WEBVTT 00:00:09.890 --> 00:00:11.320 PROFESSOR PATRICK WINSTON: I was in Washington for most of 00:00:11.320 --> 00:00:15.050 the week prospecting for gold. 00:00:15.050 --> 00:00:19.820 Another byproduct of that was that I forgot to arrange a 00:00:19.820 --> 00:00:25.330 substitute Bob Berwick for the Thursday recitations. 00:00:25.330 --> 00:00:28.070 I shall probably go to hell for this. 00:00:28.070 --> 00:00:31.830 In any event, we have many explanations, 00:00:31.830 --> 00:00:34.670 none of them good. 00:00:34.670 --> 00:00:38.680 But today we'll try to get back on track and you'll learn 00:00:38.680 --> 00:00:40.800 something fun. 00:00:40.800 --> 00:00:48.060 In particular you will learn how a graduate student of mine 00:00:48.060 --> 00:00:51.940 Mark [? Phillipson ?], together with a summer UROP 00:00:51.940 --> 00:00:55.980 student, Brett van Zuiden, one of you-- 00:00:58.710 --> 00:01:02.390 managed to pull off a tour de force and recognize in these 00:01:02.390 --> 00:01:06.610 two descriptions the pattern that we humans commonly call 00:01:06.610 --> 00:01:09.860 "revenge." It was discovered. 00:01:09.860 --> 00:01:13.090 The system didn't have a name for it, of course. 00:01:13.090 --> 00:01:16.160 It just knew that there was a pattern there and sat waiting 00:01:16.160 --> 00:01:17.340 for us to give a name to it. 00:01:17.340 --> 00:01:20.635 That's where we're going to end up. 00:01:20.635 --> 00:01:22.840 But it'll be a bit of a journey before we get there, 00:01:22.840 --> 00:01:24.039 because we've got to go through all that 00:01:24.039 --> 00:01:26.250 stuff on the outline. 00:01:26.250 --> 00:01:29.740 And in particular, we want to start off by a 00:01:29.740 --> 00:01:30.910 little tiny bit of review. 00:01:30.910 --> 00:01:33.310 Because some of the stuff we did last time 00:01:33.310 --> 00:01:36.440 went by pretty fast. 00:01:36.440 --> 00:01:39.630 In particular, you may remember they had this 00:01:39.630 --> 00:01:42.220 wonderful joint probability table, which tells us all we 00:01:42.220 --> 00:01:45.229 want to know, all we want to know. 00:01:45.229 --> 00:01:48.950 We can decide what the probability of the police 00:01:48.950 --> 00:01:52.520 being called is given the this and the that, and all that 00:01:52.520 --> 00:01:56.080 sort of stuff, by clicking the appropriate boxes. 00:01:56.080 --> 00:02:00.920 The trouble is, gee, there are only three variables there. 00:02:00.920 --> 00:02:03.560 And when there are lots of variables it gets pretty hard 00:02:03.560 --> 00:02:07.010 to make up those numbers or to even collect them. 00:02:07.010 --> 00:02:08.550 So we're driven to an alternative. 00:02:08.550 --> 00:02:11.890 And we got to that alternative just at the end of 00:02:11.890 --> 00:02:15.210 the show a week ago. 00:02:15.210 --> 00:02:21.750 And we got to the point where we were defining these 00:02:21.750 --> 00:02:25.370 inference nets, sometimes called "Bayes nets." And the 00:02:25.370 --> 00:02:29.579 one we worked with looked like this. 00:02:29.579 --> 00:02:33.060 There's a burglar, a raccoon, the possibility of a dog 00:02:33.060 --> 00:02:36.500 barking, the police being called, and a trash can being 00:02:36.500 --> 00:02:38.720 overturned. 00:02:38.720 --> 00:02:40.620 So more variables than that. 00:02:40.620 --> 00:02:41.579 That only has three. 00:02:41.579 --> 00:02:44.420 This has got five. 00:02:44.420 --> 00:02:47.590 But we're able to do some magic with this because we, as 00:02:47.590 --> 00:02:48.960 humans, when we define-- 00:02:48.960 --> 00:02:52.130 when we draw this graph we're making an assertion about how 00:02:52.130 --> 00:02:55.740 things depend or don't depend on one another. 00:02:55.740 --> 00:03:00.410 In particular, there's something to break down and 00:03:00.410 --> 00:03:03.240 memorize to the point where it rolls off your tongue. 00:03:03.240 --> 00:03:08.240 And that is that any variable on this graph is said by me to 00:03:08.240 --> 00:03:10.310 be independent of any other 00:03:10.310 --> 00:03:14.500 non-descendant given its parents. 00:03:14.500 --> 00:03:15.770 Independent of any 00:03:15.770 --> 00:03:19.690 non-descendant given its parents. 00:03:19.690 --> 00:03:21.930 So that means that the probability of the dog 00:03:21.930 --> 00:03:27.340 barking, given its parents, doesn't depend on T, the trash 00:03:27.340 --> 00:03:28.930 can being overturned. 00:03:28.930 --> 00:03:32.900 Because the intuition is all of the causality is flowing 00:03:32.900 --> 00:03:38.800 through the parents and can't get to this variable D without 00:03:38.800 --> 00:03:41.440 going through the parents. 00:03:41.440 --> 00:03:42.680 So that is [? inserted ?] 00:03:42.680 --> 00:03:44.290 property of the nets that we draw. 00:03:44.290 --> 00:03:46.960 And we tend to draw them in a way that reflects causality. 00:03:46.960 --> 00:03:49.000 So it tends to make sense. 00:03:49.000 --> 00:03:56.329 So somehow this thing is going to be-- 00:03:56.329 --> 00:03:59.710 we're going to use this thing instead of that thing. 00:03:59.710 --> 00:04:00.880 But wait. 00:04:00.880 --> 00:04:02.920 We may need that thing in order to do all the 00:04:02.920 --> 00:04:05.530 computations we want to perform. 00:04:05.530 --> 00:04:09.930 So we need to be able to show that we can get to that thing 00:04:09.930 --> 00:04:15.160 by doing calculations on this thing. 00:04:15.160 --> 00:04:18.300 So what to do? 00:04:18.300 --> 00:04:20.100 Well, we're going to use the chain rule. 00:04:20.100 --> 00:04:23.450 And remember that the chain rule came to us by way of the 00:04:23.450 --> 00:04:28.250 basic Axioms of Probability plus the definition plus a 00:04:28.250 --> 00:04:30.580 little colored chalk. 00:04:30.580 --> 00:04:33.570 So we got to the point last time where we 00:04:33.570 --> 00:04:34.750 sort of believed this. 00:04:34.750 --> 00:04:35.920 It's a really magical thing. 00:04:35.920 --> 00:04:38.670 It says that the probability of all this stuff happening 00:04:38.670 --> 00:04:42.170 together is given as the product of a bunch of 00:04:42.170 --> 00:04:44.470 conditional probabilities. 00:04:44.470 --> 00:04:47.380 And the conditional probabilities in this product 00:04:47.380 --> 00:04:49.430 are arranged such that this first guy depends 00:04:49.430 --> 00:04:51.500 on everybody else. 00:04:51.500 --> 00:04:54.050 The second guy doesn't depend on the first guy but depends 00:04:54.050 --> 00:04:55.690 on everything else. 00:04:55.690 --> 00:04:58.850 So that list of dependencies gets smaller and smaller as 00:04:58.850 --> 00:05:01.380 you go down here until it depends only one thing. 00:05:01.380 --> 00:05:05.050 There's no conditional at all. 00:05:05.050 --> 00:05:08.780 So that's going to come to our rescue because it enables us 00:05:08.780 --> 00:05:14.210 to go from calculations in here to that whole table. 00:05:14.210 --> 00:05:17.740 But first I have to show you a little bit more slowly how 00:05:17.740 --> 00:05:19.720 that comes to be. 00:05:19.720 --> 00:05:21.800 One thing I'm going to do before I think about 00:05:21.800 --> 00:05:25.650 probability is I'm going to make a linear list of all 00:05:25.650 --> 00:05:27.330 these variables. 00:05:27.330 --> 00:05:29.290 And the way I'm going to make it is I'm going to chew away 00:05:29.290 --> 00:05:32.030 at those variables from the bottom. 00:05:32.030 --> 00:05:33.810 I've taken advantage of a very important 00:05:33.810 --> 00:05:36.630 property of these nets. 00:05:36.630 --> 00:05:39.490 And that is there no loops. 00:05:39.490 --> 00:05:43.090 You can follow the arrows in any way so as 00:05:43.090 --> 00:05:45.470 you get back to yourself. 00:05:45.470 --> 00:05:48.200 So there's always going to be a bottom. 00:05:48.200 --> 00:05:50.300 So what I'm going to do is I'm going to say, well, there are 00:05:50.300 --> 00:05:53.409 two bottoms here, there's C and T. So I have a choice. 00:05:53.409 --> 00:06:02.330 I'm going to choose C. So I'm going to take that off and 00:06:02.330 --> 00:06:05.570 pretend it's not there anymore. 00:06:05.570 --> 00:06:06.980 Then I'm going to take this guy. 00:06:06.980 --> 00:06:09.430 That's now a bottom because there's nothing below it. 00:06:09.430 --> 00:06:11.430 I've already taken C out. 00:06:11.430 --> 00:06:14.590 So we'll take that out next. 00:06:14.590 --> 00:06:18.440 And now I've got this guy, this guy, and this guy. 00:06:18.440 --> 00:06:21.170 This guy no longer has anything below it. 00:06:21.170 --> 00:06:23.430 So I can list it next. 00:06:23.430 --> 00:06:26.590 Now over here I've got raccoon and trashcan. 00:06:26.590 --> 00:06:30.450 But trashcan is at the bottom. 00:06:30.450 --> 00:06:34.040 So I've got to take it next because I'm working 00:06:34.040 --> 00:06:36.011 from the bottom up. 00:06:36.011 --> 00:06:40.760 I want to ensure that there are no descendants before me 00:06:40.760 --> 00:06:42.940 in this list. 00:06:42.940 --> 00:06:45.780 So finally I get to raccoon. 00:06:45.780 --> 00:06:51.930 So the way I constructed this list like so ensures that this 00:06:51.930 --> 00:06:55.020 list arranges the elements so that for any particular 00:06:55.020 --> 00:06:57.210 element, none of its descendants 00:06:57.210 --> 00:06:58.460 appear to its left. 00:07:01.300 --> 00:07:04.260 And now that's the magical order for which I want to use 00:07:04.260 --> 00:07:06.030 the chain rule. 00:07:06.030 --> 00:07:07.960 So now I can write-- 00:07:07.960 --> 00:07:11.260 I can pick C to be my variable n. 00:07:11.260 --> 00:07:13.530 And I can say that the chain rule says that the joint 00:07:13.530 --> 00:07:17.410 probability of all these variables P of C, 00:07:17.410 --> 00:07:22.140 D, B, T, and R-- 00:07:22.140 --> 00:07:24.360 the probability of any particular combination of 00:07:24.360 --> 00:07:28.140 those things is equal to the probability of C given 00:07:28.140 --> 00:07:29.390 everybody else. 00:07:33.659 --> 00:07:38.190 Next in line is D given everybody else. 00:07:42.470 --> 00:07:45.400 Next in line is T-- 00:07:45.400 --> 00:07:49.420 next in line is B given everybody else. 00:07:57.380 --> 00:08:00.720 And next in line is T given everybody else. 00:08:03.270 --> 00:08:10.840 And finally, just R. So this combination of things has a 00:08:10.840 --> 00:08:13.145 probability that is given by this chain rule expression. 00:08:15.960 --> 00:08:17.010 Ah. 00:08:17.010 --> 00:08:19.500 But first of all, none of those expressions condition 00:08:19.500 --> 00:08:21.740 any of the variables on anything other than 00:08:21.740 --> 00:08:25.970 non-descendants, all right? 00:08:25.970 --> 00:08:28.650 That's just because of the way I've arranged the variables. 00:08:28.650 --> 00:08:30.950 And I can always do that because are no loops. 00:08:30.950 --> 00:08:33.169 I can always chew away at the bottom. 00:08:33.169 --> 00:08:35.140 That ensures that whenever I write a variable, it's going 00:08:35.140 --> 00:08:39.440 to be conditioned on stuff other than its descendants. 00:08:39.440 --> 00:08:43.190 So all of these variables in any of these conditional 00:08:43.190 --> 00:08:46.700 probabilities are non-descendants. 00:08:46.700 --> 00:08:48.740 Oh wait. 00:08:48.740 --> 00:08:54.510 When I drew this diagram, I asserted that no variable 00:08:54.510 --> 00:08:58.680 depends on any non-descendant given its parents. 00:08:58.680 --> 00:09:02.950 So if I know the parents of a variable I know that the 00:09:02.950 --> 00:09:05.895 variable is independent of all other non-descendants. 00:09:05.895 --> 00:09:07.320 All right? 00:09:07.320 --> 00:09:10.590 Now I can start scratching stuff out. 00:09:10.590 --> 00:09:11.690 Well, let's see. 00:09:11.690 --> 00:09:16.120 I know that C, from my diagram, has only one parent, 00:09:16.120 --> 00:09:21.080 D. So given its parent, it's independent of all other 00:09:21.080 --> 00:09:22.730 non-descendants. 00:09:22.730 --> 00:09:23.980 So I can scratch them out. 00:09:27.370 --> 00:09:33.470 D he has two parents, B and R. But given that, I can scratch 00:09:33.470 --> 00:09:34.720 out any other non-descendant. 00:09:37.190 --> 00:09:41.460 B is conditional on T and R. Ah, but B has no parent. 00:09:41.460 --> 00:09:45.120 So it actually is independent of those two guys. 00:09:45.120 --> 00:09:48.330 The trashcan, yeah, that's dependent on R. And R over 00:09:48.330 --> 00:09:50.640 here, the final thing in the chain, that's just a 00:09:50.640 --> 00:09:53.160 probability. 00:09:53.160 --> 00:09:56.390 So now I have a way of calculating any entry in that 00:09:56.390 --> 00:09:58.960 table because any entry in that table is going to be some 00:09:58.960 --> 00:10:02.880 combination of values for all those variables. 00:10:02.880 --> 00:10:04.440 Voila. 00:10:04.440 --> 00:10:08.320 So anything I can do with a table, I can do in principle 00:10:08.320 --> 00:10:10.580 with this little network. 00:10:10.580 --> 00:10:11.920 OK? 00:10:11.920 --> 00:10:15.480 But now the question is, I've got some probabilities I'm 00:10:15.480 --> 00:10:17.100 going to have to figure out here. 00:10:17.100 --> 00:10:19.570 So let me draw a slightly different version of it. 00:10:26.010 --> 00:10:32.030 So up here we've got the a priori probability of B. Well, 00:10:32.030 --> 00:10:46.680 that's just probability of B. Down here with the dog, I've 00:10:46.680 --> 00:10:50.350 got a bigger table because I've got probabilities that 00:10:50.350 --> 00:10:52.950 depend on the values of its parents. 00:10:52.950 --> 00:10:57.620 The probability of dog barking depends on the condition of 00:10:57.620 --> 00:11:00.560 the parents, nothing else. 00:11:00.560 --> 00:11:01.070 So let's see. 00:11:01.070 --> 00:11:03.920 I've got to have a column for B. I've got to have a column 00:11:03.920 --> 00:11:05.940 for the burglar and the raccoon. 00:11:08.660 --> 00:11:11.940 And there are a bunch of possibilities for those guys. 00:11:11.940 --> 00:11:14.050 But once I get those then I'll be able to calculate the 00:11:14.050 --> 00:11:15.670 probability of the dog barking. 00:11:18.640 --> 00:11:19.910 So there are two of these variables. 00:11:19.910 --> 00:11:22.310 So there are four combinations. 00:11:22.310 --> 00:11:30.390 There's T T. There's T R, R T, and-- 00:11:30.390 --> 00:11:32.134 whoa, what am I doing? 00:11:32.134 --> 00:11:34.050 Wake up! 00:11:34.050 --> 00:11:36.720 T false. 00:11:36.720 --> 00:11:37.930 False true. 00:11:37.930 --> 00:11:39.890 And false false. 00:11:39.890 --> 00:11:42.060 So what I really want to do is I want to calculate all of 00:11:42.060 --> 00:11:45.070 these probabilities that give the probability of the dog 00:11:45.070 --> 00:11:49.210 condition of the burglar and the raccoon. 00:11:49.210 --> 00:11:53.070 Similarly, I want to calculate the probability of B happening 00:11:53.070 --> 00:11:56.200 doesn't depend on anything else. 00:11:56.200 --> 00:11:58.590 So I don't know what to do. 00:11:58.590 --> 00:12:00.660 Well, what I'm going to actually do is I'm going to do 00:12:00.660 --> 00:12:03.410 the same thing I had to do up there. 00:12:03.410 --> 00:12:06.630 I'm going to keep track of-- 00:12:06.630 --> 00:12:07.640 I'm going to try a bunch of-- 00:12:07.640 --> 00:12:09.560 I'm going to get myself together a bunch of data. 00:12:09.560 --> 00:12:10.910 Maybe I do a bunch of experiments. 00:12:10.910 --> 00:12:12.730 Maybe somebody hands it to me. 00:12:12.730 --> 00:12:16.030 But I'm going to use that data to construct a bunch of 00:12:16.030 --> 00:12:19.990 tallies which are going to end up giving me the probabilities 00:12:19.990 --> 00:12:22.150 for all of those things. 00:12:22.150 --> 00:12:23.160 So I don't know, let's see. 00:12:23.160 --> 00:12:25.440 How should we start? 00:12:25.440 --> 00:12:27.290 Step one, find colored chalk. 00:12:27.290 --> 00:12:30.870 Step two, I'm going to extend these tables a little bit so I 00:12:30.870 --> 00:12:32.120 can keep track of the tallies. 00:12:38.840 --> 00:12:41.580 So this is going to be all the ones that end up in a 00:12:41.580 --> 00:12:42.980 particular row. 00:12:42.980 --> 00:12:47.450 And these are going to be the ones for which dog is true. 00:12:50.930 --> 00:12:53.960 Similarly, I'm going to extend this guy up here in order to 00:12:53.960 --> 00:12:56.020 keep track of some tallies. 00:12:56.020 --> 00:13:00.020 This is going to be the ones for which B is true. 00:13:02.990 --> 00:13:04.240 And this one will be all. 00:13:06.780 --> 00:13:08.680 So that's my set up. 00:13:08.680 --> 00:13:12.490 And now suppose that my first experiment comes roaring in. 00:13:12.490 --> 00:13:15.060 And it's all T's. 00:13:15.060 --> 00:13:21.720 So I have T T T. That's my first experimental result, my 00:13:21.720 --> 00:13:24.170 first data item. 00:13:24.170 --> 00:13:27.480 So let's see. 00:13:27.480 --> 00:13:31.010 The arrangement here is burglar, raccoon, dog. 00:13:31.010 --> 00:13:35.040 So burglar as a true. 00:13:35.040 --> 00:13:38.510 And there's one tally count in there. 00:13:38.510 --> 00:13:42.280 Likewise, the T T, that's the burglar and the raccoon, that 00:13:42.280 --> 00:13:45.190 brings me down to this first row. 00:13:45.190 --> 00:13:49.200 So that gives me one tally in there and dog is true so that 00:13:49.200 --> 00:13:52.475 gives me a tick mark in that one. 00:13:52.475 --> 00:13:53.190 All right? 00:13:53.190 --> 00:13:54.240 Are you with me so far? 00:13:54.240 --> 00:13:55.850 And now let's suppose that the next thing 00:13:55.850 --> 00:13:57.100 happens be all false. 00:14:01.940 --> 00:14:04.020 Well, burglar is false. 00:14:04.020 --> 00:14:05.545 But there is one experiment. 00:14:08.590 --> 00:14:09.980 Everybody's false. 00:14:09.980 --> 00:14:13.500 So we come down here to false false. 00:14:13.500 --> 00:14:14.810 And that's the row we're going to work on. 00:14:14.810 --> 00:14:16.650 We get a tally in there. 00:14:16.650 --> 00:14:19.150 Do we put one in here? 00:14:19.150 --> 00:14:21.860 No, because that's false. 00:14:21.860 --> 00:14:22.785 Dog is false. 00:14:22.785 --> 00:14:25.650 That's what our data element says. 00:14:25.650 --> 00:14:26.640 So that's cool. 00:14:26.640 --> 00:14:28.050 Maybe one more. 00:14:28.050 --> 00:14:37.210 Let's suppose we have T T F. Well in that case, we have a 00:14:37.210 --> 00:14:42.020 tick mark here and a tick mark here because the burglar 00:14:42.020 --> 00:14:44.360 element is true. 00:14:44.360 --> 00:14:48.040 Then we have T T. That brings us to the first row again. 00:14:48.040 --> 00:14:50.180 So we get a tick mark there. 00:14:50.180 --> 00:14:53.500 But dog as false, so no tick mark there. 00:14:53.500 --> 00:14:55.170 That's how it works. 00:14:55.170 --> 00:14:58.550 I suppose you'd like to see a demonstration, right? 00:14:58.550 --> 00:15:01.030 Always like to see a demonstration. 00:15:01.030 --> 00:15:02.850 So here's what it actually looks like. 00:15:05.680 --> 00:15:10.060 So on the left you see the network as we've constructed 00:15:10.060 --> 00:15:12.420 it, with a bunch probabilities there. 00:15:12.420 --> 00:15:13.850 And what I'm going to do now is I'm going to start 00:15:13.850 --> 00:15:17.850 simulating away so as to accumulate tick marks, tally 00:15:17.850 --> 00:15:20.160 marks, and see what kinds of probabilities that they 00:15:20.160 --> 00:15:21.920 indicate for the table. 00:15:21.920 --> 00:15:25.610 I happen to be using a process for which the model on the 00:15:25.610 --> 00:15:29.070 left is a correct reflection. 00:15:29.070 --> 00:15:31.540 So there's one simulation. 00:15:31.540 --> 00:15:33.460 So the dog barking-- 00:15:33.460 --> 00:15:34.800 let's see, the burglar is false. 00:15:34.800 --> 00:15:36.600 The raccoon is true. 00:15:36.600 --> 00:15:37.710 I get one tick mark. 00:15:37.710 --> 00:15:40.130 So the probability there is one. 00:15:40.130 --> 00:15:43.300 Of course, I'm not going to just go with one. 00:15:43.300 --> 00:15:44.840 I want to put a whole bunch of stuff in there. 00:15:44.840 --> 00:15:46.300 So I'll just run a bunch more simulations. 00:15:50.820 --> 00:15:51.270 No [? dice. ?] 00:15:51.270 --> 00:15:54.746 I don't even have an entry at all yet for T F here. 00:15:54.746 --> 00:15:57.210 That's because I haven't run enough data. 00:15:57.210 --> 00:16:00.280 So let me clear it instead of doing it one at a time. 00:16:00.280 --> 00:16:02.280 Let me run 100 simulations. 00:16:02.280 --> 00:16:03.450 See, it's still not too good. 00:16:03.450 --> 00:16:07.500 Because it says this T T probability true. 00:16:07.500 --> 00:16:09.910 This just because I'm feeding it data, right? 00:16:09.910 --> 00:16:12.730 And I'm keeping track of what the data elements tell me 00:16:12.730 --> 00:16:15.470 about how frequently a 00:16:15.470 --> 00:16:17.180 particular combination appears. 00:16:17.180 --> 00:16:17.680 Yes, [INAUDIBLE] 00:16:17.680 --> 00:16:20.668 STUDENT: So when you're doing one simulation, is that 00:16:20.668 --> 00:16:22.162 [INAUDIBLE] variables? 00:16:22.162 --> 00:16:23.832 PROFESSOR PATRICK WINSTON: When I'm doing one simulation, 00:16:23.832 --> 00:16:26.930 I'm just keeping track of that combination in 00:16:26.930 --> 00:16:28.800 each of these tables. 00:16:28.800 --> 00:16:30.630 Because it's going to tell me something about the 00:16:30.630 --> 00:16:33.730 probabilities that I want reflected in those tables. 00:16:33.730 --> 00:16:37.090 So it's pretty easy to see when I go up here to burglar. 00:16:37.090 --> 00:16:38.920 If I have a lot of data elements, they're all going to 00:16:38.920 --> 00:16:40.730 tell me something about the burglar as well 00:16:40.730 --> 00:16:42.280 as the other variables. 00:16:42.280 --> 00:16:44.910 So if I just look at that burglar thing, the fraction of 00:16:44.910 --> 00:16:47.850 time that it turns out true over all the data elements is 00:16:47.850 --> 00:16:50.150 going to be its probability. 00:16:50.150 --> 00:16:53.610 So now when I go down to the joint tables, I can still get 00:16:53.610 --> 00:16:54.630 these probability numbers. 00:16:54.630 --> 00:16:56.400 But now they're conditioned on reticular 00:16:56.400 --> 00:16:59.010 condition of its parents. 00:16:59.010 --> 00:17:01.430 So that's how I get these probabilities. 00:17:01.430 --> 00:17:04.700 So I didn't do too well here because that T T combination 00:17:04.700 --> 00:17:06.579 gave me an excessively high probability. 00:17:06.579 --> 00:17:10.130 So maybe 100 simulations isn't enough. 00:17:10.130 --> 00:17:15.339 Let's run 10,000. 00:17:15.339 --> 00:17:18.560 So with that much data running through, the probabilities I 00:17:18.560 --> 00:17:22.310 get-- let's see, I've got 893 here, instead of 0.9, 807 00:17:22.310 --> 00:17:25.880 instead of 0.8, 607 instead of 0.6. 00:17:25.880 --> 00:17:28.500 And that one's dead-on at 0.01. 00:17:28.500 --> 00:17:30.570 So if I run enough of these simulations, I get a pretty 00:17:30.570 --> 00:17:33.220 good idea what the probabilities ought to be 00:17:33.220 --> 00:17:37.030 given that I've got a correct model. 00:17:37.030 --> 00:17:38.860 OK, so that takes care of that one. 00:17:38.860 --> 00:17:40.880 And of course, I didn't draw the other things in here. 00:17:40.880 --> 00:17:44.800 But by extension, you can see how those would work. 00:17:44.800 --> 00:17:46.180 Oh. 00:17:46.180 --> 00:17:46.810 But you know what? 00:17:46.810 --> 00:17:48.895 I think I will put a little probability of 00:17:48.895 --> 00:17:50.145 raccoon table in here. 00:17:52.550 --> 00:17:55.390 Because the next thing I want to do is I want to 00:17:55.390 --> 00:17:56.470 go the other way. 00:17:56.470 --> 00:17:59.365 This is recoding tallies from some process so I 00:17:59.365 --> 00:18:00.730 can develop a model. 00:18:00.730 --> 00:18:03.670 But once I've got these probabilities, of course, then 00:18:03.670 --> 00:18:07.780 I can start to simulate what the model would do. 00:18:07.780 --> 00:18:08.610 All right? 00:18:08.610 --> 00:18:10.690 How would I do that? 00:18:10.690 --> 00:18:16.220 Well, do I want to use the same table? 00:18:16.220 --> 00:18:19.050 I think just to keep things sanitary, what I'll do is I'll 00:18:19.050 --> 00:18:21.340 go over here and do it again. 00:18:21.340 --> 00:18:27.260 Here's B. It's got a probability of B. Here's R. 00:18:27.260 --> 00:18:33.300 Here's a table probability of R. That comes down into a 00:18:33.300 --> 00:18:36.070 joint table for dog. 00:18:36.070 --> 00:18:37.320 And it's got four elements. 00:18:44.090 --> 00:18:46.940 Depending on the burglar condition and the raccoon 00:18:46.940 --> 00:18:51.270 condition, we get a probability of dog. 00:18:51.270 --> 00:18:53.990 And now, imagine these have all been filled in. 00:18:53.990 --> 00:18:57.290 So what do I want to do if I want to simulate this system 00:18:57.290 --> 00:19:02.130 generating some combination of values for all the variables? 00:19:02.130 --> 00:19:05.870 Well, I do the opposite of what I did when I was working 00:19:05.870 --> 00:19:09.070 around with this chain rule showing that I could go from 00:19:09.070 --> 00:19:11.480 the table to those probabilities. 00:19:11.480 --> 00:19:12.660 Now I've got the probabilities. 00:19:12.660 --> 00:19:14.450 I'm going to go the other direction. 00:19:14.450 --> 00:19:17.100 Instead of chewing away from the bottom, I'm going to chew 00:19:17.100 --> 00:19:18.660 away from the top. 00:19:18.660 --> 00:19:21.740 Because when I go into the top and chew way, everything I 00:19:21.740 --> 00:19:24.560 need to know to do a coin flip is there. 00:19:24.560 --> 00:19:28.970 So in particular, when I go up in here, I've got the 00:19:28.970 --> 00:19:31.350 probability of burglar now. 00:19:31.350 --> 00:19:36.250 So I'm going to use that probability to flip a coin. 00:19:36.250 --> 00:19:40.950 Say it produces a T. So that takes care of this guy. 00:19:40.950 --> 00:19:43.730 And I can now scratch it off since it's no longer in 00:19:43.730 --> 00:19:44.390 consideration. 00:19:44.390 --> 00:19:46.940 It's no longer a top variable. 00:19:46.940 --> 00:19:49.330 So now I go over into raccoon and I do the same thing. 00:19:49.330 --> 00:19:51.190 I take this probability. 00:19:51.190 --> 00:19:53.310 I do a flip. 00:19:53.310 --> 00:19:59.130 And say it produces an F. Whatever its probability is, I 00:19:59.130 --> 00:20:02.350 flip a biased coin and that's what I happen to get. 00:20:02.350 --> 00:20:06.340 But now, having dealt with these two guys, that uncovers 00:20:06.340 --> 00:20:08.230 this dog thing. 00:20:08.230 --> 00:20:10.330 And I've got enough information, because I've done 00:20:10.330 --> 00:20:13.286 everything above, to make the calculation for whether to dog 00:20:13.286 --> 00:20:15.510 is going to be barking or not. 00:20:15.510 --> 00:20:16.520 But wait. 00:20:16.520 --> 00:20:22.530 I have to know that I've got a T and a T and a T and an F and 00:20:22.530 --> 00:20:27.060 an F and a T and an F and an F. Because I have to select 00:20:27.060 --> 00:20:28.820 the right row. 00:20:28.820 --> 00:20:34.610 So I know that B is T. And I know that R is F. So that 00:20:34.610 --> 00:20:39.570 takes me into the table into the second row. 00:20:39.570 --> 00:20:42.360 So now I get this probability. 00:20:42.360 --> 00:20:49.850 I flip that coin and I get some result, say, T. Voila. 00:20:49.850 --> 00:20:51.810 I can do that with the other two variables. 00:20:51.810 --> 00:20:55.390 And I've got myself an experimental trial that is 00:20:55.390 --> 00:20:56.620 produced in accordance with the 00:20:56.620 --> 00:20:59.219 probabilities of the table. 00:20:59.219 --> 00:21:00.469 OK? 00:21:03.390 --> 00:21:04.640 Of course-- 00:21:08.356 --> 00:21:13.730 yeah, in fact, how did I get those numbers? 00:21:13.730 --> 00:21:16.990 Actually what I did is I used the model on the left to 00:21:16.990 --> 00:21:19.330 generate the samples that were used to compute the 00:21:19.330 --> 00:21:22.240 probabilities on the right. 00:21:22.240 --> 00:21:27.910 So you've seen that a demonstration of this already. 00:21:27.910 --> 00:21:31.330 Now of course-- 00:21:31.330 --> 00:21:36.400 I don't know, all of this sort of depends on having 00:21:36.400 --> 00:21:38.030 everything right. 00:21:38.030 --> 00:21:42.130 I've written a thing to write it one more time. 00:21:42.130 --> 00:21:51.580 Burglar, raccoon, dog, call the police, trashcan. 00:21:51.580 --> 00:21:54.440 But somebody else may say, oh, you've got it all wrong. 00:21:54.440 --> 00:21:55.690 This is what it really looks like. 00:21:58.560 --> 00:22:01.780 The dog doesn't care about the raccoon at all. 00:22:01.780 --> 00:22:04.130 So that's a correct model. 00:22:04.130 --> 00:22:06.380 Now when I do a simulation, I could fill in the tables in 00:22:06.380 --> 00:22:07.810 either model, right? 00:22:07.810 --> 00:22:10.800 I'm sure you'd like to see a demonstration. 00:22:10.800 --> 00:22:13.420 So let me show you a demonstration of that. 00:22:21.470 --> 00:22:23.040 So there are the two tables. 00:22:23.040 --> 00:22:25.060 And I can run 10,000 simulations 00:22:25.060 --> 00:22:26.310 on those guys, too. 00:22:28.530 --> 00:22:29.270 Now, look. 00:22:29.270 --> 00:22:31.150 The guy on the left is a pretty good reflection of the 00:22:31.150 --> 00:22:37.630 probabilities in a model I used to produce the data. 00:22:37.630 --> 00:22:39.253 But the guy on the right doesn't know any better. it 00:22:39.253 --> 00:22:42.960 just fills in its own tables, too. 00:22:42.960 --> 00:22:45.950 So what to do? 00:22:45.950 --> 00:22:48.060 I say this one's the right model. 00:22:48.060 --> 00:22:50.370 And you say that one's the right model. 00:22:50.370 --> 00:22:52.090 Who's right? 00:22:52.090 --> 00:22:54.120 Maybe we'll never know. 00:22:54.120 --> 00:22:57.830 And the guy on the left will get rich in the stock market 00:22:57.830 --> 00:23:00.630 and the guy on the right will go broke. 00:23:00.630 --> 00:23:01.370 I would be nice if we could actually 00:23:01.370 --> 00:23:04.520 figure out who's right. 00:23:04.520 --> 00:23:07.466 So would you to see how to figure out who's right? 00:23:07.466 --> 00:23:09.220 Yeah, so would I. What we're going to do is we're going to 00:23:09.220 --> 00:23:11.490 look at naive Bayesian inference. 00:23:11.490 --> 00:23:13.660 And that's our next chore. 00:23:13.660 --> 00:23:16.740 So here's how it works. 00:23:16.740 --> 00:23:20.800 We know, from the definition of conditional probability, we 00:23:20.800 --> 00:23:25.530 know that the probability of A given B is equal to the 00:23:25.530 --> 00:23:29.660 probability of A and B divided by the 00:23:29.660 --> 00:23:33.520 probability of B, right? 00:23:33.520 --> 00:23:36.400 Equal to by definition. 00:23:36.400 --> 00:23:43.540 So that means that the probability of A given B times 00:23:43.540 --> 00:23:45.240 the probability of B-- 00:23:45.240 --> 00:23:46.680 I'm just multiplying it out-- 00:23:46.680 --> 00:23:48.050 it equal to that joint probability. 00:23:52.690 --> 00:23:57.715 Oh, but by symmetry, there's no harm in saying I can turn 00:23:57.715 --> 00:24:02.450 that around and say that the probability of B given A times 00:24:02.450 --> 00:24:07.000 the probability of B is also equal to that joint 00:24:07.000 --> 00:24:08.690 probability, right? 00:24:08.690 --> 00:24:12.750 I've just expanded it a different and symmetric way. 00:24:12.750 --> 00:24:18.630 If I've got to write a, b on B, b, a on A. Thank you. 00:24:18.630 --> 00:24:19.590 Who was complaining? 00:24:19.590 --> 00:24:20.840 Good work. 00:24:24.200 --> 00:24:27.800 That would have been a major-league disaster. 00:24:27.800 --> 00:24:30.910 But now, having written that, I can forget about the middle. 00:24:30.910 --> 00:24:32.730 Because all I'm really interested in is how I've 00:24:32.730 --> 00:24:37.070 turned the probabilities around in that conditional. 00:24:37.070 --> 00:24:38.570 Why would I care about doing that? 00:24:38.570 --> 00:24:41.760 By the way, we're now talking about the work of 00:24:41.760 --> 00:24:43.010 the Reverend Bayes. 00:24:47.060 --> 00:24:50.420 Because we can rewrite this yet again as the probability 00:24:50.420 --> 00:25:00.160 of A given B is equal to the probability of B given A times 00:25:00.160 --> 00:25:06.420 the probability of A divided by the probability of B. 00:25:06.420 --> 00:25:09.790 That's just elementary algebra. 00:25:09.790 --> 00:25:13.460 But now I'm going to do something magical. 00:25:13.460 --> 00:25:18.670 I'm going to say I've got a classification problem. 00:25:18.670 --> 00:25:22.120 I want to know which disease you have. 00:25:22.120 --> 00:25:23.660 That's a classification problem. 00:25:23.660 --> 00:25:26.120 Maybe you've got the swine flu. 00:25:26.120 --> 00:25:29.790 Maybe you've got indigestion. 00:25:29.790 --> 00:25:30.740 Who knows. 00:25:30.740 --> 00:25:33.470 But I get all these symptoms. 00:25:33.470 --> 00:25:35.950 I get all these pieces of evidence. 00:25:35.950 --> 00:25:37.450 You've got a fever. 00:25:37.450 --> 00:25:38.140 You're throwing-- 00:25:38.140 --> 00:25:40.470 oh, well, let's not go into too much detail, there. 00:25:40.470 --> 00:25:42.060 But what I'm going to do is I'm going to say, well, let's 00:25:42.060 --> 00:25:48.490 suppose that A is equal to a class that I'm interested in, 00:25:48.490 --> 00:25:50.050 the disease you've got. 00:25:50.050 --> 00:25:56.510 And B is equal to the evidence, 00:25:56.510 --> 00:25:57.760 the symptoms I observe. 00:26:00.540 --> 00:26:02.240 Voila. 00:26:02.240 --> 00:26:04.150 I may have a pretty hard time figuring out what the 00:26:04.150 --> 00:26:07.260 probability of the class is given the evidence. 00:26:07.260 --> 00:26:08.920 But figuring out the probability of the evidence 00:26:08.920 --> 00:26:11.420 given the class might not be so hard. 00:26:11.420 --> 00:26:14.500 Let me get another board in play and show you what I mean. 00:26:20.128 --> 00:26:24.820 By plugging class and evidence into Bayes' rule, what I get 00:26:24.820 --> 00:26:31.540 is the probability of some class given the evidence is 00:26:31.540 --> 00:26:37.710 equal to the probability of the evidence given the class 00:26:37.710 --> 00:26:42.060 times the probability of the class divided by the 00:26:42.060 --> 00:26:43.690 probability of the evidence. 00:26:46.280 --> 00:26:49.230 Now you've got to let that sing to you a little bit. 00:26:49.230 --> 00:26:51.590 Suppose I've got several classes that I'm trying to 00:26:51.590 --> 00:26:54.270 decide between. 00:26:54.270 --> 00:26:58.500 I'm trying to select the best out of that batch of classes. 00:26:58.500 --> 00:26:59.800 Well, I've got the evidence. 00:26:59.800 --> 00:27:02.140 And if I know the probability of the evidence given each of 00:27:02.140 --> 00:27:06.130 those classes, and if I know, a priori, the initial 00:27:06.130 --> 00:27:09.950 probability the class, then I'm done. 00:27:09.950 --> 00:27:12.910 Because I've got the two elements in the numerator. 00:27:12.910 --> 00:27:14.590 Why am I done? 00:27:14.590 --> 00:27:18.630 Because the denominator is the same for all the classes. 00:27:18.630 --> 00:27:21.490 It's just the probability of the evidence. 00:27:21.490 --> 00:27:22.870 And then I could just sum everything up. 00:27:22.870 --> 00:27:25.440 I know it adds to 1 anyway. 00:27:25.440 --> 00:27:27.440 So that's cool. 00:27:27.440 --> 00:27:31.240 But sometimes there's evidence-- 00:27:31.240 --> 00:27:33.980 actually there's more than one piece of evidence. 00:27:33.980 --> 00:27:35.660 Let's say that there's some class. 00:27:35.660 --> 00:27:38.330 some i, and we're trying to figure out if that's the 00:27:38.330 --> 00:27:39.720 correct class. 00:27:39.720 --> 00:27:43.000 So we've got c sub i there and c sub i there. 00:27:43.000 --> 00:27:46.420 And suppose that that evidence is actually a bunch of pieces 00:27:46.420 --> 00:27:47.780 of evidence. 00:27:47.780 --> 00:27:56.580 So it could be e sub 1, e sub n, oops, 00:27:56.580 --> 00:27:58.880 premature right bracket. 00:27:58.880 --> 00:28:03.430 All that evidence, given the class i times the probability 00:28:03.430 --> 00:28:07.820 of the class i over some denominator that we don't care 00:28:07.820 --> 00:28:11.700 about because it's going to be the same for everybody. 00:28:11.700 --> 00:28:15.110 So we'll just write that as d. 00:28:15.110 --> 00:28:19.160 Now what if these pieces of evidence are all independent 00:28:19.160 --> 00:28:20.410 given the class? 00:28:22.720 --> 00:28:25.590 So if you have the swine flu, the probability you have a 00:28:25.590 --> 00:28:28.117 fever is independent of the probability you're going to 00:28:28.117 --> 00:28:31.610 throw up, say. 00:28:31.610 --> 00:28:34.280 Then can we write this another way? 00:28:34.280 --> 00:28:35.150 An easier way? 00:28:35.150 --> 00:28:36.580 Sure. 00:28:36.580 --> 00:28:39.180 Because when things are independent, the joint 00:28:39.180 --> 00:28:44.330 probability is equal to the product of the individual 00:28:44.330 --> 00:28:45.190 probabilities. 00:28:45.190 --> 00:28:46.910 So that is to say-- 00:28:46.910 --> 00:28:48.670 it's easier to see it if you write it down than if 00:28:48.670 --> 00:28:49.700 you just say it-- 00:28:49.700 --> 00:28:54.200 this probability here from these two elements here is 00:28:54.200 --> 00:28:59.280 equal to the probability of e sub 1 conditioned on c sub i 00:28:59.280 --> 00:29:04.970 times the probability of e sub 2 conditioned on c sub i, all 00:29:04.970 --> 00:29:08.080 the way down to the probability of e sub n 00:29:08.080 --> 00:29:14.800 conditioned on c sub i divided by some denominator we don't 00:29:14.800 --> 00:29:16.480 care about. 00:29:16.480 --> 00:29:18.860 See, what I'm going to try to do is I'm going to go through 00:29:18.860 --> 00:29:22.240 this for all the ci and see which one's the biggest. 00:29:22.240 --> 00:29:23.620 STUDENT: That's the [INAUDIBLE] ci, right? 00:29:26.390 --> 00:29:29.070 PROFESSOR PATRICK WINSTON: This is the probability of-- 00:29:29.070 --> 00:29:31.470 STUDENT: [INAUDIBLE] 00:29:31.470 --> 00:29:33.870 right-hand side [INAUDIBLE]. 00:29:33.870 --> 00:29:35.990 PROFESSOR PATRICK WINSTON: Right here? 00:29:35.990 --> 00:29:38.820 Oh yes, you're quite right. 00:29:38.820 --> 00:29:40.070 Oh yeah, thanks. 00:29:45.270 --> 00:29:46.775 I can't write and think at the same time. 00:29:46.775 --> 00:29:49.410 Thanks. 00:29:49.410 --> 00:29:49.770 OK. 00:29:49.770 --> 00:29:51.700 So I've just figure out which one of these is the biggest. 00:29:51.700 --> 00:29:54.150 And I've identified the class. 00:29:54.150 --> 00:29:57.490 Now you say to me, well, I would like to see an example. 00:29:57.490 --> 00:29:59.870 So-- 00:29:59.870 --> 00:30:02.050 I don't know, does anyone have any spare change? 00:30:04.660 --> 00:30:07.300 A nickel, a quarter. 00:30:07.300 --> 00:30:11.190 This is not because of infinitesimally low raises 00:30:11.190 --> 00:30:11.960 here at MIT. 00:30:11.960 --> 00:30:15.180 I just need it for a demonstration. 00:30:15.180 --> 00:30:17.180 I need two coins. 00:30:17.180 --> 00:30:19.140 Don't forget to get these back, I tend to be-- 00:30:19.140 --> 00:30:23.280 Now suppose these two coins are not exactly the same. 00:30:23.280 --> 00:30:28.200 One of these points is a legitimate, highly-prized 00:30:28.200 --> 00:30:30.160 American quarter. 00:30:30.160 --> 00:30:32.100 The other one is a fake. 00:30:32.100 --> 00:30:33.960 And with this one, the probability of heads, let us 00:30:33.960 --> 00:30:39.050 say, is 0.8 instead of 0.5. 00:30:39.050 --> 00:30:41.710 So I mix these all up. 00:30:41.710 --> 00:30:43.960 And I pick one. 00:30:43.960 --> 00:30:46.520 And I start flipping it. 00:30:46.520 --> 00:30:49.540 And I get a head. 00:30:49.540 --> 00:30:52.390 Then I flip it again. 00:30:52.390 --> 00:30:55.040 And I get a tail. 00:30:55.040 --> 00:30:58.750 Which coin did I pick? 00:30:58.750 --> 00:31:02.480 Well, we're going to use this stuff to figure it out. 00:31:02.480 --> 00:31:03.730 Here's what happens. 00:31:16.940 --> 00:31:18.500 Before I forget. 00:31:18.500 --> 00:31:20.440 Thank you very much. 00:31:20.440 --> 00:31:22.430 So what we've done is we've selected these 00:31:22.430 --> 00:31:23.810 things from my hands. 00:31:23.810 --> 00:31:24.390 And I can't draw hands. 00:31:24.390 --> 00:31:27.210 So I'll draw a little cup here. 00:31:27.210 --> 00:31:28.700 And there are two coins in here. 00:31:28.700 --> 00:31:29.610 And we're going to pick one. 00:31:29.610 --> 00:31:35.690 And one has a probability of heads equal to 0.8. 00:31:35.690 --> 00:31:40.913 And this one has a probability of a head of 0.5. 00:31:43.480 --> 00:31:45.900 So here's the draw. 00:31:45.900 --> 00:31:47.060 I pick one. 00:31:47.060 --> 00:31:48.840 Each has a probability of 0.5. 00:31:52.260 --> 00:31:56.020 This one is the one with the 0.8 as the 00:31:56.020 --> 00:31:57.300 probability of head. 00:31:57.300 --> 00:31:58.500 And this one is the one with the 00:31:58.500 --> 00:32:03.270 probability of 0.5 as a head. 00:32:03.270 --> 00:32:05.570 OK? 00:32:05.570 --> 00:32:12.910 So now suppose the first flips as it was is T. Well, that's a 00:32:12.910 --> 00:32:14.382 piece of evidence. 00:32:14.382 --> 00:32:15.310 That's here. 00:32:15.310 --> 00:32:18.230 Probably of evidence given the class. 00:32:18.230 --> 00:32:22.240 Well in the case of having drawn this biased coin, the 00:32:22.240 --> 00:32:29.740 probability of coming up with a tail-- ah, let's say a head, 00:32:29.740 --> 00:32:30.890 just to make my numbers a little easier. 00:32:30.890 --> 00:32:38.010 Probability of coming out there with a head is equal 0.8 00:32:38.010 --> 00:32:40.810 given that it's up here in this choice. 00:32:40.810 --> 00:32:46.265 The probability given that you have a fair coin is 0.5. 00:32:49.260 --> 00:32:59.230 So now if we take the next coin and take it to be a tail 00:32:59.230 --> 00:33:02.990 then the probability of this guy given 00:33:02.990 --> 00:33:06.610 that evidence is 0.2. 00:33:06.610 --> 00:33:08.900 And the probability of this guy given that evidence-- it's 00:33:08.900 --> 00:33:11.050 a fair coin, so it doesn't care. 00:33:11.050 --> 00:33:12.300 It's still 0.5. 00:33:14.250 --> 00:33:16.100 So now what's the probability of this 00:33:16.100 --> 00:33:19.710 class given this evidence? 00:33:19.710 --> 00:33:25.804 It's the product 0.5 times 0.8 times 0.2. 00:33:25.804 --> 00:33:28.590 And what's the probability of this guy? 00:33:28.590 --> 00:33:34.800 It's 05 times 0.5 times 0.5, divided by a denominator which 00:33:34.800 --> 00:33:37.520 is the same in both cases. 00:33:37.520 --> 00:33:40.230 So let's forget about this early 0.5 here. 00:33:40.230 --> 00:33:41.650 Because it's the same in both cases. 00:33:48.300 --> 00:33:50.660 And we just multiply those numbers together. 00:33:50.660 --> 00:33:52.940 That gives us 0.8 times 0.2. 00:33:52.940 --> 00:33:53.440 What's that? 00:33:53.440 --> 00:33:57.030 0.16? 00:33:57.030 --> 00:34:01.010 And this guy, 0.5 times 0.5, that's 0.25. 00:34:01.010 --> 00:34:04.610 So it looks an awful lot like-- with this combination-- 00:34:04.610 --> 00:34:08.050 that I've picked the coin that's fair. 00:34:08.050 --> 00:34:10.260 One more flip? 00:34:10.260 --> 00:34:11.870 So let's flip it again, and suppose we 00:34:11.870 --> 00:34:13.498 come up with a head. 00:34:13.498 --> 00:34:16.520 So that puts a 0.8 in here. 00:34:16.520 --> 00:34:19.170 And 0.5 in here. 00:34:19.170 --> 00:34:29.590 When you multiply those out that's 0.125. 00:34:29.590 --> 00:34:34.980 And this is 0.128. 00:34:34.980 --> 00:34:37.830 So it's about equal. 00:34:37.830 --> 00:34:40.333 So you see how that works? 00:34:40.333 --> 00:34:41.630 All right. 00:34:41.630 --> 00:34:44.560 So we're using the coin flips as evidence to figure out 00:34:44.560 --> 00:34:47.239 which class is involved. 00:34:47.239 --> 00:34:49.840 OK so I don't know, you'd probably like to see a 00:34:49.840 --> 00:34:51.370 demonstration of this, too, right? 00:34:51.370 --> 00:34:56.270 You say to me, gosh, just two kinds of coins. 00:34:56.270 --> 00:34:57.970 That's not very interesting. 00:34:57.970 --> 00:34:59.570 Let's try five kinds of coins. 00:35:03.660 --> 00:35:08.650 So what I want to show you is how the probabilities for all 00:35:08.650 --> 00:35:11.130 these coins-- there are five of them, color-coded-- 00:35:11.130 --> 00:35:15.700 how the probabilities vary with a series of flips. 00:35:15.700 --> 00:35:18.010 Let's suppose I've got a head-- 00:35:18.010 --> 00:35:22.080 the grey line, by the way, is the fraction of heads-- 00:35:22.080 --> 00:35:22.820 so that's going to be one. 00:35:22.820 --> 00:35:24.500 Because I'm just doing heads. 00:35:24.500 --> 00:35:27.830 You see that black line rising? 00:35:27.830 --> 00:35:29.250 Should look like a rocket. 00:35:29.250 --> 00:35:32.500 That's the probability that the-- 00:35:32.500 --> 00:35:35.060 that's the coin which only shows heads, the probability 00:35:35.060 --> 00:35:36.310 of head is 1. 00:35:38.760 --> 00:35:42.370 And I'm flipping a whole bunch of heads here. 00:35:42.370 --> 00:35:43.980 Isn't that cool? 00:35:43.980 --> 00:35:46.920 Now what happens if I suddenly put in a tail? 00:35:46.920 --> 00:35:50.280 By the way, you'll no doubt, here one the extreme left-- 00:35:50.280 --> 00:35:57.300 the initial probability of the P=0 coin was 0.1. 00:35:57.300 --> 00:35:59.950 As soon as I flipped a head that went to 0. 00:35:59.950 --> 00:36:02.930 And it will never get off 0, right? 00:36:02.930 --> 00:36:03.450 That makes sense. 00:36:03.450 --> 00:36:05.570 Because if the probability that you'll get a head is 1 00:36:05.570 --> 00:36:06.760 you should never see a tail. 00:36:06.760 --> 00:36:09.030 If you ever do, that isn't your coin. 00:36:09.030 --> 00:36:13.260 What happens now if I interrupt a series of heads 00:36:13.260 --> 00:36:14.957 and produce a tail? 00:36:14.957 --> 00:36:16.418 STUDENT: [INAUDIBLE]. 00:36:16.418 --> 00:36:17.392 PROFESSOR PATRICK WINSTON: What's that? 00:36:17.392 --> 00:36:18.860 STUDENT: [INAUDIBLE]. 00:36:18.860 --> 00:36:20.240 PROFESSOR PATRICK WINSTON: The black one will go to 0. 00:36:20.240 --> 00:36:21.655 What else happens? 00:36:21.655 --> 00:36:24.470 By the way, the blue one is the one with the highest 00:36:24.470 --> 00:36:27.400 probability of being a head. 00:36:27.400 --> 00:36:28.670 [INAUDIBLE] 00:36:28.670 --> 00:36:29.850 Boom! 00:36:29.850 --> 00:36:31.720 That blue one shot up. 00:36:31.720 --> 00:36:32.910 Not going up slowly. 00:36:32.910 --> 00:36:35.320 It shot up. 00:36:35.320 --> 00:36:37.640 Because now the preponderance of evidence with all those 00:36:37.640 --> 00:36:43.130 heads is that I've flipped the coin with a bias of 0.75 00:36:43.130 --> 00:36:44.820 towards heads. 00:36:44.820 --> 00:36:46.540 So let's clear this. 00:36:46.540 --> 00:36:47.680 Pick any probability you want. 00:36:47.680 --> 00:36:50.160 0.25, 0.5, and so on. 00:36:50.160 --> 00:36:52.260 I don't know, let's pick 0.25 since we've been 00:36:52.260 --> 00:36:53.510 at the upper end. 00:36:58.100 --> 00:36:59.650 So orange is 0.25. 00:36:59.650 --> 00:37:02.200 And sure enough, the probability that I've selected 00:37:02.200 --> 00:37:06.900 the 0.5 coin is going up and up and up and up after the 00:37:06.900 --> 00:37:08.540 original irregularity. 00:37:08.540 --> 00:37:10.610 The Law of Large Numbers is setting in. 00:37:10.610 --> 00:37:14.030 And a probability that I've got that 0.25 coin in play is 00:37:14.030 --> 00:37:16.660 pretty close to 1. 00:37:16.660 --> 00:37:16.910 All right. 00:37:16.910 --> 00:37:19.360 So that's cool. 00:37:19.360 --> 00:37:23.590 Now you say to me, that's awfully nice but stop. 00:37:23.590 --> 00:37:30.280 Awfully nice, but not very real-world-ish. 00:37:30.280 --> 00:37:33.520 So let me give you another problem. 00:37:33.520 --> 00:37:38.930 It's well-known that you are, with high probability, of the 00:37:38.930 --> 00:37:42.660 same political persuasion as your parents. 00:37:42.660 --> 00:37:46.620 So if I wanted to figure out which party a parent belongs 00:37:46.620 --> 00:37:50.140 to, I could look at the party that their 00:37:50.140 --> 00:37:53.490 children belong to, right? 00:37:53.490 --> 00:37:57.090 So it's just like flipping coins. 00:37:57.090 --> 00:37:59.770 The particular coin I have chosen 00:37:59.770 --> 00:38:01.680 corresponds to the parent. 00:38:01.680 --> 00:38:05.620 Individual flips correspond to the political party that the 00:38:05.620 --> 00:38:07.070 child belongs to. 00:38:07.070 --> 00:38:08.200 So let's get up a little bit-- 00:38:08.200 --> 00:38:09.890 by the way, I wrote all this stuff over the weekend. 00:38:09.890 --> 00:38:11.610 So who knows if any of it will work. 00:38:11.610 --> 00:38:13.430 But let's see. 00:38:13.430 --> 00:38:16.040 A parent party classifier. 00:38:16.040 --> 00:38:18.530 There it is, Democrats and Republicans. 00:38:18.530 --> 00:38:22.330 And now the prior for being a Republican given here is 0.5. 00:38:24.920 --> 00:38:27.300 But I don't know, this is a little bit Democratic state. 00:38:27.300 --> 00:38:31.780 So let's adjust that down a little bit. 00:38:31.780 --> 00:38:33.640 Somewhere in there might be about right But let's just, 00:38:33.640 --> 00:38:37.370 for the sake of a classroom illustration, go down here. 00:38:37.370 --> 00:38:39.650 So now the meter is showing the prior probability because 00:38:39.650 --> 00:38:41.510 that's the only thing in the formula so far. 00:38:41.510 --> 00:38:43.340 I've got no evidence. 00:38:43.340 --> 00:38:45.610 So now let's suppose that child number one is a 00:38:45.610 --> 00:38:48.380 Republican. 00:38:48.380 --> 00:38:50.850 Back to neutral. 00:38:50.850 --> 00:38:53.910 So I've got a low probability that the parent-- 00:38:53.910 --> 00:38:59.810 a priori probability that the parent is a Republican and a 00:38:59.810 --> 00:39:01.870 child who's a Republican. 00:39:01.870 --> 00:39:05.670 I notice that 0.2 and 0.8, the conditional is 0.8. 00:39:05.670 --> 00:39:06.620 And the prior is 0.2. 00:39:06.620 --> 00:39:09.690 That's why it comes out to balance each other, right? 00:39:09.690 --> 00:39:11.230 So now if we get another Republican in 00:39:11.230 --> 00:39:14.080 there it goes way up. 00:39:14.080 --> 00:39:16.860 If I have a Democratic child it goes back down. 00:39:16.860 --> 00:39:19.180 If I have an equal balance between children then it goes 00:39:19.180 --> 00:39:22.320 way back down because of that prior probability being low. 00:39:22.320 --> 00:39:27.280 So if I make that high, even though the children are 00:39:27.280 --> 00:39:29.865 balanced, I'm still going to have a high probability of 00:39:29.865 --> 00:39:32.250 being a Republican. 00:39:32.250 --> 00:39:33.140 Now let's see. 00:39:33.140 --> 00:39:36.180 If I take that slider there, the conditional probability, 00:39:36.180 --> 00:39:39.800 and drive it to the left here-- let me make that 00:39:39.800 --> 00:39:42.180 equally in. 00:39:42.180 --> 00:39:44.760 And let's make that one thing. 00:39:44.760 --> 00:39:45.620 I don't know. 00:39:45.620 --> 00:39:47.766 What am I doing now? 00:39:47.766 --> 00:39:52.460 If I make the probability less than 0.5, what's that mean? 00:39:52.460 --> 00:39:54.510 That means you're sore at your parents and you want to belong 00:39:54.510 --> 00:39:57.720 to a different party. 00:39:57.720 --> 00:40:02.200 All right, so now, what's next? 00:40:02.200 --> 00:40:04.650 Oh gosh. 00:40:04.650 --> 00:40:05.900 What's next? 00:40:09.620 --> 00:40:10.870 This is what's next. 00:40:15.720 --> 00:40:18.220 What's next to somewhere? 00:40:18.220 --> 00:40:20.440 Yeah, this is what's next. 00:40:20.440 --> 00:40:21.210 This here. 00:40:21.210 --> 00:40:22.460 We've got two models. 00:40:24.400 --> 00:40:28.070 Remember when I said we wanted to decide between them? 00:40:28.070 --> 00:40:30.930 Can we use that Bayesian hack to do that, too? 00:40:30.930 --> 00:40:31.580 Sure. 00:40:31.580 --> 00:40:34.790 Because we've got these two models. 00:40:34.790 --> 00:40:37.770 We've got the probabilities in them. 00:40:37.770 --> 00:40:42.490 So now I can take my data and calculate the probability of a 00:40:42.490 --> 00:40:45.620 left model given the data and the probability of the right 00:40:45.620 --> 00:40:48.780 model given the data, multiply that times their a priori 00:40:48.780 --> 00:40:51.860 probabilities, which I'll assume are equal. 00:40:51.860 --> 00:40:54.710 Then I can do a model selection deal much in 00:40:54.710 --> 00:40:57.250 defiance to what I was hinting at before. 00:40:57.250 --> 00:40:58.500 so let's try that. 00:41:03.330 --> 00:41:05.720 Whoa. 00:41:05.720 --> 00:41:08.915 There are my two models. 00:41:08.915 --> 00:41:10.460 Yes, there they are. 00:41:10.460 --> 00:41:11.850 We've already trained them up. 00:41:11.850 --> 00:41:13.990 And they've got their probabilities. 00:41:13.990 --> 00:41:15.630 Now what we're going to do is we're going to use the 00:41:15.630 --> 00:41:18.950 original model to simulate the data. 00:41:18.950 --> 00:41:21.350 So what we're going to do is we're going to simulate draws, 00:41:21.350 --> 00:41:25.770 simulate events, similarly combinations of all variables 00:41:25.770 --> 00:41:30.626 using a model that looks like the one on the left, that is 00:41:30.626 --> 00:41:32.530 the one on the left except for the slight differences in 00:41:32.530 --> 00:41:34.320 probabilities, OK? 00:41:34.320 --> 00:41:36.570 Then we're going to do this Bayesian thing and see where 00:41:36.570 --> 00:41:37.880 the meter goes. 00:41:37.880 --> 00:41:40.020 So we'll run one data point. 00:41:40.020 --> 00:41:41.960 Oops, went the wrong way. 00:41:41.960 --> 00:41:42.960 Makes me nervous. 00:41:42.960 --> 00:41:44.750 I just finished this at 9:15. 00:41:44.750 --> 00:41:46.610 Maybe there's a bug. 00:41:46.610 --> 00:41:49.720 Oops, two data points, swings to the left. 00:41:49.720 --> 00:41:51.140 Three data points, back to the right. 00:41:51.140 --> 00:41:54.070 Of course that's not much data. 00:41:54.070 --> 00:41:56.685 So let's put some more data in. 00:41:56.685 --> 00:41:57.130 Yeah. 00:41:57.130 --> 00:41:58.730 Boom, there it goes. 00:41:58.730 --> 00:41:59.480 Let's try that again. 00:41:59.480 --> 00:42:01.060 That was cool. 00:42:01.060 --> 00:42:07.050 So let's run 1,000 simulations and one data point. 00:42:07.050 --> 00:42:08.780 It bobbles around a little bit and goes 00:42:08.780 --> 00:42:09.640 flat over to the left. 00:42:09.640 --> 00:42:12.630 Because that is the model that reflects the one that the data 00:42:12.630 --> 00:42:15.480 is generated from. 00:42:15.480 --> 00:42:19.150 So now we got Bayesian classification, except now the 00:42:19.150 --> 00:42:21.790 classification has gone one step more and it becomes 00:42:21.790 --> 00:42:23.440 structure discovery. 00:42:23.440 --> 00:42:25.700 We've got two choices of structure. 00:42:25.700 --> 00:42:28.660 And we can use this Bayesian thing to decide which of the 00:42:28.660 --> 00:42:31.060 two structures is best. 00:42:31.060 --> 00:42:32.150 Isn't that cool? 00:42:32.150 --> 00:42:34.230 Well, it's only cool if you could do what? 00:42:38.950 --> 00:42:42.710 So if you had two choices-- 00:42:42.710 --> 00:42:46.210 you can select between them and pick the best one-- 00:42:46.210 --> 00:42:47.490 but there are-- 00:42:47.490 --> 00:42:51.120 gosh, for this number of variables, there are a whole 00:42:51.120 --> 00:42:55.540 lot of different networks that satisfy the no looping 00:42:55.540 --> 00:43:00.952 criteria and don't have very many parents. 00:43:00.952 --> 00:43:02.770 There's an awful lot of them. 00:43:02.770 --> 00:43:05.450 In fact, if you strict this network to two parents there 00:43:05.450 --> 00:43:08.260 are probably thousands and thousands of possible 00:43:08.260 --> 00:43:09.590 structures. 00:43:09.590 --> 00:43:10.840 So do I try them all? 00:43:13.390 --> 00:43:14.340 Probably not. 00:43:14.340 --> 00:43:16.370 It's too much work when you get 30 variables or 00:43:16.370 --> 00:43:19.320 something like that. 00:43:19.320 --> 00:43:20.450 So what do you do? 00:43:20.450 --> 00:43:21.740 We know what to do, right? 00:43:21.740 --> 00:43:23.390 We're almost veterans a 6034. 00:43:23.390 --> 00:43:25.750 We have to search! 00:43:25.750 --> 00:43:30.270 So what we do is we take the loser and we modified it. 00:43:30.270 --> 00:43:31.320 And then we modify it again. 00:43:31.320 --> 00:43:38.120 And we keep modifying it until we drop dead or we get 00:43:38.120 --> 00:43:40.450 something that we're happy with. 00:43:40.450 --> 00:43:43.480 So let's see what happens if we change this problem a 00:43:43.480 --> 00:43:45.030 little bit and do structure discover. 00:43:45.030 --> 00:43:47.740 We're starting out with nothing linked. 00:43:47.740 --> 00:43:50.380 And we're going to just start running this guy. 00:43:50.380 --> 00:43:51.600 So what's going to happen is that the 00:43:51.600 --> 00:43:53.690 good guy will prevail. 00:43:53.690 --> 00:43:55.990 And the bad guy will be a copy of the good guy 00:43:55.990 --> 00:43:57.150 perturbed in some way. 00:43:57.150 --> 00:44:00.280 So it's a random search. 00:44:00.280 --> 00:44:01.380 You'll notice that score-- 00:44:01.380 --> 00:44:03.170 it's too small for you to read. 00:44:03.170 --> 00:44:04.460 All these things are too small to read. 00:44:04.460 --> 00:44:05.710 Let me make it a little bigger. 00:44:08.230 --> 00:44:11.130 Too small to read, but that number on the right there is 00:44:11.130 --> 00:44:14.240 not the product of the probabilities, actually. 00:44:14.240 --> 00:44:19.850 It's the sum of the logarithms of the probabilities. 00:44:19.850 --> 00:44:21.780 They go together, right? 00:44:21.780 --> 00:44:23.440 And the reason you use this instead of the probabilities 00:44:23.440 --> 00:44:27.590 is because these numbers get so small that was a 32-bit 00:44:27.590 --> 00:44:29.740 machine, you eventually lose. 00:44:29.740 --> 00:44:34.730 So use the log of the probabilities rather than the 00:44:34.730 --> 00:44:35.630 product of the probabilities. 00:44:35.630 --> 00:44:37.660 You use the sum of the logs instead of the product of the 00:44:37.660 --> 00:44:39.840 probabilities. 00:44:39.840 --> 00:44:43.210 And eventually, you hope that this thing converges on the 00:44:43.210 --> 00:44:44.760 correct interpretation. 00:44:44.760 --> 00:44:45.470 But you know what? 00:44:45.470 --> 00:44:50.900 This thing is so flat as a space and so a large and so 00:44:50.900 --> 00:44:56.490 telephone pole-like that it's full of local maxima. 00:44:56.490 --> 00:45:00.670 So what this program is doing is every once in awhile-- 00:45:00.670 --> 00:45:02.880 I think with probability 1 and 10; I forgot what 00:45:02.880 --> 00:45:03.883 parameters I used-- 00:45:03.883 --> 00:45:08.410 every once in awhile, it'll do a total radical rearrangement 00:45:08.410 --> 00:45:09.150 of the structures. 00:45:09.150 --> 00:45:11.830 In other words, it's a random restart. 00:45:11.830 --> 00:45:13.820 It keeps track of the best guy so far. 00:45:13.820 --> 00:45:16.370 And every once in awhile it does a totally random restart 00:45:16.370 --> 00:45:18.870 in its effort to search the space. 00:45:18.870 --> 00:45:23.130 So that's how you go from probabilistic inference to 00:45:23.130 --> 00:45:25.300 structure discovery. 00:45:25.300 --> 00:45:29.560 Now when is this stuff actually useful? 00:45:29.560 --> 00:45:32.600 Well, I hinted at a medical diagnosis, right? 00:45:32.600 --> 00:45:34.890 That's a situation where you've got some symptoms. 00:45:34.890 --> 00:45:38.160 And you want to know what the disease is. 00:45:38.160 --> 00:45:42.470 So as soon as you use the keyword "diagnosis," you've 00:45:42.470 --> 00:45:45.840 got a problem for which this stuff is a candidate. 00:45:45.840 --> 00:45:48.640 So what other kinds of diagnosis problems are there? 00:45:48.640 --> 00:45:51.360 Well, you might be lying to me. 00:45:51.360 --> 00:45:53.350 So I can put a lie detector on you. 00:45:53.350 --> 00:45:55.980 And each of those variables that are measured by the lie 00:45:55.980 --> 00:45:58.370 detector are an independent indication whether you're 00:45:58.370 --> 00:45:59.910 telling the truth or not. 00:45:59.910 --> 00:46:04.300 So it's this kind of Bayesian discovery thing. 00:46:04.300 --> 00:46:07.410 Naive Bayesian Classification. 00:46:07.410 --> 00:46:09.972 What other kinds of problems speak to 00:46:09.972 --> 00:46:11.130 the issue of diagnosis? 00:46:11.130 --> 00:46:14.410 Well, we like to know how well you know the material! 00:46:14.410 --> 00:46:19.110 So we can use quizzes as pieces of evidence. 00:46:19.110 --> 00:46:21.590 Thank god we don't use exactly a naive Bayesian classifier, 00:46:21.590 --> 00:46:24.600 because then we wouldn't be able to do that combination. 00:46:24.600 --> 00:46:27.020 We have to use a slightly more complex-- 00:46:27.020 --> 00:46:30.280 what you can think of as a slightly more complex Bayesian 00:46:30.280 --> 00:46:33.430 net to do that particular kind of diagnosis. 00:46:33.430 --> 00:46:36.950 You might have a spacecraft or an airplane or other piece of 00:46:36.950 --> 00:46:38.950 equipment with all sorts of symptoms. 00:46:38.950 --> 00:46:40.530 You're trying to figure out what to do next, 00:46:40.530 --> 00:46:42.100 what the cause is. 00:46:42.100 --> 00:46:46.210 So using the evidence to go backward to the cause. 00:46:46.210 --> 00:46:49.470 So maybe you've got some program that doesn't work. 00:46:49.470 --> 00:46:51.390 Happens to me a lot. 00:46:51.390 --> 00:46:54.580 So I use the evidence from the symptoms of the misbehavior to 00:46:54.580 --> 00:46:58.480 figure out what the most probable cause is. 00:46:58.480 --> 00:47:03.020 But now to conclude the day-- last time there weren't any 00:47:03.020 --> 00:47:03.890 powerful ideas. 00:47:03.890 --> 00:47:08.090 But if you take the combination of the last 00:47:08.090 --> 00:47:14.170 lecture and this lecture to be a candidate for gold star 00:47:14.170 --> 00:47:17.740 ideas, these are the ones I'd like to leave you with. 00:47:17.740 --> 00:47:19.750 We got here is-- 00:47:19.750 --> 00:47:22.420 this Bayesian stuff, all these probabilistic calculations are 00:47:22.420 --> 00:47:23.965 the right thing to do. 00:47:23.965 --> 00:47:30.270 They're the right way to work when you don't know anything, 00:47:30.270 --> 00:47:32.600 which would make it sound like you're not very useful, 00:47:32.600 --> 00:47:34.890 because you think you always-- well, in fact, there are a lot 00:47:34.890 --> 00:47:38.020 of situations where you either can't know everything, don't 00:47:38.020 --> 00:47:40.420 have time to know everything, or don't want to take the 00:47:40.420 --> 00:47:42.630 effort to know everything. 00:47:42.630 --> 00:47:47.190 So in medical diagnosis all you've got is the symptoms. 00:47:47.190 --> 00:47:49.730 You can't go in there and figure out in a more precise 00:47:49.730 --> 00:47:50.750 way exactly what's wrong. 00:47:50.750 --> 00:47:53.730 So you use the symptoms to determine what the cause is. 00:47:53.730 --> 00:47:56.850 And then all those other kinds of cases that I mentioned. 00:47:56.850 --> 00:48:00.520 But now, what other kinds of structure discovery are there? 00:48:00.520 --> 00:48:02.760 Well, the kind of structure discovery that I hinted at in 00:48:02.760 --> 00:48:05.550 the beginning will be the subject that we'll begin with 00:48:05.550 --> 00:48:10.330 during our next and sadly final conversation here in 00:48:10.330 --> 00:48:11.700 [? 10250 ?] 00:48:11.700 --> 00:48:13.320 on Wednesday. 00:48:13.320 --> 00:48:16.210 It will feature not only a discussion of how this stuff 00:48:16.210 --> 00:48:19.890 can be used to discover patterns and stories, but 00:48:19.890 --> 00:48:24.590 we'll also talk about what's on the final, what kind of 00:48:24.590 --> 00:48:28.870 thing you could do next, that sort of thing to finish off 00:48:28.870 --> 00:48:29.420 the subject. 00:48:29.420 --> 00:48:31.710 And that's the end of the story for today.