rocket through the atmosphere rather than 00:14:38.789 --> 00:14:41.370 discrete weather states, the math shifts. Oh, 00:14:41.370 --> 00:14:43.590 so. We use things like Kalman filters, which 00:14:43.590 --> 00:14:46.129 adapt the Markov model to handle data that flows 00:14:46.129 --> 00:14:48.769 without rigid breaks. But the truly significant 00:14:48.769 --> 00:14:51.529 shift in modern AI has been the move toward discriminative 00:14:51.529 --> 00:14:54.779 approaches. source heavily emphasizes this shift. 00:14:55.279 --> 00:14:58.519 It details maximum R -SPAY Markov model's MEMS 00:14:58.519 --> 00:15:01.259 and linear chain conditional random fields, or 00:15:01.259 --> 00:15:03.700 CRFs. But let's not just name -drop the jargon. 00:15:04.340 --> 00:15:06.440 How do these actually differ from the classic 00:15:06.440 --> 00:15:09.600 hidden urns? The difference is in how they view 00:15:09.600 --> 00:15:12.840 the world. Traditional HMMs are generative models. 00:15:13.299 --> 00:15:15.899 They try to mathematically recreate exactly how 00:15:15.899 --> 00:15:17.860 the hidden states generated the observations 00:15:17.860 --> 00:15:21.200 from scratch. They model the entire joint distribution. 00:15:21.279 --> 00:15:24.580 OK, recreating the whole world. Yes. A discriminative 00:15:24.580 --> 00:15:26.820 model, like a conditional random field, doesn't 00:15:26.820 --> 00:15:29.039 care about recreating the whole world. It just 00:15:29.039 --> 00:15:31.559 models the conditional distribution. It only 00:15:31.559 --> 00:15:33.639 cares about drawing the correct boundaries to 00:15:33.639 --> 00:15:36.080 classify the observations. Think of a traditional 00:15:36.080 --> 00:15:38.519 generative HMM. as looking at the words in a 00:15:38.519 --> 00:15:40.620 sentence through a narrow straw, right? You only 00:15:40.620 --> 00:15:43.159 see one word at a time, strictly limited by that 00:15:43.159 --> 00:15:45.580 one -step Markov property we talked about. Yes, 00:15:45.779 --> 00:15:48.399 the amnesia. Right, the amnesia. Conditional 00:15:48.399 --> 00:15:51.080 random field takes the straw away. It allows 00:15:51.080 --> 00:15:53.480 the model to look at the whole sequence simultaneously. 00:15:54.120 --> 00:15:56.399 It recognizes context, like the fact that the 00:15:56.399 --> 00:15:58.649 word bank... It means something entirely different 00:15:58.649 --> 00:16:01.190 if the word river is nearby rather than if the 00:16:01.190 --> 00:16:03.850 word money is nearby. Exactly. By taking the 00:16:03.850 --> 00:16:06.750 straw away, CRFs solve what is known as the label 00:16:06.750 --> 00:16:09.769 bias problem. They allow engineers to inject 00:16:09.769 --> 00:16:12.389 domain -specific knowledge and look at combinations 00:16:12.389 --> 00:16:15.269 of nearby observations all at once. Which is 00:16:15.269 --> 00:16:18.190 huge for complex data. It is. We are also seeing 00:16:18.190 --> 00:16:20.110 the integration of recurrent neural networks, 00:16:20.509 --> 00:16:23.090 specifically reservoir networks, which feed temporal 00:16:23.090 --> 00:16:26.269 dynamics into the model. This helps the HMM handle 00:16:26.269 --> 00:16:29.350 non - stationary data. That's data where the 00:16:29.350 --> 00:16:31.730 underlying rules of the universe are constantly 00:16:31.730 --> 00:16:34.549 shifting over time. Which brings us to a massive 00:16:34.549 --> 00:16:38.330 update from the source material. In 2023, two 00:16:38.330 --> 00:16:40.750 breakthrough algorithms were introduced, the 00:16:40.750 --> 00:16:43.029 discriminative forward -backward and discriminative 00:16:43.029 --> 00:16:45.509 Viterbi algorithms. A very recent development. 00:16:45.990 --> 00:16:48.649 Yeah, super recent. They bypassed the need to 00:16:48.649 --> 00:16:51.169 model the joint distribution entirely using only 00:16:51.169 --> 00:16:55.840 conditional distributions. But wait! Uh, doesn't 00:16:55.840 --> 00:16:58.159 that rewrite the rulebook? We spent this whole 00:16:58.159 --> 00:16:59.940 deep dive saying we needed the full generative 00:16:59.940 --> 00:17:02.340 model that earns the specific emission probabilities 00:17:02.340 --> 00:17:05.079 to figure out the hidden states. And now we don't. 00:17:05.220 --> 00:17:08.019 It is a profound paradigm shift in the mathematics. 00:17:08.359 --> 00:17:10.799 By skipping the joint law, by not forcing the 00:17:10.799 --> 00:17:13.299 algorithm to learn the entire generative rulebook 00:17:13.299 --> 00:17:16.859 of the universe, These 2023 algorithms make the 00:17:16.859 --> 00:17:19.400 model vastly more computationally efficient. 00:17:19.460 --> 00:17:21.480 Oh, because it's doing less math. Precisely. 00:17:21.839 --> 00:17:24.119 You get the sequence -solving power of the Viterbi 00:17:24.119 --> 00:17:26.819 algorithm without the immense computational baggage 00:17:26.819 --> 00:17:29.400 of a traditional generative model. It makes the 00:17:29.400 --> 00:17:31.779 HMM incredibly versatile for cutting -edge AI 00:17:31.779 --> 00:17:34.279 applications where speed, scale and efficiency 00:17:34.279 --> 00:17:36.660 are paramount. You are getting the road trip 00:17:36.660 --> 00:17:38.720 map without having to mathematically simulate 00:17:38.720 --> 00:17:41.460 the engine of the car. That is wild. I mean, 00:17:41.460 --> 00:17:43.940 we've gone from a theoretical genie drawing balls 00:17:43.940 --> 00:17:47.440 from hidden urns in the 1960s to teaching Siri 00:17:47.440 --> 00:17:50.599 how to recognize the messy audio of your voice 00:17:50.599 --> 00:17:54.160 to these 2023 algorithms that slice through the 00:17:54.160 --> 00:17:56.660 underlying math faster and more efficiently than 00:17:56.660 --> 00:17:59.180 ever before. It's an incredible evolution. So, 00:17:59.220 --> 00:18:01.279 you know, the next time you are looking at a 00:18:01.279 --> 00:18:02.799 sequence of events, whether it's the erratic 00:18:02.799 --> 00:18:04.619 jumping in the stock market, the sequence of 00:18:04.619 --> 00:18:07.180 nucleotides in a DNA test, or just trying to 00:18:07.180 --> 00:18:09.450 guess if your friend is going to clean their 00:18:09.450 --> 00:18:11.930 apartment based on the weather. Remember, there 00:18:11.930 --> 00:18:14.549 is a hidden Markov chain operating right beneath 00:18:14.549 --> 00:18:17.089 the surface, calculating the odds. And before 00:18:17.089 --> 00:18:20.109 we wrap up, there is one final, almost philosophical 00:18:20.109 --> 00:18:21.990 concept from the measure theory section of the 00:18:21.990 --> 00:18:23.869 text that I think you should take with you. Ooh, 00:18:23.869 --> 00:18:26.220 go for it. Let's hear it. We established at the 00:18:26.220 --> 00:18:28.400 very beginning that the hidden part of a Markov 00:18:28.400 --> 00:18:31.000 model strictly depends only on the immediate 00:18:31.000 --> 00:18:34.000 past. It operates with mathematical amnesia. 00:18:34.220 --> 00:18:36.700 Right, the one -step rule. However, the measure 00:18:36.700 --> 00:18:38.740 theory proves that the observable sequence, the 00:18:38.740 --> 00:18:41.160 part we actually see, can be non -Markovian. 00:18:41.500 --> 00:18:45.200 Wait, meaning the shadows do have a memory, even 00:18:45.200 --> 00:18:48.039 if the object casting them doesn't. That is exactly 00:18:48.039 --> 00:18:51.160 what it means. For example, if you observe a 00:18:51.160 --> 00:18:53.539 long enough sequence of a specific outcome on 00:18:53.539 --> 00:18:56.859 the visible layer, let's call it outcome B, you 00:18:56.859 --> 00:18:59.119 might become increasingly mathematically certain 00:18:59.119 --> 00:19:02.829 that the underlying hidden state is A. This implies 00:19:02.829 --> 00:19:05.210 that the visible, observable part of the system 00:19:05.210 --> 00:19:07.930 can actually be affected by something infinitely 00:19:07.930 --> 00:19:10.869 far back in the past, even if the hidden engine 00:19:10.869 --> 00:19:14.109 driving it only looks one step back. Whoa! So 00:19:14.109 --> 00:19:15.950 even if the hidden mechanics driving the universe 00:19:15.950 --> 00:19:18.490 only care about what happened yesterday, our 00:19:18.490 --> 00:19:20.789 visible world, the shadows we see on the cave 00:19:20.789 --> 00:19:23.369 wall, might actually carry the mathematical fingerprints 00:19:23.369 --> 00:19:26.150 of the infinite past. Exactly. The observables 00:19:26.150 --> 00:19:28.250 remember what the hidden states forget. That 00:19:28.250 --> 00:19:30.630 is so cool. It's a mathematical quirk with profound 00:19:30.630 --> 00:19:33.630 philosophical weight. The ghost in the machine 00:19:33.630 --> 00:19:36.349 might only be looking one step ahead, but the 00:19:36.349 --> 00:19:38.950 shadows it casts stretch all the way back to 00:19:38.950 --> 00:19:40.809 the beginning. A perfect way to summarize it. 00:19:41.009 --> 00:19:43.009 Now that is a thought to leave you with. Until 00:19:43.009 --> 00:19:45.250 next time, keep looking for the true shapes casting 00:19:45.250 --> 00:19:45.710 the shadows.