WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.010 Commons license. 00:00:04.010 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.710 continue to offer high quality educational resources for free. 00:00:10.710 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.260 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.260 --> 00:00:17.910 at ocw.mit.edu. 00:00:20.861 --> 00:00:23.110 PROFESSOR: All right, I guess we can get started here. 00:00:23.110 --> 00:00:24.670 So welcome. 00:00:24.670 --> 00:00:28.290 Today we're going to do Bayesian updating, which 00:00:28.290 --> 00:00:30.490 is to say we're going to be using Bayes' Theorem 00:00:30.490 --> 00:00:31.560 to learn from data. 00:00:31.560 --> 00:00:33.200 We're going to update our beliefs 00:00:33.200 --> 00:00:36.950 about various hypotheses based on the data. 00:00:36.950 --> 00:00:40.190 For this week, we're going to have known priors. 00:00:40.190 --> 00:00:42.570 We're going to know our belief about the data ahead 00:00:42.570 --> 00:00:43.960 of time for certain. 00:00:43.960 --> 00:00:50.480 We'll draw coins out of drawers or pull things out 00:00:50.480 --> 00:00:52.020 of hats, things like that. 00:00:52.020 --> 00:00:54.120 Starting next week, and we discussed 00:00:54.120 --> 00:00:56.340 this a little last time, we'll have 00:00:56.340 --> 00:00:58.050 unknown priors, which is you'll have 00:00:58.050 --> 00:00:59.380 to be making up those priors. 00:00:59.380 --> 00:01:03.050 And we'll teach you techniques for making it up. 00:01:03.050 --> 00:01:10.360 So the first slide is just XKCD's view of Bayes' Theorem. 00:01:10.360 --> 00:01:10.860 Let's go on. 00:01:10.860 --> 00:01:14.645 We want to start with a clicker question here. 00:01:17.920 --> 00:01:21.027 So the question is: which treatment would you choose 00:01:21.027 --> 00:01:22.110 if you needed a treatment? 00:01:22.110 --> 00:01:25.870 Treatment 1 cured 100% of the patients. 00:01:25.870 --> 00:01:29.060 Treatment 2 cured 95% in a trial. 00:01:29.060 --> 00:01:31.930 And treatment 3 cured 90%. 00:01:31.930 --> 00:01:34.835 So with this information, which would you choose? 00:01:39.229 --> 00:01:40.770 AUDIENCE: With only this information? 00:01:40.770 --> 00:01:41.853 PROFESSOR: With only this. 00:01:41.853 --> 00:01:44.610 If this is all I told you. 00:01:44.610 --> 00:01:46.020 Which would you choose? 00:01:46.020 --> 00:01:48.785 I recognizing that you might not be too happy with just 00:01:48.785 --> 00:01:52.430 this information, but if this is their best information, what 00:01:52.430 --> 00:01:53.590 would you choose? 00:01:53.590 --> 00:01:57.110 All right, there's some holdouts for the 95% cure. 00:01:57.110 --> 00:02:00.470 I'm not sure why. 00:02:00.470 --> 00:02:02.740 Really, that 95% was just, you know, 00:02:02.740 --> 00:02:08.139 they've cured one person 19 out of 20-- 00:02:08.139 --> 00:02:11.640 they got them 19 out of 20 well, or however you say that. 00:02:11.640 --> 00:02:12.390 Yeah. 00:02:12.390 --> 00:02:16.880 All right, so there's even a few more holdouts. 00:02:16.880 --> 00:02:20.790 All right, what if I gave you this information? 00:02:20.790 --> 00:02:24.590 Treatment 1 cured 3 out of 3 patients. 00:02:24.590 --> 00:02:26.430 That's 100%. 00:02:26.430 --> 00:02:28.990 Treatment 2 cured 19 of 20 patients. 00:02:28.990 --> 00:02:32.226 That's 95%. 00:02:32.226 --> 00:02:33.600 Or, you have a standard treatment 00:02:33.600 --> 00:02:38.920 which has cured 90,000 out of 100,000 patients, 90%, 00:02:38.920 --> 00:02:41.120 in clinical practice. 00:02:41.120 --> 00:02:42.490 Now which one would you choose? 00:02:49.832 --> 00:02:51.290 PROFESSOR 2: It's very interesting. 00:02:53.706 --> 00:02:54.747 Yeah, it totally changed. 00:02:54.747 --> 00:02:55.670 PROFESSOR: What's that? 00:02:55.670 --> 00:02:56.580 PROFESSOR 2: It totally shifted. 00:02:56.580 --> 00:02:58.360 PROFESSOR: It totally shifted, really? 00:02:58.360 --> 00:03:00.734 PROFESSOR 2: We have a bunch of very conservative people. 00:03:00.734 --> 00:03:02.400 PROFESSOR: Right. 00:03:02.400 --> 00:03:06.800 So the majority would choose the third choice, 00:03:06.800 --> 00:03:09.720 90 out of 100,000. 00:03:09.720 --> 00:03:13.554 I think-- well, someone tell me, what's the intuition there? 00:03:13.554 --> 00:03:15.330 AUDIENCE: Because it's been more tested. 00:03:15.330 --> 00:03:16.705 PROFESSOR: It's been more tested. 00:03:16.705 --> 00:03:20.480 3 out of 3 is 100%, but do you really believe 100%? 00:03:20.480 --> 00:03:21.300 Not yet. 00:03:21.300 --> 00:03:23.590 Maybe it'll prove out in time. 00:03:23.590 --> 00:03:27.780 19 out of 20 is a little more, but for most people, 00:03:27.780 --> 00:03:31.620 for 79% of you, that's not enough. 00:03:31.620 --> 00:03:36.110 The 90 out of 100,000 seems like good odds, 00:03:36.110 --> 00:03:37.900 and it's well tested. 00:03:37.900 --> 00:03:44.130 19 out of 20, if the next person tested doesn't get better, 00:03:44.130 --> 00:03:46.570 now you back to 90%. 00:03:46.570 --> 00:03:48.650 So that doesn't seem like quite enough. 00:03:48.650 --> 00:03:53.372 What if were 95 out of 100 patients. 00:03:53.372 --> 00:03:54.580 We'll let you click in there. 00:03:54.580 --> 00:03:57.020 Change number 2 to 95 out of 100 and let's 00:03:57.020 --> 00:04:00.370 see what people would do. 00:04:00.370 --> 00:04:02.780 Oh, you're very obliging, 95 out of 100-- 00:04:02.780 --> 00:04:04.230 PROFESSOR 2: Now it's about 50-50. 00:04:04.230 --> 00:04:05.527 PROFESSOR: About 50-50. 00:04:05.527 --> 00:04:06.735 PROFESSOR 2: Between 2 and 3. 00:04:09.250 --> 00:04:10.750 PROFESSOR: Realistically, of course, 00:04:10.750 --> 00:04:11.833 what would you want to do? 00:04:11.833 --> 00:04:14.240 You'd want to do a little more research 00:04:14.240 --> 00:04:17.730 and find out what these trials were, how good experiments you 00:04:17.730 --> 00:04:20.820 thought they were, what people are thinking about it. 00:04:20.820 --> 00:04:23.290 But with just this data, that's what people would choose. 00:04:23.290 --> 00:04:25.040 And would people would choose differently. 00:04:25.040 --> 00:04:27.360 OK. 00:04:27.360 --> 00:04:29.810 PROFESSOR 2: Let's see. 00:04:29.810 --> 00:04:33.390 So now I'm going to give you a toy problem that 00:04:33.390 --> 00:04:35.670 actually mirrors the same kind of effect we 00:04:35.670 --> 00:04:37.470 saw on the last slide. 00:04:37.470 --> 00:04:44.100 So, suppose in this MIT mug, I have dice of two types, 00:04:44.100 --> 00:04:47.710 4-sided and 20-sided. 00:04:47.710 --> 00:04:48.210 All right? 00:04:48.210 --> 00:04:50.760 So I'm going to reach into the mug. 00:04:50.760 --> 00:04:53.380 I'm not going to look. 00:04:53.380 --> 00:04:55.880 Much. 00:04:55.880 --> 00:04:58.935 I'm going to randomly pull out a die. 00:04:58.935 --> 00:04:59.810 I'm going to roll it. 00:05:02.750 --> 00:05:05.200 All right, I got a 1. 00:05:05.200 --> 00:05:06.450 OK. 00:05:06.450 --> 00:05:07.270 I got a 1. 00:05:07.270 --> 00:05:13.190 Now, just based on this information, which type of die 00:05:13.190 --> 00:05:14.480 do you think I randomly chose? 00:05:17.340 --> 00:05:17.962 The 4-sided. 00:05:17.962 --> 00:05:20.170 Someone want to give an explanation of why they think 00:05:20.170 --> 00:05:21.728 the 4-sided is more likely? 00:05:21.728 --> 00:05:23.307 AUDIENCE: It's greater probability. 00:05:23.307 --> 00:05:24.890 PROFESSOR 2: It's greater probability. 00:05:24.890 --> 00:05:28.180 What's the probability of getting a 1 on the 4-sided die? 00:05:28.180 --> 00:05:30.230 And what about a 20-sided sided? 00:05:30.230 --> 00:05:30.960 OK. 00:05:30.960 --> 00:05:35.300 So we see that it would be more likely to roll a 1 00:05:35.300 --> 00:05:38.180 with a 4-sided die than with a 20-sided die. 00:05:38.180 --> 00:05:43.570 So, suppose I tell you that I really in this cup only 00:05:43.570 --> 00:05:46.820 have one die of each type. 00:05:46.820 --> 00:05:49.380 So does that change your analysis 00:05:49.380 --> 00:05:50.820 of which die is most likely? 00:05:50.820 --> 00:05:52.680 Don't hit forward yet. 00:05:52.680 --> 00:05:54.120 PROFESSOR: I would think of it. 00:05:54.120 --> 00:05:56.450 My hands are staying. 00:05:56.450 --> 00:06:00.130 PROFESSOR 2: Or do you want to stick with the same reasoning? 00:06:00.130 --> 00:06:00.850 All right. 00:06:00.850 --> 00:06:02.683 Let me show you what's actually in this cup. 00:06:09.002 --> 00:06:09.960 PROFESSOR: Ouch. 00:06:09.960 --> 00:06:12.630 PROFESSOR 2: Oh, you can't see it yet. 00:06:12.630 --> 00:06:15.720 Let's go with the document camera. 00:06:18.940 --> 00:06:22.610 OK, that was actually what I had in this cup. 00:06:22.610 --> 00:06:24.560 Now what? 00:06:24.560 --> 00:06:26.510 So does this change at all of your analysis 00:06:26.510 --> 00:06:29.250 of which die you think I rolled and got a 1 with? 00:06:29.250 --> 00:06:31.082 What type of die, rather? 00:06:31.082 --> 00:06:31.665 AUDIENCE: Yes. 00:06:31.665 --> 00:06:32.100 PROFESSOR 2: Yeah? 00:06:32.100 --> 00:06:33.810 Now what do you think is more likely? 00:06:33.810 --> 00:06:34.817 4-sided or 20-sided? 00:06:34.817 --> 00:06:36.400 AUDIENCE: How many 20 sided are there? 00:06:36.400 --> 00:06:38.441 PROFESSOR 2: I think I got 20 20-sided die there. 00:06:45.110 --> 00:06:49.080 All right, so raise your hand if you like a 4-sided die. 00:06:49.080 --> 00:06:51.850 Raise your hand if you like the 20-sided die. 00:06:51.850 --> 00:06:52.350 Great. 00:06:52.350 --> 00:06:55.230 So that reasoning that you're doing intuitively 00:06:55.230 --> 00:06:57.560 is something that we want to formalize this week 00:06:57.560 --> 00:06:59.930 and next week using Bayes' Theorem 00:06:59.930 --> 00:07:02.460 to do these calculations in a systematic way, 00:07:02.460 --> 00:07:06.380 to evaluate hypotheses based on data. 00:07:06.380 --> 00:07:10.230 And these numbers, for example, how many 00:07:10.230 --> 00:07:12.370 die I had of one type versus another, 00:07:12.370 --> 00:07:14.980 we can encode that information in what's called a prior, 00:07:14.980 --> 00:07:16.285 and we'll come to that today. 00:07:16.285 --> 00:07:18.410 And you should've seen, in your reading, this idea. 00:07:22.612 --> 00:07:24.820 PROFESSOR: OK so I'm going to work through an example 00:07:24.820 --> 00:07:25.420 to start with. 00:07:25.420 --> 00:07:31.090 We have 3 types of coins: type A, B, and C. However 00:07:31.090 --> 00:07:34.210 it happens, the probability of getting heads with type A 00:07:34.210 --> 00:07:35.040 is 0.5. 00:07:35.040 --> 00:07:36.130 That's a fair coin. 00:07:36.130 --> 00:07:40.820 The probability of getting heads with type B is 0.6, 00:07:40.820 --> 00:07:44.000 and of type C is 0.9. 00:07:44.000 --> 00:07:48.210 And here's a box that contains two of the fair coins 00:07:48.210 --> 00:07:49.460 and one of each of the other. 00:07:49.460 --> 00:07:54.070 So two of type A, one of type B, and one of type C. 00:07:54.070 --> 00:07:56.490 Suppose I pick one at random and I get heads. 00:07:56.490 --> 00:08:00.160 So this is like John picking a die at random, and getting a 1. 00:08:00.160 --> 00:08:02.900 I get heads. 00:08:02.900 --> 00:08:04.900 Here's the question: before I flipped it, 00:08:04.900 --> 00:08:08.930 I have a prior belief in the probability 00:08:08.930 --> 00:08:11.290 that the coin was of each type based 00:08:11.290 --> 00:08:13.610 on the number in my drawer. 00:08:13.610 --> 00:08:17.390 After flipping it, my belief changes. 00:08:17.390 --> 00:08:22.180 The probability will change based on the data I get. 00:08:22.180 --> 00:08:24.160 And so a question you could ask: what was 00:08:24.160 --> 00:08:27.590 learned by flipping the coin? 00:08:27.590 --> 00:08:31.150 What we want to teach you to do to answer these questions, 00:08:31.150 --> 00:08:33.270 we have a nice way of summarizing 00:08:33.270 --> 00:08:36.280 all of that once in a table. 00:08:36.280 --> 00:08:44.090 So here the notation, we have this nice scripty H-- 00:08:44.090 --> 00:08:44.775 that's no good. 00:08:48.110 --> 00:08:49.496 H is our hypothesis. 00:08:52.820 --> 00:08:58.740 That is, you're of type A or B or C. 00:08:58.740 --> 00:09:06.690 We're going to make a conjecture on what we think the coin was. 00:09:06.690 --> 00:09:11.330 Before we do anything, we have a prior. 00:09:11.330 --> 00:09:13.910 So what's the probability of-- how 00:09:13.910 --> 00:09:17.240 did I label this-- that the coin was of type A? 00:09:21.240 --> 00:09:24.045 Where all you know is I pulled it out of the drawer. 00:09:24.045 --> 00:09:26.385 So what's the probability it's of type A. 00:09:26.385 --> 00:09:28.135 PROFESSOR 2: Let me go back and remind you 00:09:28.135 --> 00:09:31.570 what the drawer looks like. 00:09:31.570 --> 00:09:33.850 What is it? 00:09:33.850 --> 00:09:35.040 PROFESSOR: 0.5. 00:09:35.040 --> 00:09:36.590 0.5, or I'll write it like this. 00:09:36.590 --> 00:09:39.390 2 out of 4, which is 0.5. 00:09:39.390 --> 00:09:41.240 And what's the probability-- I'll 00:09:41.240 --> 00:09:52.180 shorten this-- for just type B. Yeah, 1/4 or 0.25. 00:09:52.180 --> 00:09:58.940 The probability that we're of type C is 1/4 or 0.25. 00:09:58.940 --> 00:10:01.565 This was one of our simpler counting problems. 00:10:04.130 --> 00:10:06.380 In the table, you can see that's encoded 00:10:06.380 --> 00:10:09.860 in the line in the column marked prior. 00:10:09.860 --> 00:10:13.150 Before we take any data, this is what we'd say. 00:10:13.150 --> 00:10:16.221 This is the sort of odds we'd give if we 00:10:16.221 --> 00:10:18.440 were going to bet on this. 00:10:18.440 --> 00:10:19.770 But now we have some data. 00:10:22.790 --> 00:10:28.000 So once we have data we can get the likelihood. 00:10:32.470 --> 00:10:34.540 The likelihood is the probability 00:10:34.540 --> 00:10:37.556 of seeing the data given a hypothesis. 00:10:40.200 --> 00:10:43.590 I won't write this out, I'll just point at the slide. 00:10:43.590 --> 00:10:46.749 John has a-- you were going to make your mouse bigger. 00:10:46.749 --> 00:10:47.540 PROFESSOR 2: I was. 00:10:47.540 --> 00:10:48.123 PROFESSOR: OK. 00:10:50.790 --> 00:10:53.180 So the likelihood column-- we use 00:10:53.180 --> 00:10:56.210 this term when we are talking about maximum likelihood-- 00:10:56.210 --> 00:11:00.240 is the probability of the data given the hypothesis. 00:11:00.240 --> 00:11:06.010 And this is typically easy, or often easy, to figure out. 00:11:06.010 --> 00:11:07.800 What's the probability of seeing heads-- 00:11:07.800 --> 00:11:12.850 our data was I flipped a heads-- if my coin is a 0.5 probability 00:11:12.850 --> 00:11:13.350 coin. 00:11:17.560 --> 00:11:25.550 So the probability of seeing heads given a type A coin 00:11:25.550 --> 00:11:26.170 is what? 00:11:26.170 --> 00:11:28.660 Somebody. 00:11:28.660 --> 00:11:30.644 It's 0.5. 00:11:30.644 --> 00:11:32.310 This is particularly easy because that's 00:11:32.310 --> 00:11:33.018 what we told you. 00:11:33.018 --> 00:11:38.480 The probability of heads with a type A coin is 0.5. 00:11:38.480 --> 00:11:47.160 What about for B and C. 0.6 and 0.9. 00:11:47.160 --> 00:11:52.650 So there right here is the likelihood column in the table. 00:11:52.650 --> 00:11:56.350 Now, how do we get the posterior? 00:11:56.350 --> 00:12:02.390 Bayes' Theorem says the probability of H 00:12:02.390 --> 00:12:04.760 given D, which is what I want. 00:12:04.760 --> 00:12:06.630 I want to know, given this data, what 00:12:06.630 --> 00:12:08.670 do I believe the coin to be? 00:12:08.670 --> 00:12:11.060 What's its probability? 00:12:11.060 --> 00:12:17.610 Is the probability of D given H times the probability of H 00:12:17.610 --> 00:12:20.840 all over the probability of D. 00:12:20.840 --> 00:12:24.230 Well, notice this numerator right here. 00:12:24.230 --> 00:12:28.430 That's just our likelihood times our hypothesis, 00:12:28.430 --> 00:12:32.280 our prior belief in our hypothesis. 00:12:32.280 --> 00:12:36.390 And so what do we get when I multiply the prior column 00:12:36.390 --> 00:12:37.860 by the likelihood column? 00:12:37.860 --> 00:12:39.720 I get this column in red. 00:12:39.720 --> 00:12:41.910 And I'll call that the unnormalized posterior. 00:12:41.910 --> 00:12:42.880 Why unnormalized? 00:12:42.880 --> 00:12:46.320 Because it doesn't sum to 1. 00:12:46.320 --> 00:12:48.160 How do I get it to sum to 1? 00:12:48.160 --> 00:12:52.070 There I have to divide by the probability of the data. 00:12:52.070 --> 00:12:54.050 And how do I get the probability of the data? 00:12:58.190 --> 00:13:03.942 That's called the-- I see someone mouthing it. 00:13:03.942 --> 00:13:05.150 The Law of Total Probability. 00:13:07.920 --> 00:13:10.540 And the Law of Total Probability says 00:13:10.540 --> 00:13:14.330 you multiply, for each hypothesis, 00:13:14.330 --> 00:13:19.340 this value times this value and add them up. 00:13:19.340 --> 00:13:25.820 Which is in fact just the sum of the red column. 00:13:25.820 --> 00:13:28.340 So there I've written it: 0.625. 00:13:28.340 --> 00:13:29.750 And now, how do I normalize? 00:13:29.750 --> 00:13:34.160 I just divide all my values in my unnormalized column 00:13:34.160 --> 00:13:37.430 by that normalizing factor, 0.625, 00:13:37.430 --> 00:13:40.640 and I get these, what I call the posterior. 00:13:44.690 --> 00:13:47.117 Let's say-- well, now let's analyze this. 00:13:47.117 --> 00:13:48.200 So now what's your belief? 00:13:48.200 --> 00:13:52.130 Which coin is the most likely coin? 00:13:52.130 --> 00:13:53.400 After seeing this data. 00:13:56.760 --> 00:13:59.870 Still A. But what's happened to the probability 00:13:59.870 --> 00:14:02.770 from our prior probability? 00:14:02.770 --> 00:14:03.370 It's dropped. 00:14:03.370 --> 00:14:06.041 Which one has gained probability. 00:14:06.041 --> 00:14:06.540 AUDIENCE: C. 00:14:06.540 --> 00:14:07.998 PROFESSOR: C. Does that make sense? 00:14:07.998 --> 00:14:14.440 If I flip a heads, which coin is the most likely to flip heads? 00:14:14.440 --> 00:14:15.930 C. So I've shifted it. 00:14:15.930 --> 00:14:19.120 Suppose I flipped heads three more times. 00:14:19.120 --> 00:14:21.955 What would you see in the probability as I kept updating? 00:14:29.763 --> 00:14:30.846 PROFESSOR 2: Say it again. 00:14:30.846 --> 00:14:32.290 AUDIENCE: The probability of C gets bigger. 00:14:32.290 --> 00:14:33.664 PROFESSOR 2: The probability of C 00:14:33.664 --> 00:14:36.800 would get bigger at the expense of the probabilities 00:14:36.800 --> 00:14:37.830 of the others. 00:14:37.830 --> 00:14:39.740 If I flipped heads 9 times in a row, 00:14:39.740 --> 00:14:46.120 I'd be very convinced that I had a type C coin. 00:14:46.120 --> 00:14:49.240 One last thing to note about this table. 00:14:49.240 --> 00:14:50.920 I put totals down here. 00:14:50.920 --> 00:14:53.540 What does this total of 1 represent in the first column? 00:14:56.190 --> 00:14:56.690 Emily. 00:14:56.690 --> 00:14:59.445 AUDIENCE: The total probabilities of anything, so 00:14:59.445 --> 00:15:01.120 the total space. 00:15:01.120 --> 00:15:02.800 PROFESSOR: Yeah. 00:15:02.800 --> 00:15:04.760 When I sum up the total probabilities, 00:15:04.760 --> 00:15:07.200 they sum up to 1. 00:15:07.200 --> 00:15:08.440 Same with this last. 00:15:08.440 --> 00:15:10.030 That's a probability. 00:15:10.030 --> 00:15:12.390 This I could sum, because they were probabilities. 00:15:12.390 --> 00:15:14.550 Why didn't I sum this column? 00:15:20.980 --> 00:15:22.670 Those are the likelihoods, and what's 00:15:22.670 --> 00:15:26.010 changing as I move down this column? 00:15:26.010 --> 00:15:30.570 What's changing is H. That's not a probability function. 00:15:30.570 --> 00:15:32.970 It doesn't sum to 1. 00:15:32.970 --> 00:15:37.540 There's no reason that the likelihoods have to sum to 1, 00:15:37.540 --> 00:15:42.200 and here you can see they clearly don't sum to 1. 00:15:42.200 --> 00:15:43.330 So I don't sum that. 00:15:43.330 --> 00:15:48.390 That's an important, maybe slightly subtle point, 00:15:48.390 --> 00:15:54.110 that the likelihood function is not a probability function. 00:15:54.110 --> 00:15:55.410 OK, any questions here? 00:15:58.644 --> 00:16:00.500 PROFESSOR 2: All right. 00:16:00.500 --> 00:16:06.760 So, let's go back to this document camera. 00:16:06.760 --> 00:16:08.540 I got my trusty platonic solids. 00:16:14.680 --> 00:16:18.810 So, we're going to go through a series of board questions now, 00:16:18.810 --> 00:16:22.640 three board questions which are going to allow you to practice 00:16:22.640 --> 00:16:25.240 making these tables and getting a sense for how 00:16:25.240 --> 00:16:29.470 to do Bayesian Updating, how to iterate it, just how it works. 00:16:29.470 --> 00:16:33.780 And our toy problem is going to be involving these 5 00:16:33.780 --> 00:16:34.800 platonic dice. 00:16:34.800 --> 00:16:37.567 You all have them, but let's not break them out today, 00:16:37.567 --> 00:16:39.900 because it would be very noisy with the cameras I think. 00:16:39.900 --> 00:16:46.440 We'll use our 4-sided-- here, I got big ones. 00:16:46.440 --> 00:16:49.690 We've got our 4-sided, 8-sided, 6-sided. 00:16:49.690 --> 00:16:50.880 No 6-sided. 00:16:50.880 --> 00:16:53.546 No, I don't have a 6-sided. 00:16:53.546 --> 00:16:54.910 I have the wrong dice. 00:16:54.910 --> 00:16:55.800 Hold on. 00:16:55.800 --> 00:17:02.290 We got 20, we got 4, we got 12, we got 8, ah-- 10, 00:17:02.290 --> 00:17:03.816 what are you doing there? 00:17:03.816 --> 00:17:04.649 That's not platonic. 00:17:09.800 --> 00:17:12.829 OK, and 6. 00:17:15.670 --> 00:17:17.760 So now I'm actually going to take these. 00:17:17.760 --> 00:17:18.609 You can watch this. 00:17:18.609 --> 00:17:20.380 All right, this is empty cup. 00:17:24.079 --> 00:17:25.680 OK, they're in there right? 00:17:25.680 --> 00:17:26.180 OK. 00:17:28.950 --> 00:17:30.710 How do we do this? 00:17:30.710 --> 00:17:31.780 How do we convince them? 00:17:31.780 --> 00:17:32.810 PROFESSOR: Should we choose one at random? 00:17:32.810 --> 00:17:33.560 PROFESSOR 2: Yeah. 00:17:36.985 --> 00:17:37.860 All right, choose it. 00:17:37.860 --> 00:17:40.580 Don't let anyone see. 00:17:40.580 --> 00:17:41.810 Hand it me. 00:17:41.810 --> 00:17:43.270 OK. 00:17:43.270 --> 00:17:45.100 So I have this die. 00:17:45.100 --> 00:17:47.190 Don't look at the others. 00:17:47.190 --> 00:17:50.300 I'm going to start rolling it, all right? 00:17:50.300 --> 00:17:51.450 In this secret bin. 00:17:51.450 --> 00:17:53.700 Jerry, you can ver-- we can have a student come verify 00:17:53.700 --> 00:17:56.200 that I'm not making this up. 00:17:56.200 --> 00:17:57.640 You ready? 00:17:57.640 --> 00:17:59.596 OK, here we go. 00:17:59.596 --> 00:18:01.360 OK, what did I get? 00:18:01.360 --> 00:18:02.220 8. 00:18:02.220 --> 00:18:03.200 All right. 00:18:03.200 --> 00:18:05.430 So, which die is it so far? 00:18:05.430 --> 00:18:06.340 What do you think? 00:18:06.340 --> 00:18:08.340 Could it be 4, the 4-sided die? 00:18:08.340 --> 00:18:09.580 6-sided? 00:18:09.580 --> 00:18:10.921 8? 00:18:10.921 --> 00:18:12.262 12? 00:18:12.262 --> 00:18:13.480 20? 00:18:13.480 --> 00:18:15.460 What's most likely? 00:18:15.460 --> 00:18:17.940 All right. 00:18:17.940 --> 00:18:20.522 Come take a look at another one. 00:18:20.522 --> 00:18:21.022 Ready? 00:18:25.802 --> 00:18:27.264 Ah, got a 1. 00:18:27.264 --> 00:18:28.430 AUDIENCE: It's the same die? 00:18:28.430 --> 00:18:30.910 PROFESSOR 2: It's the same die, it's the same die. 00:18:30.910 --> 00:18:33.061 All right, so now what? 00:18:33.061 --> 00:18:35.470 How does this change, anyone want to tell me 00:18:35.470 --> 00:18:40.450 how would this, now, change from before what you believe to be? 00:18:40.450 --> 00:18:43.060 Does it change what you believe at all, in any way? 00:18:43.060 --> 00:18:44.930 You still think it's the 8-sided. 00:18:44.930 --> 00:18:48.350 Does it make the 8-sided less likely, more likely relative 00:18:48.350 --> 00:18:49.880 to the others? 00:18:49.880 --> 00:18:50.690 More likely. 00:18:50.690 --> 00:18:51.190 Wonderful. 00:18:51.190 --> 00:18:53.090 See you guys all sense this in your bones. 00:18:53.090 --> 00:18:53.690 It's great. 00:18:53.690 --> 00:18:54.340 This is going to be so easy. 00:18:54.340 --> 00:18:55.340 For all of you. 00:18:55.340 --> 00:18:56.870 All right. 00:18:56.870 --> 00:19:00.310 We've got to keep going until we figure out with this die. 00:19:00.310 --> 00:19:01.630 You, you love to participate. 00:19:01.630 --> 00:19:02.180 Come on. 00:19:05.376 --> 00:19:07.740 Ready? 00:19:07.740 --> 00:19:08.240 AUDIENCE: 3. 00:19:08.240 --> 00:19:09.621 PROFESSOR 2: 3, oh man. 00:19:09.621 --> 00:19:10.120 All right. 00:19:12.860 --> 00:19:14.805 Ready? 00:19:14.805 --> 00:19:15.782 AUDIENCE: 12. 00:19:15.782 --> 00:19:16.490 PROFESSOR 2: Ooh. 00:19:21.780 --> 00:19:24.190 Now what happened? 00:19:24.190 --> 00:19:25.060 It's a 12-sided die? 00:19:25.060 --> 00:19:27.270 But it could be a 20-sided die, right? 00:19:27.270 --> 00:19:29.811 All right, let me just do a few more, I'll tell you it I get. 00:19:29.811 --> 00:19:39.580 I get 1, I get 10, I get 10, I get 12, I get 3, I get 10. 00:19:39.580 --> 00:19:41.060 All right. 00:19:41.060 --> 00:19:43.660 Who wants to bet? 00:19:43.660 --> 00:19:45.380 What would you be willing to bet? 00:19:45.380 --> 00:19:47.930 Suppose I was willing to give you like, I don't know, 00:19:47.930 --> 00:19:50.700 10 to 1 odds that it's a 20-sided die. 00:19:55.080 --> 00:19:57.640 I mean, of course, maybe I know something. 00:19:57.640 --> 00:19:59.990 I can see the die. 00:19:59.990 --> 00:20:01.990 Right, so what we want to be able to do 00:20:01.990 --> 00:20:04.445 is precisely answer that question. 00:20:04.445 --> 00:20:06.570 All right, and it's just do this Bayesian Updating. 00:20:06.570 --> 00:20:09.800 It's not so hard, but organizing your work is really important. 00:20:09.800 --> 00:20:11.000 So, all right. 00:20:11.000 --> 00:20:13.190 It was the 12-sided after all. 00:20:13.190 --> 00:20:15.180 So you all did good. 00:20:15.180 --> 00:20:18.140 All right, so this is the first board question. 00:20:18.140 --> 00:20:18.640 All right. 00:20:18.640 --> 00:20:21.700 I want to practice making the table Jerry showed you. 00:20:21.700 --> 00:20:23.640 Make one for if it were 13. 00:20:23.640 --> 00:20:26.240 Make one if I had rolled 5 instead. 00:20:26.240 --> 00:20:28.430 Or do the same question if I had rolled a 9. 00:20:28.430 --> 00:20:31.710 In each case, compute the posterior probabilities 00:20:31.710 --> 00:20:33.090 of each of the 5 types of dice. 00:20:47.210 --> 00:20:48.960 PROFESSOR: You knew it was a 20-sided die? 00:20:48.960 --> 00:20:49.410 AUDIENCE: Yeah. 00:20:49.410 --> 00:20:51.284 PROFESSOR: All right, what about if it's a 5? 00:20:51.284 --> 00:20:52.419 AUDIENCE: Yeah. 00:20:52.419 --> 00:20:53.460 PROFESSOR: But that's it. 00:20:53.460 --> 00:20:57.580 But you see how that captures it very nicely in a table. 00:20:57.580 --> 00:20:58.540 All of your intuition. 00:20:58.540 --> 00:20:59.653 You knew it was a-- 00:21:26.780 --> 00:21:29.310 PROFESSOR 2: So right now you're doing the D equals 13 case? 00:21:29.310 --> 00:21:31.601 AUDIENCE: Yeah, or I guess we could combine the tables. 00:21:33.395 --> 00:21:34.520 PROFESSOR 2: Oh, I'm sorry. 00:21:34.520 --> 00:21:35.400 You're not combining tables. 00:21:35.400 --> 00:21:36.590 This is your likelihood column here? 00:21:36.590 --> 00:21:36.930 AUDIENCE: Yeah. 00:21:36.930 --> 00:21:38.770 PROFESSOR 2: And then this is your unnormalized posterior 00:21:38.770 --> 00:21:39.550 column. 00:21:39.550 --> 00:21:40.590 Is that right? 00:21:40.590 --> 00:21:42.640 Or, no this is actually normalized. 00:21:42.640 --> 00:21:44.800 AUDIENCE: This is for if it were like any number. 00:21:44.800 --> 00:21:46.550 Like if we didn't know what number it was. 00:21:48.434 --> 00:21:49.600 PROFESSOR 2: The likelihood? 00:21:49.600 --> 00:21:50.050 AUDIENCE: Yes. 00:21:50.050 --> 00:21:51.050 PROFESSOR 2: Absolutely. 00:21:51.050 --> 00:21:54.931 So the likelihood doesn't depend on-- wait. 00:21:54.931 --> 00:21:55.680 What are we doing? 00:21:55.680 --> 00:21:56.900 Rolling a 5 or a 13? 00:21:56.900 --> 00:21:58.757 AUDIENCE: Rolling a 13, right? 00:21:58.757 --> 00:22:01.090 PROFESSOR 2: Right, so rolling a 13, for the likelihood, 00:22:01.090 --> 00:22:03.510 you're computing the probability of data that you saw, 00:22:03.510 --> 00:22:08.344 the 13, given the hypothesis say that it's the 4-sided die. 00:22:08.344 --> 00:22:09.010 AUDIENCE: Right. 00:22:09.010 --> 00:22:10.426 PROFESSOR 2: It's the 4-sided die, 00:22:10.426 --> 00:22:12.480 what's the probability you would roll a 13? 00:22:12.480 --> 00:22:13.115 AUDIENCE: 0. 00:22:13.115 --> 00:22:13.906 PROFESSOR 2: Great. 00:22:13.906 --> 00:22:14.790 Same with this, this and this. 00:22:14.790 --> 00:22:15.850 But what about here? 00:22:15.850 --> 00:22:17.527 If it's the 20-sided die, if that's 00:22:17.527 --> 00:22:19.860 your hypothesis, what's the probability you'd roll a 13. 00:22:19.860 --> 00:22:20.840 AUDIENCE: Oh no. 00:22:20.840 --> 00:22:21.820 Oh, 1/20. 00:22:21.820 --> 00:22:22.800 PROFESSOR 2: Exactly. 00:22:22.800 --> 00:22:25.130 So that's the likelihood column. 00:22:25.130 --> 00:22:26.992 So this we're going to erase. 00:22:26.992 --> 00:22:28.840 OK. 00:22:28.840 --> 00:22:31.565 AUDIENCE 2: Then we multiply this by this and get that. 00:22:31.565 --> 00:22:32.440 PROFESSOR 2: Exactly. 00:22:32.440 --> 00:22:37.000 AUDIENCE: [INAUDIBLE] divided by [INAUDIBLE]. 00:22:37.000 --> 00:22:38.420 PROFESSOR 2: Exactly, exactly. 00:22:38.420 --> 00:22:39.757 Does that make sense? 00:22:39.757 --> 00:22:41.590 AUDIENCE: Wait, we multiply these together-- 00:22:41.590 --> 00:22:43.090 AUDIENCE 2: Multiply those together. 00:22:54.088 --> 00:23:03.474 And then we divide 1 over 100, so it's going to be 0-- 00:23:03.474 --> 00:23:04.390 PROFESSOR 2: So, yeah. 00:23:04.390 --> 00:23:06.740 So you get to just-- 00:23:06.740 --> 00:23:07.657 AUDIENCE: And then HD. 00:23:07.657 --> 00:23:08.489 PROFESSOR 2: No, no. 00:23:08.489 --> 00:23:09.610 Oh yeah, you're right. 00:23:09.610 --> 00:23:10.110 HD. 00:23:12.630 --> 00:23:17.350 One thing I want to point out is instead of using a comma, 00:23:17.350 --> 00:23:19.810 all right, this is conditional probability 00:23:19.810 --> 00:23:24.070 so it's probability of D given H, like this. 00:23:24.070 --> 00:23:24.660 OK. 00:23:24.660 --> 00:23:26.455 There you go. 00:23:26.455 --> 00:23:27.580 So that's absolutely right. 00:23:27.580 --> 00:23:29.630 At the end, we want to know how likely each 00:23:29.630 --> 00:23:31.000 die was given the data. 00:23:31.000 --> 00:23:33.362 AUDIENCE: Because this total is just P. 00:23:33.362 --> 00:23:34.445 PROFESSOR 2: That's right. 00:23:37.790 --> 00:23:47.700 Which is 1 over 100. 00:23:47.700 --> 00:23:48.200 Great. 00:23:48.200 --> 00:23:49.587 So now, do it again. 00:23:49.587 --> 00:23:50.170 Yeah, exactly. 00:23:50.170 --> 00:23:51.836 You don't even need to make a new table. 00:24:02.450 --> 00:24:04.692 PROFESSOR 2: Awesome. 00:24:04.692 --> 00:24:06.604 AUDIENCE: So this column [INAUDIBLE] 00:24:06.604 --> 00:24:08.516 to the probability, because [INAUDIBLE]. 00:24:11.980 --> 00:24:14.692 PROFESSOR 2: Yeah, you take each hypothesis times probability 00:24:14.692 --> 00:24:16.150 of data given the hypothesis, which 00:24:16.150 --> 00:24:20.210 is the same as the probability of D and H. 00:24:20.210 --> 00:24:21.210 AUDIENCE: Yeah, exactly. 00:24:21.210 --> 00:24:23.170 PROFESSOR 2: So your partitioning it up, sort of, 00:24:23.170 --> 00:24:24.170 based on the hypotheses. 00:24:24.170 --> 00:24:27.502 AUDIENCE: Because this equals that times probability of D. 00:24:27.502 --> 00:24:28.585 PROFESSOR 2: That's right. 00:24:28.585 --> 00:24:29.724 AUDIENCE: D confuses me. 00:24:29.724 --> 00:24:30.500 What is P of D? 00:24:30.500 --> 00:24:32.041 PROFESSOR 2: Probability of the data. 00:24:32.041 --> 00:24:33.770 So, right. 00:24:33.770 --> 00:24:37.020 So a priori, you don't know which die it is, right? 00:24:37.020 --> 00:24:40.810 But there's still a probability of getting a 13. 00:24:40.810 --> 00:24:41.420 Right? 00:24:41.420 --> 00:24:43.140 And what is the probability? 00:24:43.140 --> 00:24:46.140 Well first, we would have to, in this case, 00:24:46.140 --> 00:24:47.685 we could only pick the 20-sided die, 00:24:47.685 --> 00:24:49.470 that has a 1/5 chance of happening. 00:24:49.470 --> 00:24:51.970 And then, given that we picked the 20-sided die, 00:24:51.970 --> 00:24:55.010 we have to roll a 13, which is a 1/20 chance. 00:24:55.010 --> 00:24:59.736 Which is exactly why this sums to 1 in a 100. 00:24:59.736 --> 00:25:01.110 And then more generally, you have 00:25:01.110 --> 00:25:02.318 the Law of Total Probability. 00:25:09.310 --> 00:25:15.875 So here, I think you should keep in mind which is the D 00:25:15.875 --> 00:25:18.230 and which is H. Maybe write it above, 00:25:18.230 --> 00:25:24.050 otherwise it's a little confusing to know what-- great. 00:25:24.050 --> 00:25:26.610 OK, and so these are the final probabilities 00:25:26.610 --> 00:25:29.138 you got if you rolled a 9. 00:25:29.138 --> 00:25:30.620 Is that right? 00:25:30.620 --> 00:25:32.775 So there's something off here right? 00:25:32.775 --> 00:25:34.358 AUDIENCE: Sorry, these are our priors. 00:25:36.464 --> 00:25:38.750 I'm not labeling this. 00:25:38.750 --> 00:25:39.458 These are priors. 00:25:39.458 --> 00:25:40.955 This is gonna be [INAUDIBLE]. 00:25:45.132 --> 00:25:47.590 PROFESSOR 2: So actually here, I think you have likelihood. 00:25:47.590 --> 00:25:50.230 This is likelihood here, exactly. 00:25:50.230 --> 00:25:52.390 That's likelihood. 00:25:52.390 --> 00:25:54.710 Your prior should be what? 00:25:54.710 --> 00:25:55.949 AUDIENCE: 0.2. 00:25:55.949 --> 00:25:57.240 PROFESSOR 2: What should it be? 00:25:57.240 --> 00:25:58.030 AUDIENCE: 0.2 for each one. 00:25:58.030 --> 00:25:58.410 PROFESSOR 2: Yeah, 1/5. 00:25:58.410 --> 00:25:59.576 Oh, I didn't see the points. 00:25:59.576 --> 00:26:02.260 Yeah, so that's your prior, 1/5 for each. 00:26:02.260 --> 00:26:03.620 This is your likelihood. 00:26:03.620 --> 00:26:05.304 Then you multiply them, and that's 00:26:05.304 --> 00:26:06.720 the unnormalized posterior, right? 00:26:09.426 --> 00:26:11.050 Let's have a brief discussion about it, 00:26:11.050 --> 00:26:11.980 and then we're going to give you something 00:26:11.980 --> 00:26:13.063 a little more challenging. 00:26:15.910 --> 00:26:20.050 So for rolling a 13, right, the first thing you do 00:26:20.050 --> 00:26:22.510 is ask, what is the prior? 00:26:22.510 --> 00:26:23.980 And you saw, I had 5 dice. 00:26:23.980 --> 00:26:25.370 I shook it around. 00:26:25.370 --> 00:26:27.840 A student reached in randomly, not even me. 00:26:27.840 --> 00:26:31.800 So we can trust that it was 1/5 for each die. 00:26:31.800 --> 00:26:35.330 Equally likely, that's a uniform prior, right, discrete 00:26:35.330 --> 00:26:37.920 probability mass function. 00:26:37.920 --> 00:26:39.200 Great. 00:26:39.200 --> 00:26:40.140 Likelihood. 00:26:40.140 --> 00:26:42.430 Well, that's going to depend on the data, 00:26:42.430 --> 00:26:45.740 because it's the probability of the data given the hypotheses. 00:26:45.740 --> 00:26:49.310 The hypothesis in this case is how many sides on the die. 00:26:49.310 --> 00:26:53.430 So, if the data's 13, what's the probability that I 00:26:53.430 --> 00:26:57.950 would roll that data, a 13, given that it was the 4-sided 00:26:57.950 --> 00:26:59.300 die. 00:26:59.300 --> 00:26:59.890 0. 00:26:59.890 --> 00:27:00.390 Right? 00:27:00.390 --> 00:27:01.150 It's impossible. 00:27:01.150 --> 00:27:04.780 Similarly 4, 6, 8 and 12 sides. 00:27:04.780 --> 00:27:07.440 But given that I had the 20-sided die, 00:27:07.440 --> 00:27:10.850 there would be a 1/20 probability of rolling a 13. 00:27:10.850 --> 00:27:12.306 That's the likelihood column. 00:27:12.306 --> 00:27:13.280 All right. 00:27:13.280 --> 00:27:16.140 The next column, what we called the unnormalized posterior, we 00:27:16.140 --> 00:27:19.840 multiply, everything is 0, and then we get 1 out of 100 00:27:19.840 --> 00:27:25.140 as the product for the 20-sided die. 00:27:25.140 --> 00:27:29.100 If we sum this up by the Law of Total Probability, 00:27:29.100 --> 00:27:31.201 we get the probability of the data. 00:27:31.201 --> 00:27:32.950 Now, a student asked, what does that mean? 00:27:32.950 --> 00:27:34.770 We don't know which die it was? 00:27:34.770 --> 00:27:37.850 In this case, what it means is that it's 00:27:37.850 --> 00:27:42.630 reasonable to ask the question, well, if someone reaches in, 00:27:42.630 --> 00:27:45.780 grabs 1 of these 5 dice randomly and rolls it 00:27:45.780 --> 00:27:48.070 and gets-- I mean, what's the probability 00:27:48.070 --> 00:27:50.440 that the result would be a 13. 00:27:50.440 --> 00:27:50.980 Right? 00:27:50.980 --> 00:27:53.270 That's a reasonable question. 00:27:53.270 --> 00:27:56.790 And in this case, it's not so hard to analyze that directly, 00:27:56.790 --> 00:27:58.870 because the only way that could happen 00:27:58.870 --> 00:28:02.780 is if they picked the 20-sided die and then rolled a 13. 00:28:02.780 --> 00:28:06.370 Well, given that they picked the 20-sided die. 00:28:06.370 --> 00:28:09.440 Well, there's a 1/5 chance they picked the 20-sided die, 00:28:09.440 --> 00:28:13.010 and then a 1/20 chance they roll of 13 given that they 00:28:13.010 --> 00:28:15.080 picked the 20-sided die. 00:28:15.080 --> 00:28:20.880 So the probability of rolling a 13 overall is 1 in 100. 00:28:20.880 --> 00:28:23.369 The probability of the data. 00:28:23.369 --> 00:28:25.160 PROFESSOR: John, let me just say one thing. 00:28:25.160 --> 00:28:27.035 The way we would have done this before is you 00:28:27.035 --> 00:28:28.990 would have made a tree. 00:28:28.990 --> 00:28:30.690 The top branch of the tree would have 00:28:30.690 --> 00:28:33.440 gone to the 5 types of die, and then each of those die 00:28:33.440 --> 00:28:35.570 would have gone to all the possibilities. 00:28:35.570 --> 00:28:39.340 And the only branch that leads to a 13 00:28:39.340 --> 00:28:42.874 is down to the 20-sided die and then down to the 13. 00:28:42.874 --> 00:28:44.540 PROFESSOR 2: Notice also that this would 00:28:44.540 --> 00:28:47.100 be a very big tree, right? 00:28:47.100 --> 00:28:50.700 You have 5 branches, and then this would have 4 and 6, 00:28:50.700 --> 00:28:53.480 and 8, and 12, and 20 coming off of it. 00:28:53.480 --> 00:28:54.630 Finally we normalize. 00:28:54.630 --> 00:28:57.330 Using Bayes Rule, this gives us a posterior probability mass 00:28:57.330 --> 00:28:57.890 function. 00:28:57.890 --> 00:29:01.370 It's 0 for everything but the 20-sided hypothesis. 00:29:01.370 --> 00:29:06.090 Therefore, we get 100% belief, 100% probability, 00:29:06.090 --> 00:29:09.930 given that we rolled a 13, that it was the 20-sided die. 00:29:09.930 --> 00:29:12.870 Now of course you all knew that immediately. 00:29:12.870 --> 00:29:15.140 If you roll a 13, it has to be the 20-sided die. 00:29:15.140 --> 00:29:17.540 So let's look at a more interesting one that you did, 00:29:17.540 --> 00:29:18.050 the 5. 00:29:18.050 --> 00:29:18.950 So what changes? 00:29:18.950 --> 00:29:21.270 That's the big question. 00:29:21.270 --> 00:29:27.494 The only thing that changes is the likelihood. 00:29:27.494 --> 00:29:28.910 I mean, granted, the rest changes. 00:29:28.910 --> 00:29:30.540 But everything in these two columns 00:29:30.540 --> 00:29:32.250 are computed from the first two. 00:29:32.250 --> 00:29:37.080 The prior stays the same, but the likelihood function changes 00:29:37.080 --> 00:29:38.860 because we have different data. 00:29:38.860 --> 00:29:40.890 In this case, if we roll a 5, we can't 00:29:40.890 --> 00:29:42.980 get that on the 4-sided die. 00:29:42.980 --> 00:29:45.930 We have a 1/6 chance on the 6-sided, 1/8 on the 8-sided, 00:29:45.930 --> 00:29:48.970 1/12 on the 12-sided and 1/20 on the 20-sided. 00:29:48.970 --> 00:29:52.270 Multiply those columns, get the unnormalized probability, 00:29:52.270 --> 00:29:54.860 or the unnormalized posterior. 00:29:54.860 --> 00:29:58.040 That sums to the probability of getting a 5. 00:29:58.040 --> 00:30:01.160 When you divide by it, it normalizes the posterior, 00:30:01.160 --> 00:30:02.990 and what do we see? 00:30:02.990 --> 00:30:07.260 Well, of course you couldn't have used the 4-sided die, 00:30:07.260 --> 00:30:11.500 and of the remaining the most likely, the most probable, 00:30:11.500 --> 00:30:15.560 given the data, is the 6-sided die. 00:30:15.560 --> 00:30:17.310 It has a probability, that hypothesis 00:30:17.310 --> 00:30:21.470 has a probability of about 40%, compared to the 20-sided die, 00:30:21.470 --> 00:30:24.020 which is only about 12%. 00:30:24.020 --> 00:30:25.270 OK. 00:30:25.270 --> 00:30:27.450 Last one. 00:30:27.450 --> 00:30:28.620 So this is the same deal. 00:30:28.620 --> 00:30:31.530 Again, the likelihood function changes. 00:30:31.530 --> 00:30:32.890 The prior's the same. 00:30:32.890 --> 00:30:34.230 We go through the table. 00:30:34.230 --> 00:30:36.450 The only two possibilities are the 12 and 20-sided, 00:30:36.450 --> 00:30:38.850 and the 12-sided, given that you rolled a 9, 00:30:38.850 --> 00:30:42.360 is about twice as likely as the 20-sided, which 00:30:42.360 --> 00:30:45.680 should fit with your intuition. 00:30:45.680 --> 00:30:46.530 OK. 00:30:46.530 --> 00:30:50.150 So next, we want you to not erase, but go back 00:30:50.150 --> 00:30:51.810 to the boards you've already made, 00:30:51.810 --> 00:30:54.370 and we're going to explore a little bit the idea of repeated 00:30:54.370 --> 00:30:55.070 trials. 00:30:55.070 --> 00:30:55.570 Right? 00:30:55.570 --> 00:30:58.070 Often if you're collecting data, you don't just 00:30:58.070 --> 00:31:00.120 collect one data point, you collect 00:31:00.120 --> 00:31:02.860 a series of data points, maybe a series of patients 00:31:02.860 --> 00:31:03.950 in a clinical trial. 00:31:03.950 --> 00:31:05.520 You have data coming in, right? 00:31:05.520 --> 00:31:06.730 And you can update. 00:31:06.730 --> 00:31:09.110 You can update each time you get more data. 00:31:09.110 --> 00:31:12.585 And when you do this, your prior becomes a posterior 00:31:12.585 --> 00:31:13.960 in the first update, and then you 00:31:13.960 --> 00:31:17.110 use that posterior as the prior for the next update. 00:31:17.110 --> 00:31:21.680 Your beliefs are being updated as more data comes in. 00:31:21.680 --> 00:31:24.390 If you think about it, like when you walk outside this course, 00:31:24.390 --> 00:31:26.694 if you're like me you'll sort of suddenly 00:31:26.694 --> 00:31:28.610 realize you're doing this process all the time 00:31:28.610 --> 00:31:30.500 in your life, and you'll start approaching problems this way. 00:31:30.500 --> 00:31:31.874 And if you really like me, you'll 00:31:31.874 --> 00:31:34.530 change your religion on Facebook to Bayesian. 00:31:34.530 --> 00:31:37.110 But you don't have to be that extreme. 00:31:37.110 --> 00:31:39.180 So great. 00:31:39.180 --> 00:31:42.970 So here what we want you to do is pretend that I roll the die 00:31:42.970 --> 00:31:43.860 and then got a 5. 00:31:43.860 --> 00:31:45.320 And then I rolled the same die. 00:31:45.320 --> 00:31:46.200 I didn't get a different one. 00:31:46.200 --> 00:31:47.824 I just rolled the same die, just like I 00:31:47.824 --> 00:31:49.950 did in that demonstration, and then I got a 9. 00:31:49.950 --> 00:31:54.110 OK, so now we have a sequence of two pieces of data, 5 and 9. 00:31:54.110 --> 00:32:06.120 So I want to find the posterior that-- update in two steps. 00:32:06.120 --> 00:32:07.590 I'm doing this one. 00:32:07.590 --> 00:32:08.990 I'm doing this one! 00:32:08.990 --> 00:32:13.895 OK, so let me show you how to do this, and then you'll do it. 00:32:13.895 --> 00:32:15.770 OK, so first we're going to update for the 5. 00:32:15.770 --> 00:32:19.310 Then we're going to update again for the 9. 00:32:19.310 --> 00:32:21.530 Magic! 00:32:21.530 --> 00:32:24.410 The prior, 1/5 for each, right? 00:32:24.410 --> 00:32:26.030 That's where we're starting from. 00:32:26.030 --> 00:32:27.400 We roll a 5. 00:32:27.400 --> 00:32:28.890 That's our likelihood. 00:32:28.890 --> 00:32:31.410 Again, you get a 0 for the 4-sided 00:32:31.410 --> 00:32:35.070 and you get 1 over n for each of the other n-sided 00:32:35.070 --> 00:32:36.050 possibilities. 00:32:36.050 --> 00:32:37.300 We multiply. 00:32:37.300 --> 00:32:39.090 That gives us our unnormalized posterior 00:32:39.090 --> 00:32:40.630 for the first piece of data. 00:32:40.630 --> 00:32:41.420 Great. 00:32:41.420 --> 00:32:45.620 Now that is going to be our new prior. 00:32:45.620 --> 00:32:47.570 Now, you may complain, wait a minute. 00:32:47.570 --> 00:32:49.170 This doesn't add to 1. 00:32:49.170 --> 00:32:53.340 But it's OK, because we're going to do it again. 00:32:53.340 --> 00:32:56.430 At the very end we'll normalize, and we'll get the same answer 00:32:56.430 --> 00:32:59.240 as if we had normalized here and then normalized again. 00:32:59.240 --> 00:33:01.840 Because we're just multiplying the whole column 00:33:01.840 --> 00:33:03.700 by the same scalar, by the same real number. 00:33:03.700 --> 00:33:04.200 OK? 00:33:04.200 --> 00:33:06.020 So it saves you work to not bother 00:33:06.020 --> 00:33:08.951 to normalize the first time. 00:33:08.951 --> 00:33:09.450 Great. 00:33:09.450 --> 00:33:11.440 So that's our new prior. 00:33:11.440 --> 00:33:13.740 We get our next piece of data, the 9. 00:33:13.740 --> 00:33:18.460 We have 0, 0, 0, 1/12, and 1/20 as the likelihoods. 00:33:18.460 --> 00:33:22.450 We multiply this new prior by this likelihood, 00:33:22.450 --> 00:33:25.220 get the new unnormalized posterior. 00:33:25.220 --> 00:33:30.390 It only has two possible nonzero probabilities in it. 00:33:30.390 --> 00:33:35.680 The sum, the total probability of rolling a 5 and then a 9 00:33:35.680 --> 00:33:44.646 is only 0.0019, OK? 00:33:44.646 --> 00:33:47.001 PROFESSOR: That's [INAUDIBLE] unnormalized [INAUDIBLE]. 00:33:47.001 --> 00:33:48.890 PROFESSOR 2: Ah! 00:33:48.890 --> 00:33:49.580 Jerry caught me. 00:33:49.580 --> 00:33:51.730 I was hoping a student would catch me. 00:33:51.730 --> 00:33:52.470 OK, maybe not. 00:33:52.470 --> 00:33:54.550 Maybe I was fooling myself. 00:33:54.550 --> 00:33:56.130 We haven't normalized. 00:33:56.130 --> 00:33:59.250 So, it's not quite right. 00:33:59.250 --> 00:34:01.800 If we had normalized here, then by the time we got to here, 00:34:01.800 --> 00:34:05.280 we could interpret this as that total probability. 00:34:05.280 --> 00:34:07.890 But in any case, the point is we can normalize 00:34:07.890 --> 00:34:10.750 to get to the final stage, this final posterior, 00:34:10.750 --> 00:34:13.120 by dividing by the sum here. 00:34:13.120 --> 00:34:15.527 Now it adds to 1, and at the end of the day, 00:34:15.527 --> 00:34:16.360 what do we conclude? 00:34:16.360 --> 00:34:20.199 We conclude that it's almost three times more likely 00:34:20.199 --> 00:34:23.770 that it was the 12-sided die than that it 00:34:23.770 --> 00:34:26.100 was the 20-sided die. 00:34:26.100 --> 00:34:26.600 Right? 00:34:26.600 --> 00:34:29.880 So this is very similar to what happened when randomly 00:34:29.880 --> 00:34:31.600 had the 12-sided and rolled it. 00:34:31.600 --> 00:34:34.800 After two times, I had data that looked a lot like this, 00:34:34.800 --> 00:34:37.290 and if we had rigorously computed those probabilities, 00:34:37.290 --> 00:34:39.170 this is what we would have gotten. 00:34:39.170 --> 00:34:41.489 OK? 00:34:41.489 --> 00:34:44.639 So now, now it's your turn. 00:34:44.639 --> 00:34:48.190 So, we want to try the same thing, but in the other order. 00:34:48.190 --> 00:34:51.100 Suppose I first roll the 9 and then I roll the 5. 00:34:51.100 --> 00:34:52.469 So that's the first question. 00:34:52.469 --> 00:34:55.600 The second question is, can you do it in one go? 00:34:55.600 --> 00:34:59.120 Instead of doing it updating for the 9 and then for the 5, 00:34:59.120 --> 00:35:02.340 can you incorporate both the 9 and the 5 00:35:02.340 --> 00:35:05.800 into a single likelihood function 00:35:05.800 --> 00:35:07.520 and do the updating in one step? 00:35:07.520 --> 00:35:10.080 All right, so try these and think about what's similar, 00:35:10.080 --> 00:35:12.510 what's different than what I just did. 00:35:12.510 --> 00:35:13.920 You don't have to. 00:35:13.920 --> 00:35:15.747 You don't have to. 00:35:15.747 --> 00:35:17.330 You could normalize it, then you would 00:35:17.330 --> 00:35:19.000 multiply by the likelihood. 00:35:19.000 --> 00:35:20.850 You'd get something again unnormalized, 00:35:20.850 --> 00:35:22.544 and then you'd have normalize again, 00:35:22.544 --> 00:35:24.210 and then you'd get to your final answer. 00:35:24.210 --> 00:35:27.430 Or, don't bother normalizing at this stage. 00:35:27.430 --> 00:35:30.290 Just multiply this by the likelihood, and then just only 00:35:30.290 --> 00:35:31.600 normalize once at the very end. 00:35:31.600 --> 00:35:34.580 And you'll get the same answer because, you know, 00:35:34.580 --> 00:35:36.530 normalizing once and then normalizing again 00:35:36.530 --> 00:35:39.704 is the same as just normalizing at the very end. 00:35:39.704 --> 00:35:41.620 You're just multiplying a vector, this column, 00:35:41.620 --> 00:35:44.585 by rescaling it to make it unit length-- 00:35:44.585 --> 00:35:46.251 AUDIENCE: And then you're multiplying it 00:35:46.251 --> 00:35:47.043 by something else-- 00:35:47.043 --> 00:35:47.917 PROFESSOR 2: Exactly. 00:35:47.917 --> 00:35:48.890 Exactly. 00:35:48.890 --> 00:35:50.616 AUDIENCE: Isn't number 1 going to be 00:35:50.616 --> 00:35:52.945 the same answer as the example we just did, 00:35:52.945 --> 00:35:54.820 because you're going to be updating-- I mean, 00:35:54.820 --> 00:35:55.694 it's going different. 00:35:55.694 --> 00:35:58.061 OK, we just have to do it. 00:35:58.061 --> 00:35:59.060 PROFESSOR 2: Absolutely. 00:35:59.060 --> 00:36:01.699 So, yes, absolutely. 00:36:01.699 --> 00:36:03.490 All of these could give you the same answer 00:36:03.490 --> 00:36:05.156 because it's the same exact data, right? 00:36:05.156 --> 00:36:06.790 Starting from the same priors. 00:36:06.790 --> 00:36:08.400 But what we want you to really see 00:36:08.400 --> 00:36:10.702 is how the math proves that. 00:36:10.702 --> 00:36:13.160 AUDIENCE: I mean we're just multiplying in different order. 00:36:13.160 --> 00:36:14.243 PROFESSOR 2: That's right. 00:36:14.243 --> 00:36:16.900 So it all comes down to what? 00:36:16.900 --> 00:36:19.280 What property of multiplication? 00:36:19.280 --> 00:36:20.944 AUDIENCE: Associative? 00:36:20.944 --> 00:36:21.860 AUDIENCE: Commutative. 00:36:21.860 --> 00:36:22.820 PROFESSOR 2: Commutative. 00:36:22.820 --> 00:36:24.486 I guess you're also using associativity. 00:36:24.486 --> 00:36:27.290 It's just that multiplication is commutative, right? 00:36:27.290 --> 00:36:30.640 So it doesn't matter what order you multiply these columns in, 00:36:30.640 --> 00:36:32.776 the final result's going to be the same. 00:36:32.776 --> 00:36:35.860 So just about how the order changes depending 00:36:35.860 --> 00:36:38.740 on the order of the 5 versus the 9, and also, 00:36:38.740 --> 00:36:40.490 what happens if you think of both at once, 00:36:40.490 --> 00:36:42.837 how would you then make a table directly for that? 00:36:42.837 --> 00:36:43.670 [INTERPOSING VOICES] 00:37:03.650 --> 00:37:04.220 PROFESSOR: 5. 00:37:04.220 --> 00:37:04.720 Oh, I see. 00:37:04.720 --> 00:37:08.602 So you multiplied-- here, is this your likelihood? 00:37:08.602 --> 00:37:09.560 So you multiplied that. 00:37:09.560 --> 00:37:11.890 Where's your plus times this? 00:37:11.890 --> 00:37:14.850 Oh, you even normalized. 00:37:14.850 --> 00:37:15.357 Right. 00:37:15.357 --> 00:37:16.940 So did you understand John's statement 00:37:16.940 --> 00:37:19.016 that you didn't actually have to normalize? 00:37:19.016 --> 00:37:19.750 AUDIENCE: Right. 00:37:19.750 --> 00:37:20.960 I mean, if you'd already done it. 00:37:20.960 --> 00:37:22.126 PROFESSOR: No, I understand. 00:37:22.126 --> 00:37:23.430 It's a nicer number than 1/60. 00:37:23.430 --> 00:37:24.860 But if you didn't have it, you could 00:37:24.860 --> 00:37:27.068 have just used this column and normalized at the end. 00:37:27.068 --> 00:37:27.867 That make sense? 00:37:27.867 --> 00:37:28.367 Excellent. 00:37:35.049 --> 00:37:35.840 I'm not trying to-- 00:37:35.840 --> 00:37:37.820 AUDIENCE: This is just if you do it in one step. 00:37:37.820 --> 00:37:38.286 PROFESSOR: I see. 00:37:38.286 --> 00:37:39.000 AUDIENCE: You just multiply. 00:37:39.000 --> 00:37:40.040 PROFESSOR: Did you get different answers? 00:37:40.040 --> 00:37:40.740 AUDIENCE: No, it's the same. 00:37:40.740 --> 00:37:41.250 PROFESSOR: It's the same. 00:37:41.250 --> 00:37:41.920 It makes sense. 00:37:41.920 --> 00:37:42.670 You have the data. 00:37:42.670 --> 00:37:44.852 That's the evidence you have. 00:37:44.852 --> 00:37:46.620 Right. 00:37:46.620 --> 00:37:48.342 So was this all clear? 00:37:48.342 --> 00:37:49.800 Was John's statement about not only 00:37:49.800 --> 00:37:51.440 needing the unnormalized posterior 00:37:51.440 --> 00:37:54.310 in the middle reasonably clear? 00:37:54.310 --> 00:37:55.810 AUDIENCE: What were talking about is 00:37:55.810 --> 00:37:59.394 if you were to use the normalized one, 00:37:59.394 --> 00:38:01.386 would then your unnormalized posterior 00:38:01.386 --> 00:38:03.690 be the actual probabilities? 00:38:03.690 --> 00:38:04.860 PROFESSOR: No. 00:38:04.860 --> 00:38:09.360 No, because in order to use the Law of Total Probability, 00:38:09.360 --> 00:38:10.860 you have to multiply your likelihood 00:38:10.860 --> 00:38:12.730 by genuine probabilities. 00:38:12.730 --> 00:38:15.407 And the unnormalized posterior is not a genuine probability. 00:38:15.407 --> 00:38:17.240 AUDIENCE: But if you use the normalized one. 00:38:17.240 --> 00:38:18.990 PROFESSOR: If you used the normalized one, 00:38:18.990 --> 00:38:24.210 then it would be-- that would be the total probability of that. 00:38:24.210 --> 00:38:25.235 That's exactly right. 00:38:25.235 --> 00:38:26.610 AUDIENCE: But, in this case, it's 00:38:26.610 --> 00:38:28.050 this that's the total probability, right? 00:38:28.050 --> 00:38:28.716 PROFESSOR: Yeah. 00:38:28.716 --> 00:38:31.440 This is now your posterior probability. 00:38:31.440 --> 00:38:32.690 These are now the probability. 00:38:32.690 --> 00:38:34.110 So, make a distinction. 00:38:34.110 --> 00:38:36.970 The total probability is about the data, 00:38:36.970 --> 00:38:42.249 and these probabilities here are about the hypotheses. 00:38:42.249 --> 00:38:45.182 AUDIENCE: Sort of like the fit of the hypothesis to the data. 00:38:45.182 --> 00:38:46.640 PROFESSOR: You could think of that. 00:38:46.640 --> 00:38:47.890 They're genuine probabilities. 00:38:47.890 --> 00:38:52.700 This is the probability that you chose the 12-sided die given 00:38:52.700 --> 00:38:53.200 the data. 00:38:53.200 --> 00:38:59.410 That is, if I did this a million times and every time, 00:38:59.410 --> 00:39:02.360 I only looked at the times I rolled a 9 and then the 5, 00:39:02.360 --> 00:39:05.160 you would find this fraction of those times 00:39:05.160 --> 00:39:07.460 would be-- it would be a 12-sided 00:39:07.460 --> 00:39:09.060 and this fraction would be a 20-sided. 00:39:09.060 --> 00:39:12.320 So, I forget what the numbers were, like 3 to 1. 00:39:12.320 --> 00:39:12.920 Yeah. 00:39:12.920 --> 00:39:13.942 Makes sense? 00:39:13.942 --> 00:39:15.270 Excellent. 00:39:15.270 --> 00:39:19.010 AUDIENCE: So we multiply them together, but then-- 00:39:19.010 --> 00:39:21.170 PROFESSOR 2: You get the same exact answer, right? 00:39:21.170 --> 00:39:23.332 AUDIENCE: So we get these, this answer. 00:39:23.332 --> 00:39:25.790 AUDIENCE 2: That's right, so it's unnormalized. 00:39:25.790 --> 00:39:26.520 PROFESSOR: And then you normalize 00:39:26.520 --> 00:39:27.770 it and you get this, right. 00:39:27.770 --> 00:39:31.015 AUDIENCE: There's no way to normalize it without-- 00:39:31.015 --> 00:39:33.390 PROFESSOR 2: All right, let's come back together and take 00:39:33.390 --> 00:39:35.070 a look at what happened here. 00:39:38.470 --> 00:39:39.910 I saw some aha moments. 00:39:39.910 --> 00:39:40.590 That was great. 00:39:43.410 --> 00:39:45.160 Some people cited The Commutative Property 00:39:45.160 --> 00:39:46.040 of Multiplication. 00:39:46.040 --> 00:39:47.040 That made me very happy. 00:39:49.510 --> 00:39:52.100 So, what do we do? 00:39:52.100 --> 00:39:54.175 So when we do the 9 and then the 5, 00:39:54.175 --> 00:39:56.300 it looks just like what I did with the 5 and the 9, 00:39:56.300 --> 00:39:59.240 except the likelihood columns are flipped. 00:39:59.240 --> 00:40:04.490 So we have our prior, the likelihood for the 9 00:40:04.490 --> 00:40:07.050 gives us this column. 00:40:07.050 --> 00:40:10.550 The product gives us an, first, unnormalized posterior. 00:40:10.550 --> 00:40:14.540 Then we do the likelihood for rolling a 5. 00:40:14.540 --> 00:40:17.000 The product of the unnormalized posterior 00:40:17.000 --> 00:40:18.750 and the second likelihood gives us 00:40:18.750 --> 00:40:22.410 our final unnormalized posterior. 00:40:22.410 --> 00:40:25.550 And notice the sum is still exactly what it was before, 00:40:25.550 --> 00:40:27.910 0.019. 00:40:27.910 --> 00:40:30.940 When we normalize, we get exactly the same answer 00:40:30.940 --> 00:40:31.810 that we had before. 00:40:31.810 --> 00:40:34.720 Now, if you contrast this with the table 00:40:34.720 --> 00:40:36.680 we looked at when we did the 5 and then 00:40:36.680 --> 00:40:41.940 the 9, what changed is these two columns were flipped around. 00:40:41.940 --> 00:40:44.210 And so I suppose this column was different 00:40:44.210 --> 00:40:45.960 and this column was different as a result. 00:40:45.960 --> 00:40:50.740 And then the final result, however, was exactly the same. 00:40:50.740 --> 00:40:53.674 Now, what if we do them both at once? 00:40:53.674 --> 00:40:54.590 What should that mean? 00:40:57.340 --> 00:40:59.500 If we do them both at once, then, 00:40:59.500 --> 00:41:02.220 well we start from the same uniform prior, 00:41:02.220 --> 00:41:05.780 and our likelihood-- now, we should think of the data 00:41:05.780 --> 00:41:09.700 as both rolls, the 5 and the 9. 00:41:09.700 --> 00:41:10.790 OK. 00:41:10.790 --> 00:41:15.100 Now what's important here is that if we condition 00:41:15.100 --> 00:41:16.400 on a given hypothesis. 00:41:16.400 --> 00:41:18.970 For example, if we condition on the hypothesis 00:41:18.970 --> 00:41:22.980 that it's a 12-sided die, then the first roll 00:41:22.980 --> 00:41:27.130 and the second roll are independent events, right? 00:41:27.130 --> 00:41:31.260 It means that we can figure out the probability of a 5 00:41:31.260 --> 00:41:33.430 and then a 9 on the 12-sided die. 00:41:33.430 --> 00:41:35.610 It's just the product of the probability 00:41:35.610 --> 00:41:38.820 of a 5 and a 12-sided die and the probability of a 9 00:41:38.820 --> 00:41:40.140 on a 12-sided die. 00:41:40.140 --> 00:41:42.270 That's why we're justified in taking the product. 00:41:42.270 --> 00:41:45.890 If we condition on the die that it is, on the hypothesis, 00:41:45.890 --> 00:41:50.280 12-sided, then the rolls are all independent of each other. 00:41:50.280 --> 00:41:55.070 In particular, we get 1/12 times 1/12 for rolling a 5 00:41:55.070 --> 00:41:56.080 and then rolling a 9. 00:41:56.080 --> 00:41:58.820 And for the 20-sided die, we get 1/20 times 1/20. 00:42:01.750 --> 00:42:05.260 So now in one step, we update. 00:42:05.260 --> 00:42:12.100 Same exact sum, we normalize, same exact answer. 00:42:12.100 --> 00:42:16.300 And I mentioned that some groups, maybe when slightly 00:42:16.300 --> 00:42:21.290 prompted by me, said oh, multiplication is commutative. 00:42:21.290 --> 00:42:22.020 Right? 00:42:22.020 --> 00:42:24.390 That's all that's going on here. 00:42:24.390 --> 00:42:26.260 In the end, all we're doing is taking 00:42:26.260 --> 00:42:29.300 our prior, the first likelihood column, the second likelihood 00:42:29.300 --> 00:42:31.660 column, multiplying them together and then normalizing 00:42:31.660 --> 00:42:32.180 it. 00:42:32.180 --> 00:42:33.182 Right? 00:42:33.182 --> 00:42:35.390 It doesn't matter what order those likelihood columns 00:42:35.390 --> 00:42:35.680 are in. 00:42:35.680 --> 00:42:37.263 It doesn't matter if we think of the 5 00:42:37.263 --> 00:42:39.310 and then the 9, the 9 and the 5, or the 5 and 9 00:42:39.310 --> 00:42:41.900 multiplied together to make their own likelihood column. 00:42:41.900 --> 00:42:44.240 You get the same exact unnormalized result no matter 00:42:44.240 --> 00:42:45.870 what. 00:42:45.870 --> 00:42:46.580 OK. 00:42:46.580 --> 00:42:50.380 So this is all there is to why it doesn't matter 00:42:50.380 --> 00:42:53.900 what order you update the data in sequentially, 00:42:53.900 --> 00:42:56.110 it just matters what the data is. 00:42:56.110 --> 00:42:57.960 Now I also had a question. 00:42:57.960 --> 00:43:00.880 Why, if you could do it all once, 00:43:00.880 --> 00:43:04.740 would you even bother with doing it once and then 00:43:04.740 --> 00:43:06.280 doing it again? 00:43:06.280 --> 00:43:08.180 It's a very good question. 00:43:08.180 --> 00:43:11.079 Does anyone want to suggest a reason? 00:43:11.079 --> 00:43:12.745 Maybe other than the group that I talked 00:43:12.745 --> 00:43:16.594 to specifically about it Yeah. 00:43:16.594 --> 00:43:18.010 AUDIENCE: Well, I'm just thinking, 00:43:18.010 --> 00:43:20.146 as you're saying, if you have ongoing patients 00:43:20.146 --> 00:43:22.450 and you have to like continuously update, 00:43:22.450 --> 00:43:25.866 then the first data set may be all 00:43:25.866 --> 00:43:27.616 that you have at that moment, but then you 00:43:27.616 --> 00:43:29.004 have to keeping adding on to it. 00:43:29.004 --> 00:43:30.420 PROFESSOR 2: That's exactly right. 00:43:30.420 --> 00:43:31.370 That's exactly right. 00:43:31.370 --> 00:43:36.280 So, in life and in clinical trials, 00:43:36.280 --> 00:43:38.680 you don't get all the data necessarily at once. 00:43:38.680 --> 00:43:42.579 It sort of comes in continuously. 00:43:42.579 --> 00:43:44.370 When I wake up in the morning, I don't know 00:43:44.370 --> 00:43:46.900 what temperature it is outside. 00:43:46.900 --> 00:43:49.004 I have a prior, based on what temperature 00:43:49.004 --> 00:43:50.670 it was yesterday at about the same time, 00:43:50.670 --> 00:43:52.227 like I have a memory of that. 00:43:52.227 --> 00:43:53.810 So it gives me some prior distribution 00:43:53.810 --> 00:43:55.140 on the temperature. 00:43:55.140 --> 00:43:57.640 OK, that might be a continuous one, like maybe a bell curve. 00:43:57.640 --> 00:44:00.441 We're going to get to that next week, these continuous updating 00:44:00.441 --> 00:44:00.940 problems. 00:44:00.940 --> 00:44:02.110 OK. 00:44:02.110 --> 00:44:03.590 And then, when I get out of bed, I 00:44:03.590 --> 00:44:06.940 see that the window is covered in condensation. 00:44:06.940 --> 00:44:08.310 That's some data. 00:44:08.310 --> 00:44:09.440 I can update my beliefs. 00:44:09.440 --> 00:44:12.790 Probably means it's kind of cold. 00:44:12.790 --> 00:44:13.910 Great. 00:44:13.910 --> 00:44:16.640 Then I can open the window to let the cat out, 00:44:16.640 --> 00:44:18.870 and a burst of freezing cold air comes in, 00:44:18.870 --> 00:44:21.328 like it did this morning-- well, actually not this morning. 00:44:21.328 --> 00:44:22.960 This morning was great. 00:44:22.960 --> 00:44:25.700 Yesterday it was cold. 00:44:25.700 --> 00:44:28.841 And again, now I have more data and I update my beliefs. 00:44:28.841 --> 00:44:29.340 All right. 00:44:29.340 --> 00:44:30.715 You can go through life this way. 00:44:30.715 --> 00:44:32.810 It'll give you a headache, but you can do it. 00:44:32.810 --> 00:44:36.070 Now, with clinical trials, the same thing can happen. 00:44:36.070 --> 00:44:37.320 Maybe you do a pilot study. 00:44:37.320 --> 00:44:44.930 Maybe you have some belief that eating almonds causes 00:44:44.930 --> 00:44:48.220 something, stomach cramps. 00:44:48.220 --> 00:44:48.970 Great. 00:44:48.970 --> 00:44:50.940 And maybe it's just for you. 00:44:50.940 --> 00:44:52.582 But you want to do a study about that. 00:44:52.582 --> 00:44:55.040 So maybe you know for yourself that eating almonds give you 00:44:55.040 --> 00:44:56.500 stomach cramps. 00:44:56.500 --> 00:44:58.550 You're at the Harvard School of Public Health 00:44:58.550 --> 00:45:00.820 and you really want to investigate this further, 00:45:00.820 --> 00:45:03.130 so you gather a few willing subjects, 00:45:03.130 --> 00:45:05.780 you feed them some almonds, you see if they get stomach cramps, 00:45:05.780 --> 00:45:07.080 how many of them. 00:45:07.080 --> 00:45:09.800 And maybe your prior ahead was you just kind of 00:45:09.800 --> 00:45:12.024 felt like you had this really strong belief that 00:45:12.024 --> 00:45:13.690 almonds cause stomach cramps, because it 00:45:13.690 --> 00:45:14.670 does for you and you really haven't even 00:45:14.670 --> 00:45:15.650 asked a lot of other people. 00:45:15.650 --> 00:45:17.483 So you might have a lot of prior probability 00:45:17.483 --> 00:45:19.630 on that hypothesis being true. 00:45:19.630 --> 00:45:21.060 Once you collect a few more people 00:45:21.060 --> 00:45:22.930 and maybe you ask your friends, maybe you 00:45:22.930 --> 00:45:25.054 find that most of them don't have the same problem. 00:45:25.054 --> 00:45:27.920 Suddenly your posterior belief changes 00:45:27.920 --> 00:45:30.880 and now you're not quite so convinced that it's true. 00:45:30.880 --> 00:45:32.470 And then, if you're really ambitious 00:45:32.470 --> 00:45:35.010 and you have a lot of grant money from the NIH, 00:45:35.010 --> 00:45:36.880 you recruit 1,000 people and you do 00:45:36.880 --> 00:45:39.020 a randomized controlled trial with placebo 00:45:39.020 --> 00:45:43.760 and you find out that exactly 10% of the folks 00:45:43.760 --> 00:45:46.310 complain about the stomach cramps with almonds. 00:45:46.310 --> 00:45:48.069 Maybe 10% on placebo also, so you're 00:45:48.069 --> 00:45:49.360 not quite sure what's going on. 00:45:49.360 --> 00:45:51.528 But the point is, you're getting data 00:45:51.528 --> 00:45:52.736 a little more, a little more. 00:45:52.736 --> 00:45:57.410 And you can keep incorporating it, updating your beliefs. 00:45:57.410 --> 00:45:58.810 So that's the principle. 00:45:58.810 --> 00:46:00.660 One last thing I'll point out is, 00:46:00.660 --> 00:46:02.640 what if you have a prior belief that 00:46:02.640 --> 00:46:07.210 is so strong, so you are so sure, for example, 00:46:07.210 --> 00:46:10.025 that almonds don't cause stomach pain, 00:46:10.025 --> 00:46:12.400 because it doesn't for you and you're just sure that that 00:46:12.400 --> 00:46:14.136 would be ridiculous, right? 00:46:14.136 --> 00:46:15.885 So you've put your prior probability at 0. 00:46:18.557 --> 00:46:20.640 How much evidence will it take, how much data will 00:46:20.640 --> 00:46:22.680 take for you to be convinced? 00:46:25.280 --> 00:46:26.150 Infinite. 00:46:26.150 --> 00:46:30.480 No amount of data is going to convince you otherwise. 00:46:30.480 --> 00:46:33.580 This is why whenever you do this kind of Bayesian Updating, 00:46:33.580 --> 00:46:37.620 it's generally best to leave a little bit of possibility 00:46:37.620 --> 00:46:39.580 for Santa Claus, OK. 00:46:39.580 --> 00:46:41.300 Like a little bit. 00:46:41.300 --> 00:46:46.040 Just so that, should a fat white man come down your chimney 00:46:46.040 --> 00:46:48.170 and deliver the presents one year, 00:46:48.170 --> 00:46:52.300 you have a possibility to adjust. 00:46:52.300 --> 00:46:55.340 So those are some things to keep in mind. 00:46:55.340 --> 00:46:57.770 I think we've got one more board question 00:46:57.770 --> 00:46:59.860 that Jerry's going to lead. 00:46:59.860 --> 00:47:05.080 PROFESSOR: Right, so it's nice to get probabilities 00:47:05.080 --> 00:47:08.110 for hypotheses, but one thing you also want to do 00:47:08.110 --> 00:47:10.850 is predict what's going to come next. 00:47:10.850 --> 00:47:15.320 And so that's what we call probabilistic prediction. 00:47:15.320 --> 00:47:19.870 If I pick one of the John's 5 die, and I'm about to roll it, 00:47:19.870 --> 00:47:23.200 you could tell me using the Law of Total Probability 00:47:23.200 --> 00:47:27.740 the probability that I get a 5. 00:47:27.740 --> 00:47:30.530 And we've done that, we did that in the first unit 00:47:30.530 --> 00:47:32.890 on probability in this class. 00:47:32.890 --> 00:47:37.960 But suppose he rolls a 5 first. 00:47:37.960 --> 00:47:41.650 Now I can update my probabilities for which 00:47:41.650 --> 00:47:43.200 die it is, which is going to update 00:47:43.200 --> 00:47:48.200 my estimation of the probability of rolling, say a 4 00:47:48.200 --> 00:47:49.779 on the next roll. 00:47:49.779 --> 00:47:51.820 This is what we'll call probabilistic prediction. 00:47:51.820 --> 00:47:54.240 We can predict what's going to happen next, 00:47:54.240 --> 00:47:59.104 at least with probabilities, based on the data 00:47:59.104 --> 00:47:59.770 that we've seen. 00:47:59.770 --> 00:48:03.660 We update our predictions based on that. 00:48:03.660 --> 00:48:05.220 So here's a set up. 00:48:05.220 --> 00:48:06.510 D_1 is the first roll. 00:48:06.510 --> 00:48:07.380 D_2 is the second. 00:48:10.840 --> 00:48:12.202 I'll do this example. 00:48:12.202 --> 00:48:13.410 Oh no, this a board question. 00:48:13.410 --> 00:48:15.280 You're going to do this. 00:48:15.280 --> 00:48:16.430 You're on your own here. 00:48:16.430 --> 00:48:18.860 So first, you don't know anything 00:48:18.860 --> 00:48:23.510 about the die except that 1 of the 5 was chosen at random. 00:48:23.510 --> 00:48:27.320 What's the probability that the first roll will be a 5? 00:48:27.320 --> 00:48:31.690 Now you've rolled that once, and you're going to roll again. 00:48:31.690 --> 00:48:34.030 And you can ask, well now I've seen some data. 00:48:34.030 --> 00:48:38.350 I have some information about which die was chosen. 00:48:38.350 --> 00:48:41.810 What's the probability that I'll see a 4 on the next die. 00:48:41.810 --> 00:48:45.280 So that's your job to find that for this problem. 00:48:52.368 --> 00:48:54.826 PROFESSOR 2: Cool, all right, so this is your first answer? 00:48:54.826 --> 00:48:55.492 AUDIENCE: Right. 00:48:55.492 --> 00:48:58.810 And then, now we're just multiplying across. 00:48:58.810 --> 00:49:01.798 You get 0. 00:49:01.798 --> 00:49:14.248 1 over 180, 1 over 320, 1 over 740, [INAUDIBLE]. 00:49:14.248 --> 00:49:19.726 And then add those up. 00:49:19.726 --> 00:49:22.150 PROFESSOR 2: Yeah, get out your calculator. 00:49:22.150 --> 00:49:23.775 What are you dividing by to normalize? 00:49:23.775 --> 00:49:25.397 AUDIENCE: Is this-- I don't know. 00:49:25.397 --> 00:49:26.480 PROFESSOR 2: That's right. 00:49:26.480 --> 00:49:31.190 So this is, when you add these up, exactly the probability 00:49:31.190 --> 00:49:32.249 of the 5, right? 00:49:32.249 --> 00:49:34.040 So now, if you want to repeat this process, 00:49:34.040 --> 00:49:36.920 you need to use your normalized posterior instead 00:49:36.920 --> 00:49:37.920 of the original prior. 00:49:37.920 --> 00:49:38.826 AUDIENCE: That's what we were-- 00:49:38.826 --> 00:49:39.732 PROFESSOR 2: Yeah. 00:49:39.732 --> 00:49:41.550 Great. 00:49:41.550 --> 00:49:43.100 Now, you just need to do division. 00:49:43.100 --> 00:49:46.140 That I can't help you with. 00:49:46.140 --> 00:49:48.780 Calculator. 00:49:48.780 --> 00:49:50.226 Division skills, either way. 00:49:58.207 --> 00:49:59.040 [INTERPOSING VOICES] 00:50:13.597 --> 00:50:15.555 PROFESSOR 2: Now, is this the probability of 4, 00:50:15.555 --> 00:50:18.184 or the probability of 4 given 5? 00:50:18.184 --> 00:50:18.725 That's right. 00:50:21.350 --> 00:50:25.700 So tell me what you did to get these. 00:50:25.700 --> 00:50:28.676 AUDIENCE: So this column [INAUDIBLE] 00:50:28.676 --> 00:50:31.156 which we calculated for, and then you 00:50:31.156 --> 00:50:34.628 sum it up to get the total probability of getting a 5. 00:50:34.628 --> 00:50:36.116 And then [INAUDIBLE] normalized it, 00:50:36.116 --> 00:50:40.100 and then took the new [INAUDIBLE] times the same-- 00:50:40.100 --> 00:50:41.475 PROFESSOR 2: The same likelihood. 00:50:41.475 --> 00:50:43.183 AUDIENCE: And then you get the new column 00:50:43.183 --> 00:50:44.265 and then you add it up. 00:50:44.265 --> 00:50:46.425 [INAUDIBLE] 00:50:46.425 --> 00:50:47.300 PROFESSOR 2: Perfect. 00:50:47.300 --> 00:50:49.832 Now you said, just for the heck of it you normalized it, 00:50:49.832 --> 00:50:52.040 but it's actually really important that you normalize 00:50:52.040 --> 00:50:54.060 it, because if you're going to use the Law of Total 00:50:54.060 --> 00:50:55.640 Probability again, right, you've got 00:50:55.640 --> 00:50:58.170 to be working with a probability mass function, something 00:50:58.170 --> 00:50:59.362 that adds to 1 here. 00:50:59.362 --> 00:51:01.320 If you didn't normalize, then your final answer 00:51:01.320 --> 00:51:03.300 would get multiplied by something, right? 00:51:03.300 --> 00:51:05.220 Whatever that normalization constant was, 00:51:05.220 --> 00:51:05.710 or its reciprocal. 00:51:05.710 --> 00:51:06.210 Perfect. 00:51:12.955 --> 00:51:13.830 Check these guys out. 00:51:20.110 --> 00:51:23.617 PROFESSOR: And this right here is not a probability, 00:51:23.617 --> 00:51:24.700 because it's unnormalized. 00:51:24.700 --> 00:51:26.670 So what should you normalize this by? 00:51:26.670 --> 00:51:29.230 AUDIENCE: [INAUDIBLE] 00:51:29.230 --> 00:51:29.920 PROFESSOR: Yeah. 00:51:29.920 --> 00:51:32.410 In other words, if you made a normalized probability here, 00:51:32.410 --> 00:51:34.810 it would be this divided by that, this divided by that. 00:51:34.810 --> 00:51:36.010 AUDIENCE: So all we need to do is divide this-- 00:51:36.010 --> 00:51:37.051 PROFESSOR: Exactly right. 00:51:54.042 --> 00:51:55.208 PROFESSOR 2: How's it going? 00:51:55.208 --> 00:51:57.580 AUDIENCE: It's going fine. 00:51:57.580 --> 00:51:58.587 So we're just saying-- 00:51:58.587 --> 00:51:59.420 [INTERPOSING VOICES] 00:52:17.379 --> 00:52:19.170 PROFESSOR 2: So, what did you get for this? 00:52:19.170 --> 00:52:20.490 How did you find it? 00:52:20.490 --> 00:52:23.480 So you summed up this, right? 00:52:23.480 --> 00:52:26.189 Great, and then what did you do next? 00:52:26.189 --> 00:52:27.730 AUDIENCE: We do the same thing for 4. 00:52:27.730 --> 00:52:29.354 PROFESSOR 2: So, starting from scratch. 00:52:29.354 --> 00:52:31.940 AUDIENCE: Yeah, starting from scratch if you roll a 4. 00:52:31.940 --> 00:52:32.800 AUDIENCE: OK. 00:52:32.800 --> 00:52:38.840 So how does that incorporate the fact that you first rolled a 5? 00:52:38.840 --> 00:52:41.429 AUDIENCE: So we're trying to go back to the original Bayes, 00:52:41.429 --> 00:52:43.470 but that didn't work out so well because we don't 00:52:43.470 --> 00:52:44.677 know how to calculate this. 00:52:44.677 --> 00:52:46.010 PROFESSOR 2: Exactly, all right. 00:52:46.010 --> 00:52:47.990 So this is no easier than that. 00:52:47.990 --> 00:52:54.230 So here's the idea: so you start with this belief, right? 00:52:54.230 --> 00:52:55.660 And you have your likelihood. 00:52:55.660 --> 00:52:57.493 You multiply them, Law of Total Probability, 00:52:57.493 --> 00:52:59.850 and you get the probability of rolling a 5. 00:52:59.850 --> 00:53:03.070 Now at that point, after you've rolled this 5, 00:53:03.070 --> 00:53:05.070 your beliefs have changed, based on that, 00:53:05.070 --> 00:53:08.370 about the probability that you have each of the 5 dice. 00:53:08.370 --> 00:53:12.500 So if we take that and you use that as your prior 00:53:12.500 --> 00:53:16.320 here, then the likelihood, multiply and sum, 00:53:16.320 --> 00:53:18.304 that should give you-- 00:53:18.304 --> 00:53:19.220 AUDIENCE: I like that. 00:53:19.220 --> 00:53:20.130 PROFESSOR 2: OK? 00:53:20.130 --> 00:53:21.857 AUDIENCE: Oh, but now we have decimals. 00:53:21.857 --> 00:53:23.190 AUDIENCE: Now you have decimals. 00:53:23.190 --> 00:53:24.555 I can't make your life too easy. 00:53:31.093 --> 00:53:31.926 AUDIENCE: Use these. 00:53:35.430 --> 00:53:38.010 PROFESSOR: Right, so you multiply this number 00:53:38.010 --> 00:53:39.984 by this, this number by this. 00:53:39.984 --> 00:53:41.900 So in the end, you'll get all of these numbers 00:53:41.900 --> 00:53:44.440 except they're all multiplied by, divided by. 00:53:44.440 --> 00:53:48.540 So you should divide this by-- to get a probability. 00:53:48.540 --> 00:53:50.940 That make sense? 00:53:50.940 --> 00:53:55.010 So you have to be careful for figuring out 00:53:55.010 --> 00:53:57.420 probabilities of hypotheses, we can 00:53:57.420 --> 00:53:59.760 avoid using the unnormalized. 00:53:59.760 --> 00:54:01.450 We don't have to keep normalizing. 00:54:01.450 --> 00:54:03.890 But when you want to do posterior prediction, 00:54:03.890 --> 00:54:07.280 you want real probabilities of data. 00:54:07.280 --> 00:54:09.280 So you're going to have to normalize 00:54:09.280 --> 00:54:11.700 these numbers at some point. 00:54:11.700 --> 00:54:13.384 Either first or at the end. 00:54:13.384 --> 00:54:15.300 It's probably, if you're going to do it a lot, 00:54:15.300 --> 00:54:16.345 easier to just keep doing it. 00:54:16.345 --> 00:54:17.220 AUDIENCE: OK, Thanks. 00:54:30.700 --> 00:54:32.182 AUDIENCE: How are we doing? 00:54:32.182 --> 00:54:33.140 PROFESSOR: You tell me. 00:54:36.090 --> 00:54:37.520 It's a beautiful use of color. 00:54:40.339 --> 00:54:41.172 [INTERPOSING VOICES] 00:54:48.735 --> 00:54:50.860 PROFESSOR: So is this normalized or not normalized? 00:54:50.860 --> 00:54:51.980 AUDIENCE: This is normalized. 00:54:51.980 --> 00:54:52.896 PROFESSOR: Normalized. 00:54:52.896 --> 00:54:54.960 Good, because you want real probabilities there. 00:54:54.960 --> 00:54:55.670 Yes, thank you. 00:54:55.670 --> 00:54:56.545 AUDIENCE: [INAUDIBLE] 00:54:59.553 --> 00:55:00.928 AUDIENCE: And so then we multiply 00:55:00.928 --> 00:55:03.400 by what are the likelihoods. 00:55:03.400 --> 00:55:04.234 PROFESSOR: Yeah. 00:55:04.234 --> 00:55:06.646 AUDIENCE: [INAUDIBLE] 00:55:06.646 --> 00:55:08.520 PROFESSOR: And now you're going to sum it up. 00:55:08.520 --> 00:55:09.452 AUDIENCE: Yeah. 00:55:09.452 --> 00:55:10.850 PROFESSOR: That's perfect. 00:55:10.850 --> 00:55:12.222 AUDIENCE: Way to go Caitlin! 00:55:12.222 --> 00:55:13.680 PROFESSOR: Because the Law of Total 00:55:13.680 --> 00:55:16.044 Probabilities says multiply this by this. 00:55:16.044 --> 00:55:17.710 That's just what you did with the trees, 00:55:17.710 --> 00:55:19.460 do you see the trees when I wave my hands? 00:55:24.452 --> 00:55:24.952 Beautiful. 00:55:24.952 --> 00:55:25.827 AUDIENCE: [INAUDIBLE] 00:55:32.182 --> 00:55:34.700 AUDIENCE: Our new prior is given the data we had before. 00:55:34.700 --> 00:55:35.325 PROFESSOR: Yes. 00:55:40.210 --> 00:55:42.182 That's good. 00:55:42.182 --> 00:55:44.880 PROFESSOR 2: You're adding them up? 00:55:44.880 --> 00:55:46.170 OK, so that looks right. 00:55:49.050 --> 00:55:51.530 So let's think, how does this number 00:55:51.530 --> 00:55:53.900 compare to what you would have, if you didn't 00:55:53.900 --> 00:55:55.230 know this other information? 00:55:55.230 --> 00:55:59.250 So suppose you knew nothing going in, 00:55:59.250 --> 00:56:03.169 do you think that the fact that you rolled a 5 the first time 00:56:03.169 --> 00:56:04.710 makes it more or less likely that you 00:56:04.710 --> 00:56:05.840 roll a 4 the second time. 00:56:05.840 --> 00:56:07.329 AUDIENCE: It should be less likely. 00:56:07.329 --> 00:56:08.620 PROFESSOR 2: Is it less likely? 00:56:08.620 --> 00:56:10.580 AUDIENCE: Because you get rid of the 4-- 00:56:10.580 --> 00:56:13.280 the die has only 4 sides. 00:56:13.280 --> 00:56:15.370 PROFESSOR 2: So it might make you go down, right? 00:56:15.370 --> 00:56:17.703 On the other hand, you're going to give more probability 00:56:17.703 --> 00:56:20.660 to the 6-sided, let's say that, than it had originally. 00:56:20.660 --> 00:56:22.160 So they're sort of competing forces. 00:56:22.160 --> 00:56:23.201 So I'm actually not sure. 00:56:23.201 --> 00:56:25.239 I'm not actually sure which is the right answer. 00:56:25.239 --> 00:56:27.697 But anyway, it's something you can easily check or compute. 00:56:30.704 --> 00:56:32.120 PROFESSOR: This is a standard day. 00:56:32.120 --> 00:56:33.880 We always finish ahead of time. 00:56:38.480 --> 00:56:40.055 I don't need that right here. 00:56:46.020 --> 00:56:48.350 Like with all the other answers today, we'll 00:56:48.350 --> 00:56:52.130 use the screens to show the table. 00:56:54.850 --> 00:56:58.540 So when we go through it, we do the usual thing we did before. 00:56:58.540 --> 00:57:02.030 We started with a 5, so we'll update 00:57:02.030 --> 00:57:05.860 our probabilities of the hypotheses based on that 5. 00:57:05.860 --> 00:57:08.970 So that's the first column is the prior, then 00:57:08.970 --> 00:57:12.260 the likelihood of getting a 5, which 00:57:12.260 --> 00:57:16.850 is 0, if you have the 4-sided die, 1/6, 1/8, 1/12, and 1/20. 00:57:16.850 --> 00:57:18.810 We get our unnormalized posterior. 00:57:18.810 --> 00:57:20.580 And then in this table, I actually 00:57:20.580 --> 00:57:22.480 got the normalized posterior, because when 00:57:22.480 --> 00:57:25.570 you're computing predictive probabilities, 00:57:25.570 --> 00:57:27.790 you need to use normalized probabilities. 00:57:27.790 --> 00:57:30.430 They should be true probabilities. 00:57:30.430 --> 00:57:35.750 So we use this column here of the posterior probability. 00:57:35.750 --> 00:57:38.160 There's different orders you could do this calculation. 00:57:38.160 --> 00:57:42.700 I think this is one of the straightforward ones. 00:57:42.700 --> 00:57:50.400 Then we look at the likelihood of D_2 in the next column. 00:57:50.400 --> 00:57:52.500 Now I put a star for H_4 because there's 00:57:52.500 --> 00:57:56.990 no way you could get a probability there 00:57:56.990 --> 00:57:59.180 if you have D_1 and D_2. 00:57:59.180 --> 00:58:01.880 If I had left off the D_1, which most people did 00:58:01.880 --> 00:58:04.580 and which is OK because of the independence 00:58:04.580 --> 00:58:07.100 John talked about-- given the hypothesis 00:58:07.100 --> 00:58:09.220 the rolls are independent-- I would have had 00:58:09.220 --> 00:58:11.230 a 1/4 where I have the star. 00:58:11.230 --> 00:58:14.990 But I'm going to multiply it by 0 anyway, so who cares. 00:58:14.990 --> 00:58:19.580 So I have my new priors-- excuse me, 00:58:19.580 --> 00:58:23.210 my new likelihoods for the second bit of data, 00:58:23.210 --> 00:58:27.060 which are the same as the old ones: 1/6, 1/8, 1/12, 1/20. 00:58:27.060 --> 00:58:30.000 Now my total probability is done the same way. 00:58:30.000 --> 00:58:33.200 You multiply in this new language posterior 00:58:33.200 --> 00:58:37.610 1, which is your prior for likelihood 2, 00:58:37.610 --> 00:58:42.810 times likelihood 2 and sum them up, and I get this last column. 00:58:47.720 --> 00:58:51.730 So again, once I have my new, my posterior 00:58:51.730 --> 00:58:53.470 for the first bit of data, I can use 00:58:53.470 --> 00:58:56.210 that in the Law of Total Probability 00:58:56.210 --> 00:58:57.751 to make a prediction. 00:58:57.751 --> 00:58:59.000 So what's happened right here? 00:58:59.000 --> 00:59:00.230 What's become more likely? 00:59:05.120 --> 00:59:10.240 This 0.124 seems a little bit larger 00:59:10.240 --> 00:59:12.460 than I would have guessed ahead of time. 00:59:15.790 --> 00:59:18.016 What's happened with rolling a 5 and then a 4 00:59:18.016 --> 00:59:19.390 to make that a little bit bigger? 00:59:25.620 --> 00:59:26.120 Any ideas? 00:59:33.700 --> 00:59:35.340 AUDIENCE: When you roll a 5 first, 00:59:35.340 --> 00:59:37.252 it increases the probability that it's 00:59:37.252 --> 00:59:38.836 one of the lower dice. 00:59:38.836 --> 00:59:39.710 PROFESSOR: Excellent. 00:59:39.710 --> 00:59:41.578 AUDIENCE: So then it makes it more likely 00:59:41.578 --> 00:59:42.925 that you'll get a 4. 00:59:42.925 --> 00:59:43.800 PROFESSOR: Excellent. 00:59:43.800 --> 00:59:45.850 That's what I would think too. 00:59:45.850 --> 00:59:48.400 Seeing a 5-- a 4, what did roll first? 00:59:48.400 --> 00:59:51.820 The 5-- makes the smaller die more likely. 00:59:51.820 --> 00:59:53.540 If the smaller die is more likely, 00:59:53.540 --> 00:59:57.170 then you're more likely to roll a 4 and a 5. 00:59:57.170 --> 01:00:00.780 For instance, if I had the 6-sided die, 01:00:00.780 --> 01:00:02.960 what's the probability I'd roll a 4 and a 5 01:00:02.960 --> 01:00:04.150 with the 6-sided die? 01:00:07.520 --> 01:00:11.030 1 in 36 which is about 3%. 01:00:11.030 --> 01:00:13.010 So this doesn't get up to 3%, but it's 01:00:13.010 --> 01:00:15.610 a little bigger than if I had the 20-sided 01:00:15.610 --> 01:00:19.045 die where it would be-- what's 1/400. 01:00:19.045 --> 01:00:20.380 PROFESSOR 2: Very small. 01:00:20.380 --> 01:00:23.030 PROFESSOR: Small, John says it's small. 01:00:23.030 --> 01:00:25.580 PROFESSOR 2: So, this is one argument you can make, right? 01:00:25.580 --> 01:00:30.470 That rolling the 5 makes it likely that it's a smaller die, 01:00:30.470 --> 01:00:32.290 and therefore more likely to roll a 4 next. 01:00:32.290 --> 01:00:34.380 But is there another argument you 01:00:34.380 --> 01:00:37.009 can make for why maybe rolling the 5 01:00:37.009 --> 01:00:38.300 would have the opposite effect? 01:00:38.300 --> 01:00:42.910 It might make it less likely to roll a 4 next? 01:00:42.910 --> 01:00:45.776 AUDIENCE: My initial thought was if you roll a 5, 01:00:45.776 --> 01:00:47.950 you know it's not the 4-sided die. 01:00:47.950 --> 01:00:52.207 And that's your die that you're most likely to roll a 4. 01:00:52.207 --> 01:00:53.290 PROFESSOR 2: That's right. 01:00:53.290 --> 01:00:56.880 So I'm not as sure as you as to whether this goes up or down. 01:00:56.880 --> 01:00:59.680 I think there are two competing arguments. 01:00:59.680 --> 01:01:02.190 That on the one hand, it makes it more likely 01:01:02.190 --> 01:01:05.150 it's a 6 or 8-sided, but it also makes it completely impossible 01:01:05.150 --> 01:01:06.005 that it's a 4-sided. 01:01:06.005 --> 01:01:08.213 PROFESSOR: I'll give you 3 to 1 odds that it goes up. 01:01:08.213 --> 01:01:09.834 PROFESSOR 2: Aw crap. 01:01:09.834 --> 01:01:11.250 PROFESSOR: So what he doesn't know 01:01:11.250 --> 01:01:13.550 is whether I've computed it already or not. 01:01:13.550 --> 01:01:15.770 PROFESSOR 2: He probably has. 01:01:15.770 --> 01:01:18.730 But he probably did it at like 3 AM, so I don't trust it. 01:01:18.730 --> 01:01:20.530 PROFESSOR: I simulated it in R. 01:01:20.530 --> 01:01:22.460 PROFESSOR 2: I see. 01:01:22.460 --> 01:01:23.869 Are there any questions? 01:01:23.869 --> 01:01:25.660 That's what we have for today, so are there 01:01:25.660 --> 01:01:27.340 any questions about Bayesian Updating 01:01:27.340 --> 01:01:28.506 or doing these computations? 01:01:32.190 --> 01:01:34.240 All right, great.