WEBVTT 00:00:00.000 --> 00:00:00.040 align:middle line:90% 00:00:00.040 --> 00:00:02.460 align:middle line:84% The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 align:middle line:90% Commons license. 00:00:03.870 --> 00:00:06.910 align:middle line:84% Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 align:middle line:84% offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.460 align:middle line:84% To make a donation or view additional materials from 00:00:13.460 --> 00:00:19.290 align:middle line:84% hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:21.708 align:middle line:90% ocw.mit.edu. 00:00:21.708 --> 00:00:25.380 align:middle line:84% PROFESSOR: It involves real phenomena out there. 00:00:25.380 --> 00:00:28.960 align:middle line:84% So we have real stuff that happens. 00:00:28.960 --> 00:00:33.630 align:middle line:84% So it might be an arrival process to a bank that we're 00:00:33.630 --> 00:00:35.790 align:middle line:90% trying to model. 00:00:35.790 --> 00:00:38.230 align:middle line:84% This is a reality, but this is what we have 00:00:38.230 --> 00:00:39.660 align:middle line:90% been doing so far. 00:00:39.660 --> 00:00:41.910 align:middle line:84% We have been playing with models of 00:00:41.910 --> 00:00:43.770 align:middle line:90% probabilistic phenomena. 00:00:43.770 --> 00:00:46.730 align:middle line:84% And somehow we need to tie the two together. 00:00:46.730 --> 00:00:50.930 align:middle line:84% The way these are tied is that we observe the real world and 00:00:50.930 --> 00:00:53.530 align:middle line:90% this gives us data. 00:00:53.530 --> 00:00:58.590 align:middle line:84% And then based on these data, we try to come up with a model 00:00:58.590 --> 00:01:01.930 align:middle line:90% of what exactly is going on. 00:01:01.930 --> 00:01:05.290 align:middle line:84% For example, for an arrival process, you might ask the 00:01:05.290 --> 00:01:08.680 align:middle line:84% model in question, is my arrival process Poisson or is 00:01:08.680 --> 00:01:10.300 align:middle line:90% it something different? 00:01:10.300 --> 00:01:14.630 align:middle line:84% If it is Poisson, what is the rate of the arrival process? 00:01:14.630 --> 00:01:17.460 align:middle line:84% Once you come up with your model and you come up with the 00:01:17.460 --> 00:01:21.710 align:middle line:84% parameters of the model, then you can use it to make 00:01:21.710 --> 00:01:27.520 align:middle line:84% predictions about reality or to figure out certain hidden 00:01:27.520 --> 00:01:31.890 align:middle line:84% things, certain hidden aspects of reality, that you do not 00:01:31.890 --> 00:01:35.560 align:middle line:84% observe directly, but you try to infer what they are. 00:01:35.560 --> 00:01:38.900 align:middle line:84% So that's where the usefulness of the model comes in. 00:01:38.900 --> 00:01:43.330 align:middle line:84% Now this field is of course tremendously useful. 00:01:43.330 --> 00:01:46.650 align:middle line:84% And it shows up pretty much everywhere. 00:01:46.650 --> 00:01:50.000 align:middle line:84% So we talked about the polling examples in the 00:01:50.000 --> 00:01:51.280 align:middle line:90% last couple of lectures. 00:01:51.280 --> 00:01:53.520 align:middle line:84% This is, of course, a real application. 00:01:53.520 --> 00:01:57.525 align:middle line:84% You sample and on the basis of the sample that you have, you 00:01:57.525 --> 00:02:00.400 align:middle line:84% try to make some inferences about, let's say, the 00:02:00.400 --> 00:02:03.060 align:middle line:84% preferences in a given population. 00:02:03.060 --> 00:02:06.230 align:middle line:84% Let's say in the medical field, you want to try whether 00:02:06.230 --> 00:02:08.919 align:middle line:84% a certain drug makes a difference or not. 00:02:08.919 --> 00:02:14.380 align:middle line:84% So people would do medical trials, get some results, and 00:02:14.380 --> 00:02:17.640 align:middle line:84% then from the data somehow you need to make sense of them and 00:02:17.640 --> 00:02:18.530 align:middle line:90% make a decision. 00:02:18.530 --> 00:02:21.360 align:middle line:84% Is the new drug useful or is it not? 00:02:21.360 --> 00:02:23.460 align:middle line:84% How do we go systematically about the 00:02:23.460 --> 00:02:24.710 align:middle line:90% question of this type? 00:02:24.710 --> 00:02:27.770 align:middle line:90% 00:02:27.770 --> 00:02:32.170 align:middle line:84% A sexier, more recent topic, there's this famous Netflix 00:02:32.170 --> 00:02:37.510 align:middle line:84% competition where Netflix gives you a huge table of 00:02:37.510 --> 00:02:41.450 align:middle line:90% movies and people. 00:02:41.450 --> 00:02:45.860 align:middle line:84% And people have rated the movies, but not everyone has 00:02:45.860 --> 00:02:47.850 align:middle line:84% watched all of the movies in there. 00:02:47.850 --> 00:02:49.460 align:middle line:90% You have some of the ratings. 00:02:49.460 --> 00:02:53.250 align:middle line:84% For example, this person gave a 4 to that particular movie. 00:02:53.250 --> 00:02:56.300 align:middle line:84% So you get the table that's partially filled. 00:02:56.300 --> 00:02:58.300 align:middle line:84% And the Netflix asks you to make 00:02:58.300 --> 00:02:59.860 align:middle line:90% recommendations to people. 00:02:59.860 --> 00:03:02.410 align:middle line:90% So this means trying to guess. 00:03:02.410 --> 00:03:06.100 align:middle line:84% This person here, how much would they like this 00:03:06.100 --> 00:03:07.610 align:middle line:90% particular movie? 00:03:07.610 --> 00:03:11.130 align:middle line:84% And you can start thinking, well, maybe this person has 00:03:11.130 --> 00:03:14.860 align:middle line:84% given somewhat similar ratings with another person. 00:03:14.860 --> 00:03:18.440 align:middle line:84% And if that other person has also seen that movie, maybe 00:03:18.440 --> 00:03:21.290 align:middle line:84% the rating of that other person is relevant. 00:03:21.290 --> 00:03:24.230 align:middle line:84% But of course it's a lot more complicated than that. 00:03:24.230 --> 00:03:26.650 align:middle line:84% And this has been a serious competition where people have 00:03:26.650 --> 00:03:30.230 align:middle line:84% been using every heavy, wet machinery that there is in 00:03:30.230 --> 00:03:32.540 align:middle line:84% statistics, trying to come up with good 00:03:32.540 --> 00:03:35.140 align:middle line:90% recommendation systems. 00:03:35.140 --> 00:03:37.870 align:middle line:84% Then the other people, of course, are trying to analyze 00:03:37.870 --> 00:03:39.010 align:middle line:90% financial data. 00:03:39.010 --> 00:03:43.680 align:middle line:84% Somebody gives you the sequence of the values, let's 00:03:43.680 --> 00:03:45.840 align:middle line:90% say of the SMP index. 00:03:45.840 --> 00:03:47.850 align:middle line:90% You look at something like this 00:03:47.850 --> 00:03:49.770 align:middle line:90% and you can ask questions. 00:03:49.770 --> 00:03:55.030 align:middle line:84% How do I model these data using any of the models that 00:03:55.030 --> 00:03:57.060 align:middle line:90% we have in our bag of tools? 00:03:57.060 --> 00:04:00.230 align:middle line:84% How can I make predictions about what's going to happen 00:04:00.230 --> 00:04:03.310 align:middle line:90% afterwards, and so on? 00:04:03.310 --> 00:04:09.700 align:middle line:84% On the engineering side, anywhere where you have noise 00:04:09.700 --> 00:04:11.590 align:middle line:90% inference comes in. 00:04:11.590 --> 00:04:13.810 align:middle line:84% Signal processing, in some sense, is just 00:04:13.810 --> 00:04:14.960 align:middle line:90% an inference problem. 00:04:14.960 --> 00:04:18.730 align:middle line:84% You observe signals that are noisy and you try to figure 00:04:18.730 --> 00:04:21.750 align:middle line:84% out exactly what's happening out there or what kind of 00:04:21.750 --> 00:04:24.130 align:middle line:90% signal has been sent. 00:04:24.130 --> 00:04:28.830 align:middle line:84% Maybe the beginning of the field could be traced a few 00:04:28.830 --> 00:04:32.060 align:middle line:84% hundred years ago where people would observe, make 00:04:32.060 --> 00:04:35.420 align:middle line:84% astronomical observations of the position of the 00:04:35.420 --> 00:04:37.550 align:middle line:90% planets in the sky. 00:04:37.550 --> 00:04:41.130 align:middle line:84% They would have some beliefs that perhaps the orbits of 00:04:41.130 --> 00:04:44.070 align:middle line:90% planets is an ellipse. 00:04:44.070 --> 00:04:47.840 align:middle line:84% Or if it's a comet, maybe it's a parabola, hyperbola, don't 00:04:47.840 --> 00:04:48.640 align:middle line:90% know what it is. 00:04:48.640 --> 00:04:51.320 align:middle line:84% But they would have a model of that. 00:04:51.320 --> 00:04:53.840 align:middle line:84% But, of course, astronomical measurements would not be 00:04:53.840 --> 00:04:55.300 align:middle line:90% perfectly exact. 00:04:55.300 --> 00:05:00.690 align:middle line:84% And they would try to find the curve that fits these data. 00:05:00.690 --> 00:05:05.580 align:middle line:84% How do you go about choosing this particular curve on the 00:05:05.580 --> 00:05:07.960 align:middle line:84% base of noisy data and try to do it in a 00:05:07.960 --> 00:05:11.274 align:middle line:90% somewhat principled way? 00:05:11.274 --> 00:05:13.890 align:middle line:84% OK, so questions of this type-- clearly the 00:05:13.890 --> 00:05:17.100 align:middle line:84% applications are all over the place. 00:05:17.100 --> 00:05:20.830 align:middle line:84% But how is this related conceptually with what we have 00:05:20.830 --> 00:05:22.480 align:middle line:90% been doing so far? 00:05:22.480 --> 00:05:25.960 align:middle line:84% What's the relation between the field of inference and the 00:05:25.960 --> 00:05:28.130 align:middle line:84% field of probability as we have been 00:05:28.130 --> 00:05:30.650 align:middle line:90% practicing until now? 00:05:30.650 --> 00:05:33.620 align:middle line:84% Well, mathematically speaking, what's going to happen in the 00:05:33.620 --> 00:05:38.780 align:middle line:84% next few lectures could be just exercises or homework 00:05:38.780 --> 00:05:44.880 align:middle line:84% problems in the class in based on what we have done so far. 00:05:44.880 --> 00:05:48.560 align:middle line:84% That means you're not going to get any new facts about 00:05:48.560 --> 00:05:50.200 align:middle line:90% probability theory. 00:05:50.200 --> 00:05:53.930 align:middle line:84% Everything we're going to do will be simple applications of 00:05:53.930 --> 00:05:57.110 align:middle line:84% things that you already do know. 00:05:57.110 --> 00:06:00.140 align:middle line:84% So in some sense, statistics and inference is just an 00:06:00.140 --> 00:06:02.780 align:middle line:84% applied exercise in probability. 00:06:02.780 --> 00:06:08.310 align:middle line:84% But actually, things are not that simple in 00:06:08.310 --> 00:06:09.550 align:middle line:90% the following sense. 00:06:09.550 --> 00:06:12.510 align:middle line:84% If you get a probability problem, 00:06:12.510 --> 00:06:14.040 align:middle line:90% there's a correct answer. 00:06:14.040 --> 00:06:15.450 align:middle line:90% There's a correct solution. 00:06:15.450 --> 00:06:18.170 align:middle line:84% And that correct solution is unique. 00:06:18.170 --> 00:06:20.550 align:middle line:90% There's no ambiguity. 00:06:20.550 --> 00:06:23.380 align:middle line:84% The theory of probability has clearly defined rules. 00:06:23.380 --> 00:06:24.570 align:middle line:90% These are the axioms. 00:06:24.570 --> 00:06:27.550 align:middle line:84% You're given some information about probability 00:06:27.550 --> 00:06:28.280 align:middle line:90% distributions. 00:06:28.280 --> 00:06:31.000 align:middle line:84% You're asked to calculate certain other things. 00:06:31.000 --> 00:06:32.190 align:middle line:90% There's no ambiguity. 00:06:32.190 --> 00:06:34.230 align:middle line:90% Answers are always unique. 00:06:34.230 --> 00:06:39.180 align:middle line:84% In statistical questions, it's no longer the case that the 00:06:39.180 --> 00:06:41.420 align:middle line:90% question has a unique answer. 00:06:41.420 --> 00:06:44.990 align:middle line:84% If I give you data and I ask you what's the best way of 00:06:44.990 --> 00:06:49.710 align:middle line:84% estimating the motion of that planet, reasonable people can 00:06:49.710 --> 00:06:53.370 align:middle line:84% come up with different methods. 00:06:53.370 --> 00:06:56.790 align:middle line:84% And reasonable people will try to argue that's my method has 00:06:56.790 --> 00:07:00.140 align:middle line:84% these desirable properties but somebody else may say, here's 00:07:00.140 --> 00:07:03.740 align:middle line:84% another method that has certain desirable properties. 00:07:03.740 --> 00:07:08.220 align:middle line:84% And it's not clear what the best method is. 00:07:08.220 --> 00:07:11.330 align:middle line:84% So it's good to have some understanding of what the 00:07:11.330 --> 00:07:16.910 align:middle line:84% issues are and to know at least what is the general 00:07:16.910 --> 00:07:20.150 align:middle line:84% class of methods that one tries to consider, how does 00:07:20.150 --> 00:07:22.380 align:middle line:90% one go about such problems. 00:07:22.380 --> 00:07:24.360 align:middle line:84% So we're going to see lots and lots of 00:07:24.360 --> 00:07:25.880 align:middle line:90% different inference methods. 00:07:25.880 --> 00:07:27.350 align:middle line:84% We're not going to tell you that one is 00:07:27.350 --> 00:07:28.730 align:middle line:90% better than the other. 00:07:28.730 --> 00:07:30.940 align:middle line:84% But it's important to understand what are the 00:07:30.940 --> 00:07:33.980 align:middle line:84% concepts between those different methods. 00:07:33.980 --> 00:07:38.710 align:middle line:84% And finally, statistics can be misused really badly. 00:07:38.710 --> 00:07:41.870 align:middle line:84% That is, one can come up with methods that you think are 00:07:41.870 --> 00:07:48.650 align:middle line:84% sound, but in fact they're not quite that. 00:07:48.650 --> 00:07:52.830 align:middle line:84% I will bring some examples next time and talk a little 00:07:52.830 --> 00:07:54.290 align:middle line:90% more about this. 00:07:54.290 --> 00:07:58.540 align:middle line:84% So, they want to say, you have some data, you want to make 00:07:58.540 --> 00:08:02.590 align:middle line:84% some inference from them, what many people will do is to go 00:08:02.590 --> 00:08:06.340 align:middle line:84% to Wikipedia, find a statistical test that they 00:08:06.340 --> 00:08:08.990 align:middle line:84% think it applies to that situation, plug in numbers, 00:08:08.990 --> 00:08:10.880 align:middle line:90% and present results. 00:08:10.880 --> 00:08:14.220 align:middle line:84% Are the conclusions that they get really justified or are 00:08:14.220 --> 00:08:16.400 align:middle line:84% they misusing statistical methods? 00:08:16.400 --> 00:08:20.520 align:middle line:84% Well, too many people actually do misuse statistics and 00:08:20.520 --> 00:08:24.530 align:middle line:84% conclusions that people get are often false. 00:08:24.530 --> 00:08:29.840 align:middle line:84% So it's important to, besides just being able to copy 00:08:29.840 --> 00:08:32.600 align:middle line:84% statistical tests and use them, to understand what are 00:08:32.600 --> 00:08:35.860 align:middle line:84% the assumptions between the different methods and what 00:08:35.860 --> 00:08:40.559 align:middle line:84% kind of guarantees they have, if any. 00:08:40.559 --> 00:08:44.420 align:middle line:84% All right, so we'll try to do a quick tour through the field 00:08:44.420 --> 00:08:47.600 align:middle line:84% of inference in this lecture and the next few lectures that 00:08:47.600 --> 00:08:51.700 align:middle line:84% we have left this semester and try to highlight at the very 00:08:51.700 --> 00:08:53.940 align:middle line:84% high level the main concept skills, and 00:08:53.940 --> 00:08:56.990 align:middle line:90% techniques that come in. 00:08:56.990 --> 00:08:59.840 align:middle line:84% Let's start with some generalities and some general 00:08:59.840 --> 00:09:01.090 align:middle line:90% statements. 00:09:01.090 --> 00:09:03.090 align:middle line:90% 00:09:03.090 --> 00:09:07.090 align:middle line:84% One first statement is that statistics or inference 00:09:07.090 --> 00:09:11.800 align:middle line:84% problems come up in very different guises. 00:09:11.800 --> 00:09:16.490 align:middle line:84% And they may look as if they are of very different forms. 00:09:16.490 --> 00:09:20.190 align:middle line:84% Although, at some fundamental level, the basic issues turn 00:09:20.190 --> 00:09:23.320 align:middle line:84% out to be always pretty much the same. 00:09:23.320 --> 00:09:27.880 align:middle line:90% So let's look at this example. 00:09:27.880 --> 00:09:31.420 align:middle line:84% There's an unknown signal that's being sent. 00:09:31.420 --> 00:09:35.840 align:middle line:84% It's sent through some medium, and that medium just takes the 00:09:35.840 --> 00:09:39.180 align:middle line:84% signal and amplifies it by a certain number. 00:09:39.180 --> 00:09:41.340 align:middle line:84% So you can think of somebody shouting. 00:09:41.340 --> 00:09:42.920 align:middle line:90% There's the air out there. 00:09:42.920 --> 00:09:46.420 align:middle line:84% What you shouted will be attenuated through the air 00:09:46.420 --> 00:09:48.040 align:middle line:90% until it gets to a receiver. 00:09:48.040 --> 00:09:51.730 align:middle line:84% And that receiver then observes this, but together 00:09:51.730 --> 00:09:53.110 align:middle line:90% with some random noise. 00:09:53.110 --> 00:09:56.040 align:middle line:90% 00:09:56.040 --> 00:10:00.390 align:middle line:84% Here I meant S. S is the signal that's being sent. 00:10:00.390 --> 00:10:06.280 align:middle line:90% And what you observe is an X. 00:10:06.280 --> 00:10:09.240 align:middle line:84% You observe X, so what kind of inference problems 00:10:09.240 --> 00:10:11.240 align:middle line:90% could we have here? 00:10:11.240 --> 00:10:15.400 align:middle line:84% In some cases, you want to build a model of the physical 00:10:15.400 --> 00:10:17.450 align:middle line:84% phenomenon that you're dealing with. 00:10:17.450 --> 00:10:21.180 align:middle line:84% So for example, you don't know the attenuation of your signal 00:10:21.180 --> 00:10:25.190 align:middle line:84% and you try to find out what this number is based on the 00:10:25.190 --> 00:10:26.980 align:middle line:90% observations that you have. 00:10:26.980 --> 00:10:30.240 align:middle line:84% So the way this is done in engineering systems is that 00:10:30.240 --> 00:10:35.020 align:middle line:84% you design a certain signal, you know what it is, you shout 00:10:35.020 --> 00:10:39.560 align:middle line:84% a particular word, and then the receiver listens. 00:10:39.560 --> 00:10:43.460 align:middle line:84% And based on the intensity of the signal that they get, they 00:10:43.460 --> 00:10:48.380 align:middle line:84% try to make a guess about A. So you don't know A, but you 00:10:48.380 --> 00:10:52.460 align:middle line:84% know S. And by observing X, you get some information 00:10:52.460 --> 00:10:54.270 align:middle line:90% about what A is. 00:10:54.270 --> 00:10:57.810 align:middle line:84% So in this case, you're trying to build a model of the medium 00:10:57.810 --> 00:11:01.170 align:middle line:84% through which your signal is propagating. 00:11:01.170 --> 00:11:04.600 align:middle line:84% So sometimes one would call problems of this kind, let's 00:11:04.600 --> 00:11:07.990 align:middle line:90% say, system identification. 00:11:07.990 --> 00:11:11.980 align:middle line:84% In a different version of an inference problem that comes 00:11:11.980 --> 00:11:15.300 align:middle line:84% with this picture, you've done your modeling. 00:11:15.300 --> 00:11:18.160 align:middle line:84% You know your A. You know the medium through which the 00:11:18.160 --> 00:11:22.330 align:middle line:84% signal is going, but it's a communication system. 00:11:22.330 --> 00:11:24.190 align:middle line:84% This person is trying to communicate 00:11:24.190 --> 00:11:26.140 align:middle line:90% something to that person. 00:11:26.140 --> 00:11:30.250 align:middle line:84% So you send the signal S, but that person receives a noisy 00:11:30.250 --> 00:11:35.430 align:middle line:84% version of S. So that person tries to reconstruct S based 00:11:35.430 --> 00:11:36.930 align:middle line:90% on X. 00:11:36.930 --> 00:11:42.210 align:middle line:84% So in both cases, we have a linear relation between X and 00:11:42.210 --> 00:11:43.490 align:middle line:90% the unknown quantity. 00:11:43.490 --> 00:11:47.360 align:middle line:84% In one version, A is the unknown and we know S. In the 00:11:47.360 --> 00:11:51.670 align:middle line:84% other version, A is known, and so we try to infer S. 00:11:51.670 --> 00:11:54.300 align:middle line:84% Mathematically, you can see that this is essentially the 00:11:54.300 --> 00:11:57.060 align:middle line:84% same kind of problem in both cases. 00:11:57.060 --> 00:12:03.590 align:middle line:84% Although, the kind of practical problem that you're 00:12:03.590 --> 00:12:07.580 align:middle line:84% trying to solve is a little different. 00:12:07.580 --> 00:12:11.880 align:middle line:84% So we will not be making any distinctions between problems 00:12:11.880 --> 00:12:15.940 align:middle line:84% of the model building type as opposed to models where you 00:12:15.940 --> 00:12:19.260 align:middle line:84% try to estimate some unknown signal and so on. 00:12:19.260 --> 00:12:22.400 align:middle line:84% Because conceptually, the tools that one uses for both 00:12:22.400 --> 00:12:26.850 align:middle line:84% types of problems are essentially the same. 00:12:26.850 --> 00:12:30.430 align:middle line:84% OK, next a very useful classification 00:12:30.430 --> 00:12:31.680 align:middle line:90% of inference problems-- 00:12:31.680 --> 00:12:34.170 align:middle line:90% 00:12:34.170 --> 00:12:37.760 align:middle line:84% the unknown quantity that you're trying to estimate 00:12:37.760 --> 00:12:40.770 align:middle line:84% could be either a discrete one that takes a 00:12:40.770 --> 00:12:43.040 align:middle line:90% small number of values. 00:12:43.040 --> 00:12:45.605 align:middle line:84% So this could be discrete problems, such as the airplane 00:12:45.605 --> 00:12:48.080 align:middle line:84% radar problem we encountered back a long 00:12:48.080 --> 00:12:50.120 align:middle line:90% time ago in this class. 00:12:50.120 --> 00:12:52.120 align:middle line:90% So there's two possibilities-- 00:12:52.120 --> 00:12:55.450 align:middle line:84% an airplane is out there or an airplane is not out there. 00:12:55.450 --> 00:12:57.050 align:middle line:84% And you're trying to make a decision 00:12:57.050 --> 00:12:58.940 align:middle line:90% between these two options. 00:12:58.940 --> 00:13:01.570 align:middle line:84% Or you can have other problems would you have, let's say, 00:13:01.570 --> 00:13:03.380 align:middle line:90% four possible options. 00:13:03.380 --> 00:13:05.970 align:middle line:84% You don't know which one is true, but you get data and you 00:13:05.970 --> 00:13:09.040 align:middle line:84% try to figure out which one is true. 00:13:09.040 --> 00:13:12.050 align:middle line:84% In problems of these kind, usually you want to make a 00:13:12.050 --> 00:13:14.050 align:middle line:90% decision based on your data. 00:13:14.050 --> 00:13:17.000 align:middle line:84% And you're interested in the probability of making a 00:13:17.000 --> 00:13:18.040 align:middle line:90% correct decision. 00:13:18.040 --> 00:13:19.430 align:middle line:84% You would like that probability to 00:13:19.430 --> 00:13:21.830 align:middle line:90% be as high as possible. 00:13:21.830 --> 00:13:24.000 align:middle line:84% Estimation problems are a little different. 00:13:24.000 --> 00:13:28.540 align:middle line:84% Here you have some continuous quantity that's not known. 00:13:28.540 --> 00:13:31.860 align:middle line:84% And you try to make a good guess of that quantity. 00:13:31.860 --> 00:13:36.050 align:middle line:84% And you would like your guess to be as close as possible to 00:13:36.050 --> 00:13:37.310 align:middle line:90% the true quantity. 00:13:37.310 --> 00:13:40.270 align:middle line:84% So the polling problem was of this type. 00:13:40.270 --> 00:13:44.720 align:middle line:84% There was an unknown fraction f of the population that had 00:13:44.720 --> 00:13:45.870 align:middle line:90% some property. 00:13:45.870 --> 00:13:50.040 align:middle line:84% And you try to estimate f as accurately as you can. 00:13:50.040 --> 00:13:53.420 align:middle line:84% So the distinction here is that usually here the unknown 00:13:53.420 --> 00:13:56.440 align:middle line:84% quantity takes on discrete set of values. 00:13:56.440 --> 00:13:57.890 align:middle line:84% Here the unknown quantity takes a 00:13:57.890 --> 00:14:00.030 align:middle line:90% continuous set of values. 00:14:00.030 --> 00:14:02.980 align:middle line:84% Here we're interested in the probability of error. 00:14:02.980 --> 00:14:07.400 align:middle line:84% Here we're interested in the size of the error. 00:14:07.400 --> 00:14:11.000 align:middle line:84% Broadly speaking, most inference problems fall either 00:14:11.000 --> 00:14:13.940 align:middle line:84% in this category or in that category. 00:14:13.940 --> 00:14:17.230 align:middle line:84% Although, if you want to complicate life, you can also 00:14:17.230 --> 00:14:20.250 align:middle line:84% think or construct problems where both of these aspects 00:14:20.250 --> 00:14:24.410 align:middle line:90% are simultaneously present. 00:14:24.410 --> 00:14:28.530 align:middle line:84% OK, finally since we're in classification mode, there is 00:14:28.530 --> 00:14:33.670 align:middle line:84% a very big, important dichotomy into how one goes 00:14:33.670 --> 00:14:35.940 align:middle line:90% about inference problems. 00:14:35.940 --> 00:14:39.150 align:middle line:84% And here there's two fundamentally different 00:14:39.150 --> 00:14:46.070 align:middle line:84% philosophical points of view, which is how do we model the 00:14:46.070 --> 00:14:50.270 align:middle line:90% quantity that is unknown? 00:14:50.270 --> 00:14:54.530 align:middle line:84% In one approach, you say there's a certain quantity 00:14:54.530 --> 00:14:57.590 align:middle line:90% that has a definite value. 00:14:57.590 --> 00:15:00.010 align:middle line:84% It just happens that they don't know it. 00:15:00.010 --> 00:15:01.320 align:middle line:90% But it's a number. 00:15:01.320 --> 00:15:03.290 align:middle line:84% There's nothing random about it. 00:15:03.290 --> 00:15:05.945 align:middle line:84% So think of trying to estimate some physical quantity. 00:15:05.945 --> 00:15:10.630 align:middle line:90% 00:15:10.630 --> 00:15:13.350 align:middle line:84% You're making measurements, you try to estimate the mass 00:15:13.350 --> 00:15:15.820 align:middle line:84% of an electron, which is a sort of 00:15:15.820 --> 00:15:18.270 align:middle line:90% universal physical constant. 00:15:18.270 --> 00:15:20.320 align:middle line:84% There's nothing random about it. 00:15:20.320 --> 00:15:22.340 align:middle line:90% It's a fixed number. 00:15:22.340 --> 00:15:29.120 align:middle line:84% You get data, because you have some measuring apparatus. 00:15:29.120 --> 00:15:33.020 align:middle line:84% And that measuring apparatus, depending on what that results 00:15:33.020 --> 00:15:37.160 align:middle line:84% that you get are affected by the true mass of the electron, 00:15:37.160 --> 00:15:39.340 align:middle line:90% but there's also some noise. 00:15:39.340 --> 00:15:42.200 align:middle line:84% You take the data out of your measuring apparatus and you 00:15:42.200 --> 00:15:44.465 align:middle line:84% try to come up with some estimate of 00:15:44.465 --> 00:15:47.220 align:middle line:90% that quantity theta. 00:15:47.220 --> 00:15:49.760 align:middle line:84% So this is definitely a legitimate picture, but the 00:15:49.760 --> 00:15:52.370 align:middle line:84% important thing in this picture is that this theta is 00:15:52.370 --> 00:15:54.570 align:middle line:90% written as lowercase. 00:15:54.570 --> 00:15:58.110 align:middle line:84% And that's to make the point that it's a real number, not a 00:15:58.110 --> 00:16:00.900 align:middle line:90% random variable. 00:16:00.900 --> 00:16:03.230 align:middle line:84% There's a different philosophical approach which 00:16:03.230 --> 00:16:08.180 align:middle line:84% says, well, anything that I don't know I should model it 00:16:08.180 --> 00:16:10.190 align:middle line:90% as a random variable. 00:16:10.190 --> 00:16:11.130 align:middle line:90% Yes, I know. 00:16:11.130 --> 00:16:14.500 align:middle line:84% The mass of the electron is not really random. 00:16:14.500 --> 00:16:15.690 align:middle line:90% It's a constant. 00:16:15.690 --> 00:16:17.920 align:middle line:90% But I don't know what it is. 00:16:17.920 --> 00:16:22.510 align:middle line:84% I have some vague sense, perhaps, what it is perhaps 00:16:22.510 --> 00:16:24.290 align:middle line:84% because of the experiments that some other 00:16:24.290 --> 00:16:25.940 align:middle line:90% people carried out. 00:16:25.940 --> 00:16:30.560 align:middle line:84% So perhaps I have a prior distribution on the possible 00:16:30.560 --> 00:16:32.160 align:middle line:90% values of Theta. 00:16:32.160 --> 00:16:34.990 align:middle line:84% And that prior distribution doesn't mean that the nature 00:16:34.990 --> 00:16:39.320 align:middle line:84% is random, but it's more of a subjective description of my 00:16:39.320 --> 00:16:44.570 align:middle line:84% subjective beliefs of where do I think this constant number 00:16:44.570 --> 00:16:46.200 align:middle line:90% happens to be. 00:16:46.200 --> 00:16:50.140 align:middle line:84% So even though it's not truly random, I model my initial 00:16:50.140 --> 00:16:52.600 align:middle line:84% beliefs before the experiment starts. 00:16:52.600 --> 00:16:55.790 align:middle line:84% In terms of a prior distribution, I view it as a 00:16:55.790 --> 00:16:57.470 align:middle line:90% random variable. 00:16:57.470 --> 00:17:01.850 align:middle line:84% Then I observe another related random variable through some 00:17:01.850 --> 00:17:02.930 align:middle line:90% measuring apparatus. 00:17:02.930 --> 00:17:05.920 align:middle line:84% And then I use this again to create an estimate. 00:17:05.920 --> 00:17:08.819 align:middle line:90% 00:17:08.819 --> 00:17:12.069 align:middle line:84% So these two pictures philosophically are very 00:17:12.069 --> 00:17:13.589 align:middle line:90% different from each other. 00:17:13.589 --> 00:17:17.130 align:middle line:84% Here we treat the unknown quantities as unknown numbers. 00:17:17.130 --> 00:17:20.589 align:middle line:84% Here we treat them as random variables. 00:17:20.589 --> 00:17:24.829 align:middle line:84% When we treat them as a random variables, then we know pretty 00:17:24.829 --> 00:17:27.109 align:middle line:84% much already what we should be doing. 00:17:27.109 --> 00:17:29.470 align:middle line:84% We should just use the Bayes rule. 00:17:29.470 --> 00:17:31.850 align:middle line:84% Based on X, find the conditional 00:17:31.850 --> 00:17:33.670 align:middle line:90% distribution of Theta. 00:17:33.670 --> 00:17:37.520 align:middle line:84% And that's what we will be doing mostly over this lecture 00:17:37.520 --> 00:17:40.010 align:middle line:90% and the next lecture. 00:17:40.010 --> 00:17:44.660 align:middle line:84% Now in both cases, what you end up getting at the end is 00:17:44.660 --> 00:17:47.240 align:middle line:90% an estimate. 00:17:47.240 --> 00:17:52.120 align:middle line:84% But actually, that estimate is what kind of object is it? 00:17:52.120 --> 00:17:55.170 align:middle line:84% It's a random variable in both cases. 00:17:55.170 --> 00:17:56.000 align:middle line:90% Why? 00:17:56.000 --> 00:17:58.130 align:middle line:84% Even in this case where theta was a 00:17:58.130 --> 00:18:01.060 align:middle line:90% constant, my data are random. 00:18:01.060 --> 00:18:02.860 align:middle line:90% I do my data processing. 00:18:02.860 --> 00:18:06.050 align:middle line:84% So I calculate a function of the data, the 00:18:06.050 --> 00:18:07.580 align:middle line:90% data are random variables. 00:18:07.580 --> 00:18:11.390 align:middle line:84% So out here we output something which is a function 00:18:11.390 --> 00:18:12.770 align:middle line:90% of a random variable. 00:18:12.770 --> 00:18:15.830 align:middle line:84% So this quantity here will be also random. 00:18:15.830 --> 00:18:18.400 align:middle line:84% It's affected by the noise and the experiment that I have 00:18:18.400 --> 00:18:19.650 align:middle line:90% been doing. 00:18:19.650 --> 00:18:22.330 align:middle line:84% That's why these estimators will be denoted 00:18:22.330 --> 00:18:24.920 align:middle line:90% by uppercase Thetas. 00:18:24.920 --> 00:18:26.740 align:middle line:90% And we will be using hats. 00:18:26.740 --> 00:18:29.030 align:middle line:84% Hat, usually in estimation, means 00:18:29.030 --> 00:18:32.990 align:middle line:90% an estimate of something. 00:18:32.990 --> 00:18:35.380 align:middle line:84% All right, so this is the big picture. 00:18:35.380 --> 00:18:38.690 align:middle line:84% We're going to start with the Bayesian version. 00:18:38.690 --> 00:18:42.830 align:middle line:84% And then the last few lectures we're going to talk about the 00:18:42.830 --> 00:18:45.690 align:middle line:84% non-Bayesian version or the classical one. 00:18:45.690 --> 00:18:48.610 align:middle line:84% By the way, I should say that statisticians have been 00:18:48.610 --> 00:18:52.500 align:middle line:84% debating fiercely for 100 years whether the right way to 00:18:52.500 --> 00:18:56.030 align:middle line:84% approach statistics is to go the classical way or the 00:18:56.030 --> 00:18:57.420 align:middle line:90% Bayesian way. 00:18:57.420 --> 00:19:00.530 align:middle line:84% And there have been tides going back and forth between 00:19:00.530 --> 00:19:02.260 align:middle line:90% the two sides. 00:19:02.260 --> 00:19:05.330 align:middle line:84% These days, Bayesian methods tend to become a little more 00:19:05.330 --> 00:19:07.320 align:middle line:90% popular for various reasons. 00:19:07.320 --> 00:19:11.730 align:middle line:84% We're going to come back to this later. 00:19:11.730 --> 00:19:14.610 align:middle line:84% All right, so in Bayesian estimation, what we got in our 00:19:14.610 --> 00:19:16.610 align:middle line:90% hands is Bayes rule. 00:19:16.610 --> 00:19:19.380 align:middle line:84% And if you have Bayes rule, there's not a lot 00:19:19.380 --> 00:19:21.340 align:middle line:90% that's left to do. 00:19:21.340 --> 00:19:24.190 align:middle line:84% We have different forms of the Bayes rule, depending on 00:19:24.190 --> 00:19:27.920 align:middle line:84% whether we're dealing with discrete data, And discrete 00:19:27.920 --> 00:19:32.310 align:middle line:84% quantities to estimate, or continuous data, and so on. 00:19:32.310 --> 00:19:36.020 align:middle line:84% In the hypothesis testing problem, the unknown quantity 00:19:36.020 --> 00:19:38.210 align:middle line:90% Theta is discrete. 00:19:38.210 --> 00:19:42.890 align:middle line:84% So in both cases here, we have a P of Theta. 00:19:42.890 --> 00:19:45.530 align:middle line:90% We obtain data, the X's. 00:19:45.530 --> 00:19:49.040 align:middle line:84% And on the basis of the X that we observe, we can calculate 00:19:49.040 --> 00:19:53.340 align:middle line:84% the posterior distribution of Theta, given the data. 00:19:53.340 --> 00:19:59.840 align:middle line:84% So to use Bayesian inference, what do we start with? 00:19:59.840 --> 00:20:03.160 align:middle line:90% We start with some priors. 00:20:03.160 --> 00:20:05.910 align:middle line:84% These are our initial beliefs about what 00:20:05.910 --> 00:20:07.890 align:middle line:90% Theta that might be. 00:20:07.890 --> 00:20:10.440 align:middle line:84% That's before we do the experiment. 00:20:10.440 --> 00:20:13.840 align:middle line:84% We have a model of the experimental aparatus. 00:20:13.840 --> 00:20:17.520 align:middle line:90% 00:20:17.520 --> 00:20:21.550 align:middle line:84% And the model of the experimental apparatus tells 00:20:21.550 --> 00:20:28.040 align:middle line:84% us if this Theta is true, I'm going to see X's of that kind. 00:20:28.040 --> 00:20:31.480 align:middle line:84% If that other Theta is true, I'm going to see X's that they 00:20:31.480 --> 00:20:33.130 align:middle line:90% are somewhere else. 00:20:33.130 --> 00:20:35.200 align:middle line:90% That models my apparatus. 00:20:35.200 --> 00:20:39.150 align:middle line:84% And based on that knowledge, once I observe I have these 00:20:39.150 --> 00:20:41.975 align:middle line:84% two functions in my hands, we have already seen that if you 00:20:41.975 --> 00:20:44.760 align:middle line:84% know those two functions, you can also calculate the 00:20:44.760 --> 00:20:46.550 align:middle line:90% denominator here. 00:20:46.550 --> 00:20:50.900 align:middle line:84% So all of these functions are available, so you can compute, 00:20:50.900 --> 00:20:54.170 align:middle line:84% you can find a formula for this function as well. 00:20:54.170 --> 00:20:58.780 align:middle line:84% And as soon as you observe the data, that X's, you plug in 00:20:58.780 --> 00:21:02.220 align:middle line:84% here the numerical value of those X's. 00:21:02.220 --> 00:21:04.720 align:middle line:84% And you get a function of Theta. 00:21:04.720 --> 00:21:07.870 align:middle line:84% And this is the posterior distribution of Theta, given 00:21:07.870 --> 00:21:09.680 align:middle line:90% the data that you have seen. 00:21:09.680 --> 00:21:11.930 align:middle line:84% So you've already done a fair number of 00:21:11.930 --> 00:21:13.760 align:middle line:90% exercises of these kind. 00:21:13.760 --> 00:21:17.320 align:middle line:90% So we not say more about this. 00:21:17.320 --> 00:21:20.470 align:middle line:84% And there's a similar formula as you know for the case where 00:21:20.470 --> 00:21:22.460 align:middle line:90% we have continuous data. 00:21:22.460 --> 00:21:25.140 align:middle line:84% If the X's are continuous random variable, then the 00:21:25.140 --> 00:21:28.620 align:middle line:84% formula is the same, except that X's are described by 00:21:28.620 --> 00:21:31.630 align:middle line:84% densities instead of being described by a probability 00:21:31.630 --> 00:21:32.880 align:middle line:90% mass functions. 00:21:32.880 --> 00:21:35.170 align:middle line:90% 00:21:35.170 --> 00:21:40.200 align:middle line:84% OK, now if Theta is continuous, then we're dealing 00:21:40.200 --> 00:21:42.160 align:middle line:90% with estimation problems. 00:21:42.160 --> 00:21:44.880 align:middle line:84% But the story is once more the same. 00:21:44.880 --> 00:21:47.920 align:middle line:84% You're going to use the Bayes rule to come up with the 00:21:47.920 --> 00:21:51.090 align:middle line:84% posterior density of Theta, given the data 00:21:51.090 --> 00:21:53.300 align:middle line:90% that you have observed. 00:21:53.300 --> 00:21:57.250 align:middle line:84% Now just for the sake of the example, let's come back to 00:21:57.250 --> 00:21:58.900 align:middle line:90% this picture here. 00:21:58.900 --> 00:22:03.490 align:middle line:84% Suppose that something is flying in the air, and maybe 00:22:03.490 --> 00:22:07.800 align:middle line:84% this is just an object in the air close to the Earth. 00:22:07.800 --> 00:22:10.820 align:middle line:84% So because of gravity, the trajectory that it's going to 00:22:10.820 --> 00:22:15.170 align:middle line:84% follow it's going to be a parabola. 00:22:15.170 --> 00:22:18.014 align:middle line:84% So this is the general equation of a parabola. 00:22:18.014 --> 00:22:23.450 align:middle line:84% Zt is the position of my objects at time t. 00:22:23.450 --> 00:22:26.310 align:middle line:90% 00:22:26.310 --> 00:22:29.500 align:middle line:84% But I don't know exactly which parabola it is. 00:22:29.500 --> 00:22:32.690 align:middle line:84% So the parameters of the parabola are unknown 00:22:32.690 --> 00:22:34.040 align:middle line:90% quantities. 00:22:34.040 --> 00:22:37.710 align:middle line:84% What I can do is to go and measure the position of my 00:22:37.710 --> 00:22:41.880 align:middle line:90% objects at different times. 00:22:41.880 --> 00:22:44.575 align:middle line:84% But unfortunately, my measurements are noisy. 00:22:44.575 --> 00:22:47.380 align:middle line:90% 00:22:47.380 --> 00:22:51.070 align:middle line:84% What I want to do is to model the motion of my object. 00:22:51.070 --> 00:22:56.260 align:middle line:84% So I guess in the picture, the axis would be t going this way 00:22:56.260 --> 00:22:59.980 align:middle line:90% and Z going this way. 00:22:59.980 --> 00:23:02.470 align:middle line:84% And on the basis of the data that they get, 00:23:02.470 --> 00:23:05.020 align:middle line:90% these are my X's. 00:23:05.020 --> 00:23:07.390 align:middle line:84% I want to figure out the Thetas. 00:23:07.390 --> 00:23:09.570 align:middle line:84% That is, I want to figure out the exact 00:23:09.570 --> 00:23:11.840 align:middle line:90% equation of this parabola. 00:23:11.840 --> 00:23:14.940 align:middle line:84% Now if somebody gives you probability distributions for 00:23:14.940 --> 00:23:18.490 align:middle line:84% Theta, these would be your priors. 00:23:18.490 --> 00:23:19.840 align:middle line:90% So this is given. 00:23:19.840 --> 00:23:23.200 align:middle line:90% 00:23:23.200 --> 00:23:26.200 align:middle line:84% We need the conditional distribution of the X's given 00:23:26.200 --> 00:23:27.360 align:middle line:90% the Thetas. 00:23:27.360 --> 00:23:30.870 align:middle line:84% Well, we have the conditional distribution of Z, given the 00:23:30.870 --> 00:23:32.920 align:middle line:90% Thetas from this equation. 00:23:32.920 --> 00:23:36.040 align:middle line:84% And then by playing with this equation, you can also find 00:23:36.040 --> 00:23:42.460 align:middle line:84% how is X distributed if Theta takes a particular value. 00:23:42.460 --> 00:23:46.420 align:middle line:84% So you do have all of the densities that you might need. 00:23:46.420 --> 00:23:48.790 align:middle line:84% And you can apply the Bayes rule. 00:23:48.790 --> 00:23:53.620 align:middle line:84% And at the end, your end result would be a formula for 00:23:53.620 --> 00:23:57.270 align:middle line:84% the distribution of Theta, given to the X 00:23:57.270 --> 00:23:59.130 align:middle line:90% that you have observed-- 00:23:59.130 --> 00:24:03.000 align:middle line:84% except for one sort of computation, or to make things 00:24:03.000 --> 00:24:04.470 align:middle line:90% more interesting. 00:24:04.470 --> 00:24:07.680 align:middle line:84% Instead of these X's and Theta's being single random 00:24:07.680 --> 00:24:11.070 align:middle line:84% variables that we have here, typically those X's and 00:24:11.070 --> 00:24:13.400 align:middle line:84% Theta's will be multi-dimensional random 00:24:13.400 --> 00:24:16.490 align:middle line:84% variables or will correspond to multiple ones. 00:24:16.490 --> 00:24:19.920 align:middle line:84% So this little Theta here actually stands for a triplet 00:24:19.920 --> 00:24:22.880 align:middle line:90% of Theta0, Theta1, and Theta2. 00:24:22.880 --> 00:24:26.820 align:middle line:84% And that X here stands here for the entire sequence of X's 00:24:26.820 --> 00:24:28.410 align:middle line:90% that we have observed. 00:24:28.410 --> 00:24:31.060 align:middle line:84% So in reality, the object that you're going to get at to the 00:24:31.060 --> 00:24:35.900 align:middle line:84% end after inference is done is a function that you plug in 00:24:35.900 --> 00:24:39.430 align:middle line:84% the values of the data and you get the function of the 00:24:39.430 --> 00:24:43.240 align:middle line:84% Theta's that tells you the relative likelihoods of 00:24:43.240 --> 00:24:46.780 align:middle line:90% different Theta triplets. 00:24:46.780 --> 00:24:49.760 align:middle line:84% So what I'm saying is that this is no harder than the 00:24:49.760 --> 00:24:53.720 align:middle line:84% problems that you have dealt with so far, except perhaps 00:24:53.720 --> 00:24:56.020 align:middle line:84% for the complication that's usually in interesting 00:24:56.020 --> 00:24:57.490 align:middle line:90% inference problems. 00:24:57.490 --> 00:25:01.940 align:middle line:84% Your Theta's and X's are often the vectors of random 00:25:01.940 --> 00:25:05.490 align:middle line:84% variables instead of individual random variables. 00:25:05.490 --> 00:25:09.630 align:middle line:84% Now if you are to do estimation in a case where you 00:25:09.630 --> 00:25:13.520 align:middle line:84% have discrete data, again the situation is no different. 00:25:13.520 --> 00:25:17.020 align:middle line:84% We still have a Bayes rule of the same kind, except that 00:25:17.020 --> 00:25:19.540 align:middle line:84% densities gets replaced by PMF's. 00:25:19.540 --> 00:25:23.680 align:middle line:84% If X is discrete, you put a P here instead of putting an f. 00:25:23.680 --> 00:25:27.990 align:middle line:84% So an example of an estimation problem with discrete data is 00:25:27.990 --> 00:25:29.740 align:middle line:84% similar to the polling problem. 00:25:29.740 --> 00:25:31.600 align:middle line:90% You have a coin. 00:25:31.600 --> 00:25:33.500 align:middle line:84% It has an unknown parameter Theta. 00:25:33.500 --> 00:25:35.230 align:middle line:84% This is the probability of obtaining heads. 00:25:35.230 --> 00:25:37.410 align:middle line:90% You flip the coin many times. 00:25:37.410 --> 00:25:41.560 align:middle line:84% What can you tell me about the true value of Theta? 00:25:41.560 --> 00:25:46.200 align:middle line:84% A classical statistician, at this point, would say, OK, I'm 00:25:46.200 --> 00:25:48.900 align:middle line:84% going to use an estimator, the most reasonable 00:25:48.900 --> 00:25:50.950 align:middle line:90% one, which is this. 00:25:50.950 --> 00:25:54.200 align:middle line:84% How many heads did they obtain in n trials? 00:25:54.200 --> 00:25:56.440 align:middle line:84% Divide by the total number of trials. 00:25:56.440 --> 00:26:00.700 align:middle line:84% This is my estimate of the bias of my coin. 00:26:00.700 --> 00:26:02.860 align:middle line:84% And then the classical statistician would continue 00:26:02.860 --> 00:26:07.610 align:middle line:84% from here and try to prove some properties and argue that 00:26:07.610 --> 00:26:10.030 align:middle line:90% this estimate is a good one. 00:26:10.030 --> 00:26:12.850 align:middle line:84% For example, we have the weak law of large numbers that 00:26:12.850 --> 00:26:15.630 align:middle line:84% tells us that this particular estimate converges in 00:26:15.630 --> 00:26:17.990 align:middle line:84% probability to the true parameter. 00:26:17.990 --> 00:26:21.000 align:middle line:84% This is a kind of guarantee that's useful to have. 00:26:21.000 --> 00:26:23.410 align:middle line:84% And the classical statistician would pretty much close the 00:26:23.410 --> 00:26:24.660 align:middle line:90% subject in this way. 00:26:24.660 --> 00:26:27.340 align:middle line:90% 00:26:27.340 --> 00:26:30.160 align:middle line:84% What would the Bayesian person do differently? 00:26:30.160 --> 00:26:35.040 align:middle line:84% The Bayesian person would start by assuming a prior 00:26:35.040 --> 00:26:37.100 align:middle line:90% distribution of Theta. 00:26:37.100 --> 00:26:39.820 align:middle line:84% Instead of treating Theta as an unknown constant, they 00:26:39.820 --> 00:26:44.340 align:middle line:84% would say that Theta would speak randomly or pretend that 00:26:44.340 --> 00:26:47.360 align:middle line:84% it would speak randomly and assume a 00:26:47.360 --> 00:26:49.300 align:middle line:90% distribution on Theta. 00:26:49.300 --> 00:26:54.290 align:middle line:84% So for example, if you don't know they need anything more, 00:26:54.290 --> 00:26:57.510 align:middle line:84% you might assume that any value for the bias of the coin 00:26:57.510 --> 00:27:01.460 align:middle line:84% is as likely as any other value of the bias of the coin. 00:27:01.460 --> 00:27:04.150 align:middle line:84% And this way so the probability distribution 00:27:04.150 --> 00:27:05.720 align:middle line:90% that's uniform. 00:27:05.720 --> 00:27:09.840 align:middle line:84% Or if you have a little more faith in the manufacturing 00:27:09.840 --> 00:27:13.270 align:middle line:84% processes that's created that coin, you might choose your 00:27:13.270 --> 00:27:17.660 align:middle line:84% prior to be a distribution that's centered around 1/2 and 00:27:17.660 --> 00:27:21.860 align:middle line:84% sits fairly narrowly centered around 1/2. 00:27:21.860 --> 00:27:24.500 align:middle line:84% That would be a prior distribution in which you say, 00:27:24.500 --> 00:27:27.920 align:middle line:84% well, I believe that the manufacturer tried to make my 00:27:27.920 --> 00:27:29.410 align:middle line:90% coin to be fair. 00:27:29.410 --> 00:27:33.070 align:middle line:84% But they often makes some mistakes, so it's going to be, 00:27:33.070 --> 00:27:36.600 align:middle line:84% I believe, it's approximately 1/2 but not quite. 00:27:36.600 --> 00:27:40.050 align:middle line:84% So depending on your beliefs, you would choose an 00:27:40.050 --> 00:27:43.630 align:middle line:84% appropriate prior for the distribution of Theta. 00:27:43.630 --> 00:27:48.610 align:middle line:84% And then you would use the Bayes rule to find the 00:27:48.610 --> 00:27:52.270 align:middle line:84% probabilities of different values of Theta, based on the 00:27:52.270 --> 00:27:53.520 align:middle line:90% data that you have observed. 00:27:53.520 --> 00:27:59.620 align:middle line:90% 00:27:59.620 --> 00:28:04.640 align:middle line:84% So no matter which version of the Bayes rule that you use, 00:28:04.640 --> 00:28:10.540 align:middle line:84% the end product of the Bayes rule is going to be either a 00:28:10.540 --> 00:28:14.400 align:middle line:84% plot of this kind or a plot of that kind. 00:28:14.400 --> 00:28:16.740 align:middle line:90% So what am I plotting here? 00:28:16.740 --> 00:28:19.810 align:middle line:90% This axis is the Theta axis. 00:28:19.810 --> 00:28:23.830 align:middle line:84% These are the possible values of the unknown quantity that 00:28:23.830 --> 00:28:26.670 align:middle line:90% we're trying to estimate. 00:28:26.670 --> 00:28:28.990 align:middle line:84% In the continuous case, theta is a 00:28:28.990 --> 00:28:30.800 align:middle line:90% continuous random variable. 00:28:30.800 --> 00:28:32.560 align:middle line:90% I obtain my data. 00:28:32.560 --> 00:28:36.430 align:middle line:84% And I plot for the posterior probability distribution after 00:28:36.430 --> 00:28:37.940 align:middle line:90% observing my data. 00:28:37.940 --> 00:28:42.220 align:middle line:84% And I'm plotting here the probability density for Theta. 00:28:42.220 --> 00:28:45.500 align:middle line:84% So this is a plot of that density. 00:28:45.500 --> 00:28:49.210 align:middle line:84% In the discrete case, theta can take finitely many values 00:28:49.210 --> 00:28:51.570 align:middle line:90% or a discrete set of values. 00:28:51.570 --> 00:28:54.470 align:middle line:84% And for each one of those values, I'm telling you how 00:28:54.470 --> 00:28:58.080 align:middle line:84% likely is that the value to be the correct one, given the 00:28:58.080 --> 00:29:01.040 align:middle line:90% data that I have observed. 00:29:01.040 --> 00:29:04.990 align:middle line:84% And in general, what you would go back to your boss and 00:29:04.990 --> 00:29:08.520 align:middle line:84% report after you've done all your inference work would be 00:29:08.520 --> 00:29:10.870 align:middle line:84% either a plot of this kinds or of that kind. 00:29:10.870 --> 00:29:14.180 align:middle line:84% So you go to your boss who asks you, what is 00:29:14.180 --> 00:29:15.190 align:middle line:90% the value of Theta? 00:29:15.190 --> 00:29:17.490 align:middle line:84% And you say, well, I only have limited data. 00:29:17.490 --> 00:29:19.420 align:middle line:90% That I don't know what it is. 00:29:19.420 --> 00:29:22.920 align:middle line:84% It could be this, with so much probability. 00:29:22.920 --> 00:29:24.640 align:middle line:90% There's probability. 00:29:24.640 --> 00:29:27.220 align:middle line:84% OK, let's throw in some numbers here. 00:29:27.220 --> 00:29:32.250 align:middle line:84% There's probability 0.3 that Theta is this value. 00:29:32.250 --> 00:29:36.100 align:middle line:84% There's probability 0.2 that Theta is this value, 0.1 that 00:29:36.100 --> 00:29:39.420 align:middle line:84% it's this one, 0.1 that it's this one, 0.2 that it's that 00:29:39.420 --> 00:29:40.830 align:middle line:90% one, and so on. 00:29:40.830 --> 00:29:44.890 align:middle line:84% OK, now bosses often want simple answers. 00:29:44.890 --> 00:29:48.480 align:middle line:84% They say, OK, you're talking too much. 00:29:48.480 --> 00:29:51.770 align:middle line:90% What do you think Theta is? 00:29:51.770 --> 00:29:55.920 align:middle line:84% And now you're forced to make a decision. 00:29:55.920 --> 00:30:00.680 align:middle line:84% If that was the situation and you have to make a decision, 00:30:00.680 --> 00:30:02.370 align:middle line:90% how would you make it? 00:30:02.370 --> 00:30:06.880 align:middle line:84% Well, I'm going to make a decision that's most likely to 00:30:06.880 --> 00:30:09.120 align:middle line:90% be correct. 00:30:09.120 --> 00:30:13.060 align:middle line:84% If I make this decision, what's going to happen? 00:30:13.060 --> 00:30:17.670 align:middle line:84% Theta is this value with probability 0.2, which means 00:30:17.670 --> 00:30:21.150 align:middle line:84% there's probably 0.8 that they make an error 00:30:21.150 --> 00:30:23.280 align:middle line:90% if I make that guess. 00:30:23.280 --> 00:30:29.370 align:middle line:84% If I make that decision, this decision has probably 0.3 of 00:30:29.370 --> 00:30:30.750 align:middle line:90% being the correct one. 00:30:30.750 --> 00:30:34.530 align:middle line:84% So I have probably of error 0.7. 00:30:34.530 --> 00:30:38.460 align:middle line:84% So if you want to just maximize the probability of 00:30:38.460 --> 00:30:41.730 align:middle line:84% giving the correct decision, or if you want to minimize the 00:30:41.730 --> 00:30:44.780 align:middle line:84% probability of making an incorrect decision, what 00:30:44.780 --> 00:30:48.790 align:middle line:84% you're going to choose to report is that value of Theta 00:30:48.790 --> 00:30:51.450 align:middle line:84% for which the probability is highest. 00:30:51.450 --> 00:30:54.230 align:middle line:84% So in this case, I would choose to report this 00:30:54.230 --> 00:30:58.210 align:middle line:84% particular value, the most likely value of Theta, given 00:30:58.210 --> 00:31:00.120 align:middle line:90% what I have observed. 00:31:00.120 --> 00:31:04.640 align:middle line:84% And that value is called them maximum a posteriori 00:31:04.640 --> 00:31:07.550 align:middle line:90% probability estimate. 00:31:07.550 --> 00:31:11.550 align:middle line:84% It's going to be this one in our case. 00:31:11.550 --> 00:31:16.830 align:middle line:84% So picking the point in the posterior PMF that has the 00:31:16.830 --> 00:31:19.040 align:middle line:90% highest probability. 00:31:19.040 --> 00:31:20.720 align:middle line:84% That's the reasonable thing to do. 00:31:20.720 --> 00:31:23.850 align:middle line:84% This is the optimal thing to do if you want to minimize the 00:31:23.850 --> 00:31:27.340 align:middle line:84% probability of an incorrect inference. 00:31:27.340 --> 00:31:31.400 align:middle line:84% And that's what people do usually if they need to report 00:31:31.400 --> 00:31:35.280 align:middle line:84% a single answer, if they need to report a single decision. 00:31:35.280 --> 00:31:39.530 align:middle line:84% How about in the estimation context? 00:31:39.530 --> 00:31:43.250 align:middle line:84% If that's what you know about Theta, Theta could be around 00:31:43.250 --> 00:31:46.670 align:middle line:84% here, but there's also some sharp probability that it is 00:31:46.670 --> 00:31:48.720 align:middle line:90% around here. 00:31:48.720 --> 00:31:52.380 align:middle line:84% What's the single answer that you would give to your boss? 00:31:52.380 --> 00:31:56.310 align:middle line:84% One option is to use the same philosophy and say, OK, I'm 00:31:56.310 --> 00:32:00.135 align:middle line:84% going to find the Theta at which this posterior density 00:32:00.135 --> 00:32:01.690 align:middle line:90% is highest. 00:32:01.690 --> 00:32:06.010 align:middle line:84% So I would pick this point here and report this 00:32:06.010 --> 00:32:06.920 align:middle line:90% particular Theta. 00:32:06.920 --> 00:32:11.110 align:middle line:84% So this would be my Theta, again, Theta MAP, the Theta 00:32:11.110 --> 00:32:15.290 align:middle line:84% that has the highest a posteriori probability, just 00:32:15.290 --> 00:32:19.100 align:middle line:84% because it corresponds to the peak of the density. 00:32:19.100 --> 00:32:23.810 align:middle line:84% But in this context, the maximum a posteriori 00:32:23.810 --> 00:32:27.120 align:middle line:84% probability theta was the one that was most 00:32:27.120 --> 00:32:28.600 align:middle line:90% likely to be true. 00:32:28.600 --> 00:32:32.460 align:middle line:84% In the continuous case, you cannot really say that this is 00:32:32.460 --> 00:32:34.940 align:middle line:84% the most likely value of Theta. 00:32:34.940 --> 00:32:38.340 align:middle line:84% In a continuous setting, any value of Theta has zero 00:32:38.340 --> 00:32:41.530 align:middle line:84% probability, so when we talk about densities. 00:32:41.530 --> 00:32:43.260 align:middle line:90% So it's not the most likely. 00:32:43.260 --> 00:32:48.240 align:middle line:84% It's the one for which the density, so the probabilities 00:32:48.240 --> 00:32:51.820 align:middle line:84% of that neighborhoods, are highest. 00:32:51.820 --> 00:32:56.390 align:middle line:84% So the rationale for picking this particular estimate in 00:32:56.390 --> 00:33:00.050 align:middle line:84% the continuous case is much less compelling than the 00:33:00.050 --> 00:33:02.210 align:middle line:90% rationale that we had in here. 00:33:02.210 --> 00:33:05.590 align:middle line:84% So in this case, reasonable people might choose different 00:33:05.590 --> 00:33:07.460 align:middle line:90% quantities to report. 00:33:07.460 --> 00:33:11.810 align:middle line:84% And the very popular one would be to report instead the 00:33:11.810 --> 00:33:13.700 align:middle line:90% conditional expectation. 00:33:13.700 --> 00:33:15.990 align:middle line:84% So I don't know quite what Theta is. 00:33:15.990 --> 00:33:19.600 align:middle line:84% Given the data that I have, Theta has this distribution. 00:33:19.600 --> 00:33:23.320 align:middle line:84% Let me just report the average over that distribution. 00:33:23.320 --> 00:33:27.090 align:middle line:84% Let me report to the center of gravity of this figure. 00:33:27.090 --> 00:33:30.340 align:middle line:84% And in this figure, the center of gravity would probably be 00:33:30.340 --> 00:33:32.230 align:middle line:90% somewhere around here. 00:33:32.230 --> 00:33:35.690 align:middle line:84% And that would be a different estimate that you 00:33:35.690 --> 00:33:37.520 align:middle line:90% might choose to report. 00:33:37.520 --> 00:33:40.340 align:middle line:84% So center of gravity is something around here. 00:33:40.340 --> 00:33:43.580 align:middle line:84% And this is a conditional expectation of Theta, given 00:33:43.580 --> 00:33:46.010 align:middle line:90% the data that you have. 00:33:46.010 --> 00:33:51.190 align:middle line:84% So these are two, in some sense, fairly reasonable ways 00:33:51.190 --> 00:33:53.850 align:middle line:84% of choosing what to report to your boss. 00:33:53.850 --> 00:33:55.690 align:middle line:84% Some people might choose to report this. 00:33:55.690 --> 00:33:58.630 align:middle line:84% Some people might choose to report that. 00:33:58.630 --> 00:34:03.230 align:middle line:84% And a priori, if there's no compelling reason why one 00:34:03.230 --> 00:34:08.639 align:middle line:84% would be preferable than other one, unless you set some rules 00:34:08.639 --> 00:34:12.350 align:middle line:84% for the game and you describe a little more precisely what 00:34:12.350 --> 00:34:14.090 align:middle line:90% your objectives are. 00:34:14.090 --> 00:34:19.070 align:middle line:84% But no matter which one you report, a single answer, a 00:34:19.070 --> 00:34:24.350 align:middle line:84% point estimate, doesn't really tell you the whole story. 00:34:24.350 --> 00:34:28.159 align:middle line:84% There's a lot more information conveyed by this posterior 00:34:28.159 --> 00:34:31.060 align:middle line:84% distribution plot than any single number 00:34:31.060 --> 00:34:32.159 align:middle line:90% that you might report. 00:34:32.159 --> 00:34:36.510 align:middle line:84% So in general, you may wish to convince your boss that's it's 00:34:36.510 --> 00:34:40.310 align:middle line:84% worth their time to look at the entire plot, because that 00:34:40.310 --> 00:34:43.100 align:middle line:84% plot sort of covers all the possibilities. 00:34:43.100 --> 00:34:47.060 align:middle line:84% It tells your boss most likely we're in that range, but 00:34:47.060 --> 00:34:51.620 align:middle line:84% there's also a distinct change that our Theta happens to lie 00:34:51.620 --> 00:34:54.080 align:middle line:90% in that range. 00:34:54.080 --> 00:34:58.400 align:middle line:84% All right, now let us try to perhaps differentiate between 00:34:58.400 --> 00:35:02.570 align:middle line:84% these two and see under what circumstances this one might 00:35:02.570 --> 00:35:05.530 align:middle line:84% be the better estimate to perform. 00:35:05.530 --> 00:35:07.320 align:middle line:90% Better with respect to what? 00:35:07.320 --> 00:35:08.830 align:middle line:90% We need some rules. 00:35:08.830 --> 00:35:10.730 align:middle line:84% So we're going to throw in some rules. 00:35:10.730 --> 00:35:14.320 align:middle line:90% 00:35:14.320 --> 00:35:17.450 align:middle line:84% As a warm up, we're going to deal with the problem of 00:35:17.450 --> 00:35:22.000 align:middle line:84% making an estimation if you had no information at all, 00:35:22.000 --> 00:35:24.670 align:middle line:84% except for a prior distribution. 00:35:24.670 --> 00:35:27.650 align:middle line:84% So this is a warm up for what's coming next, which 00:35:27.650 --> 00:35:32.970 align:middle line:84% would be estimation that takes into account some information. 00:35:32.970 --> 00:35:34.860 align:middle line:90% So we have a Theta. 00:35:34.860 --> 00:35:38.500 align:middle line:84% And because of your subjective beliefs or models by others, 00:35:38.500 --> 00:35:41.780 align:middle line:84% you believe that Theta is uniformly distributed between, 00:35:41.780 --> 00:35:46.250 align:middle line:90% let's say, 4 and 10. 00:35:46.250 --> 00:35:48.120 align:middle line:84% You want to come up with a point estimate. 00:35:48.120 --> 00:35:51.770 align:middle line:90% 00:35:51.770 --> 00:35:54.900 align:middle line:84% Let's try to look for an estimate. 00:35:54.900 --> 00:35:57.580 align:middle line:90% Call it c, in this case. 00:35:57.580 --> 00:36:00.090 align:middle line:84% I want to pick a number with which to estimate 00:36:00.090 --> 00:36:01.340 align:middle line:90% the value of Theta. 00:36:01.340 --> 00:36:04.030 align:middle line:90% 00:36:04.030 --> 00:36:08.260 align:middle line:84% I will be interested in the size of the error that I make. 00:36:08.260 --> 00:36:12.310 align:middle line:84% And I really dislike large errors, so I'm going to focus 00:36:12.310 --> 00:36:15.500 align:middle line:84% on the square of the error that they make. 00:36:15.500 --> 00:36:19.140 align:middle line:90% So I pick c. 00:36:19.140 --> 00:36:21.340 align:middle line:84% Theta that has a random value that I don't know. 00:36:21.340 --> 00:36:25.900 align:middle line:84% But whatever it is, once it becomes known, it results into 00:36:25.900 --> 00:36:28.640 align:middle line:84% a squared error between what it is and what I 00:36:28.640 --> 00:36:30.660 align:middle line:90% guessed that it was. 00:36:30.660 --> 00:36:35.770 align:middle line:84% And I'm interested in making a small air on the average, 00:36:35.770 --> 00:36:38.170 align:middle line:84% where the average is taken with respect to all the 00:36:38.170 --> 00:36:42.350 align:middle line:84% possible and unknown values of Theta. 00:36:42.350 --> 00:36:47.220 align:middle line:84% So the problem, this is a least squares formulation of 00:36:47.220 --> 00:36:49.240 align:middle line:84% the problem, where we try to minimize the 00:36:49.240 --> 00:36:51.150 align:middle line:90% least squares errors. 00:36:51.150 --> 00:36:53.900 align:middle line:90% How do you find the optimal c? 00:36:53.900 --> 00:36:57.200 align:middle line:84% Well, we take that expression and expand it. 00:36:57.200 --> 00:37:00.930 align:middle line:90% 00:37:00.930 --> 00:37:05.650 align:middle line:84% And it is, using linearity of expectations-- 00:37:05.650 --> 00:37:11.460 align:middle line:84% square minus 2c expected Theta plus c squared-- 00:37:11.460 --> 00:37:13.620 align:middle line:84% that's the quantity that we want to minimize, 00:37:13.620 --> 00:37:16.670 align:middle line:90% with respect to c. 00:37:16.670 --> 00:37:19.670 align:middle line:84% To do the minimization, take the derivative with respect to 00:37:19.670 --> 00:37:21.950 align:middle line:90% c and set it to 0. 00:37:21.950 --> 00:37:27.320 align:middle line:84% So that differentiation gives us from here minus 2 expected 00:37:27.320 --> 00:37:32.420 align:middle line:84% value of Theta plus 2c is equal to 0. 00:37:32.420 --> 00:37:36.550 align:middle line:84% And the answer that you get by solving this equation is that 00:37:36.550 --> 00:37:39.350 align:middle line:84% c is the expected value of Theta. 00:37:39.350 --> 00:37:42.860 align:middle line:84% So when you do this optimization, you find that 00:37:42.860 --> 00:37:45.170 align:middle line:84% the optimal estimate, the things you should be 00:37:45.170 --> 00:37:47.970 align:middle line:84% reporting, is the expected value of Theta. 00:37:47.970 --> 00:37:51.630 align:middle line:84% So in this particular example, you would choose your estimate 00:37:51.630 --> 00:37:55.500 align:middle line:84% c to be just the middle of these values, 00:37:55.500 --> 00:37:57.980 align:middle line:90% which would be 7. 00:37:57.980 --> 00:38:02.642 align:middle line:90% 00:38:02.642 --> 00:38:06.640 align:middle line:84% OK, and in case your boss asks you, how 00:38:06.640 --> 00:38:08.610 align:middle line:90% good is your estimate? 00:38:08.610 --> 00:38:11.390 align:middle line:84% How big is your error going to be? 00:38:11.390 --> 00:38:14.910 align:middle line:90% 00:38:14.910 --> 00:38:19.870 align:middle line:84% What you could report is the average size of the estimation 00:38:19.870 --> 00:38:22.570 align:middle line:90% error that you are making. 00:38:22.570 --> 00:38:26.760 align:middle line:84% We picked our estimates to be the expected value of Theta. 00:38:26.760 --> 00:38:29.450 align:middle line:84% So for this particular way that I'm choosing to do my 00:38:29.450 --> 00:38:33.610 align:middle line:84% estimation, this is the mean squared error that I get. 00:38:33.610 --> 00:38:35.330 align:middle line:84% And this is a familiar quantity. 00:38:35.330 --> 00:38:38.370 align:middle line:84% It's just the variance of the distribution. 00:38:38.370 --> 00:38:41.890 align:middle line:84% So the expectation is that best way to estimate a 00:38:41.890 --> 00:38:45.550 align:middle line:84% quantity, if you're interested in the mean squared error. 00:38:45.550 --> 00:38:50.430 align:middle line:84% And the resulting mean squared error is the variance itself. 00:38:50.430 --> 00:38:56.380 align:middle line:84% How will this story change if we now have data as well? 00:38:56.380 --> 00:39:01.290 align:middle line:84% Now having data means that we can compute posterior 00:39:01.290 --> 00:39:05.150 align:middle line:84% distributions or conditional distributions. 00:39:05.150 --> 00:39:10.400 align:middle line:84% So we get transported into a new universe where instead the 00:39:10.400 --> 00:39:14.740 align:middle line:84% working with the original distribution of Theta, the 00:39:14.740 --> 00:39:18.860 align:middle line:84% prior distribution, now we work with the condition of 00:39:18.860 --> 00:39:22.280 align:middle line:84% distribution of Theta, given the data 00:39:22.280 --> 00:39:24.860 align:middle line:90% that we have observed. 00:39:24.860 --> 00:39:30.430 align:middle line:84% Now remember our old slogan that conditional models and 00:39:30.430 --> 00:39:33.570 align:middle line:84% conditional probabilities are no different than ordinary 00:39:33.570 --> 00:39:38.880 align:middle line:84% probabilities, except that we live now in a new universe 00:39:38.880 --> 00:39:42.690 align:middle line:84% where the new information has been taken into account. 00:39:42.690 --> 00:39:47.860 align:middle line:84% So if you use that philosophy and you're asked to minimize 00:39:47.860 --> 00:39:53.310 align:middle line:84% the squared error but now that you live in a new universe 00:39:53.310 --> 00:39:56.910 align:middle line:84% where X has been fixed to something, what would the 00:39:56.910 --> 00:39:59.210 align:middle line:90% optimal solution be? 00:39:59.210 --> 00:40:03.540 align:middle line:84% It would again be the expectation of theta, but 00:40:03.540 --> 00:40:04.730 align:middle line:90% which expectation? 00:40:04.730 --> 00:40:08.910 align:middle line:84% It's the expectation which applies in the new conditional 00:40:08.910 --> 00:40:12.350 align:middle line:84% universe in which we live right now. 00:40:12.350 --> 00:40:16.330 align:middle line:84% So because of what we did before, by the same 00:40:16.330 --> 00:40:20.330 align:middle line:84% calculation, we would find that the optimal estimates is 00:40:20.330 --> 00:40:24.970 align:middle line:84% the expected value of X of Theta, but the optimal 00:40:24.970 --> 00:40:26.730 align:middle line:84% estimate that takes into account the 00:40:26.730 --> 00:40:29.170 align:middle line:90% information that we have. 00:40:29.170 --> 00:40:33.600 align:middle line:84% So the conclusion, once you get your data, if you want to 00:40:33.600 --> 00:40:40.480 align:middle line:84% minimize the mean squared error, you should just report 00:40:40.480 --> 00:40:43.870 align:middle line:84% the conditional estimation of this unknown quantity based on 00:40:43.870 --> 00:40:46.640 align:middle line:90% the data that you have. 00:40:46.640 --> 00:40:53.050 align:middle line:84% So the picture here is that Theta is unknown. 00:40:53.050 --> 00:41:00.710 align:middle line:84% You have your apparatus that creates measurements. 00:41:00.710 --> 00:41:07.880 align:middle line:84% So this creates an X. You take an X, and here you have a box 00:41:07.880 --> 00:41:10.203 align:middle line:90% that does calculations. 00:41:10.203 --> 00:41:13.490 align:middle line:90% 00:41:13.490 --> 00:41:18.180 align:middle line:84% It does calculations and it spits out the conditional 00:41:18.180 --> 00:41:22.230 align:middle line:84% expectation of Theta, given the particular data that you 00:41:22.230 --> 00:41:24.750 align:middle line:90% have observed. 00:41:24.750 --> 00:41:28.680 align:middle line:84% And what we have done in this class so far is, to some 00:41:28.680 --> 00:41:33.450 align:middle line:84% extent, developing the computational tools and skills 00:41:33.450 --> 00:41:36.020 align:middle line:84% to do with this particular calculation-- 00:41:36.020 --> 00:41:39.780 align:middle line:84% how to calculate the posterior density for Theta and how to 00:41:39.780 --> 00:41:42.750 align:middle line:84% calculate expectations, conditional expectations. 00:41:42.750 --> 00:41:45.330 align:middle line:84% So in principle, we know how to do this. 00:41:45.330 --> 00:41:50.040 align:middle line:84% In principle, we can program a computer to take the data and 00:41:50.040 --> 00:41:51.670 align:middle line:84% to spit out condition expectations. 00:41:51.670 --> 00:41:56.140 align:middle line:90% 00:41:56.140 --> 00:42:04.390 align:middle line:84% Somebody who doesn't think like us might instead design a 00:42:04.390 --> 00:42:09.940 align:middle line:84% calculating machine that does something differently and 00:42:09.940 --> 00:42:16.490 align:middle line:90% produces some other estimate. 00:42:16.490 --> 00:42:20.000 align:middle line:84% So we went through this argument and we decided to 00:42:20.000 --> 00:42:23.110 align:middle line:84% program our computer to calculate conditional 00:42:23.110 --> 00:42:24.490 align:middle line:90% expectations. 00:42:24.490 --> 00:42:28.460 align:middle line:84% Somebody else came up with some other crazy idea for how 00:42:28.460 --> 00:42:30.590 align:middle line:84% to estimate the random variable. 00:42:30.590 --> 00:42:34.460 align:middle line:84% They came up with some function g and the programmed 00:42:34.460 --> 00:42:38.700 align:middle line:84% it, and they designed a machine that estimates Theta's 00:42:38.700 --> 00:42:43.000 align:middle line:84% by outputting a certain g of X. 00:42:43.000 --> 00:42:47.690 align:middle line:84% That could be an alternative estimator. 00:42:47.690 --> 00:42:50.280 align:middle line:90% Which one is better? 00:42:50.280 --> 00:42:56.350 align:middle line:84% Well, we convinced ourselves that this is the optimal one 00:42:56.350 --> 00:42:59.780 align:middle line:84% in a universe where we have fixed the particular 00:42:59.780 --> 00:43:01.420 align:middle line:90% value of the data. 00:43:01.420 --> 00:43:06.030 align:middle line:84% So what we have proved so far is a relation of this kind. 00:43:06.030 --> 00:43:09.670 align:middle line:84% In this conditional universe, the mean squared 00:43:09.670 --> 00:43:11.920 align:middle line:90% error that I get-- 00:43:11.920 --> 00:43:15.170 align:middle line:84% I'm the one who's using this estimator-- 00:43:15.170 --> 00:43:18.850 align:middle line:84% is less than or equal than the mean squared error that this 00:43:18.850 --> 00:43:23.960 align:middle line:84% person will get, the person who uses that estimator. 00:43:23.960 --> 00:43:28.040 align:middle line:84% For any particular value of the data, I'm going to do 00:43:28.040 --> 00:43:30.190 align:middle line:90% better than the other person. 00:43:30.190 --> 00:43:32.760 align:middle line:84% Now the data themselves are random. 00:43:32.760 --> 00:43:38.050 align:middle line:84% If I average over all possible values of the data, I should 00:43:38.050 --> 00:43:40.240 align:middle line:90% still be better off. 00:43:40.240 --> 00:43:45.120 align:middle line:84% If I'm better off for any possible value X, then I 00:43:45.120 --> 00:43:49.140 align:middle line:84% should be better off on the average over all possible 00:43:49.140 --> 00:43:50.640 align:middle line:90% values of X. 00:43:50.640 --> 00:43:55.670 align:middle line:84% So let us average both sides of this quantity with respect 00:43:55.670 --> 00:43:58.990 align:middle line:84% to the probability distribution of X. If you want 00:43:58.990 --> 00:44:03.350 align:middle line:84% to do it formally, you can write this inequality between 00:44:03.350 --> 00:44:06.520 align:middle line:84% numbers as an inequality between random variables. 00:44:06.520 --> 00:44:10.240 align:middle line:84% And it tells that no matter what that random variable 00:44:10.240 --> 00:44:14.010 align:middle line:84% turns out to be, this quantity is better than that quantity. 00:44:14.010 --> 00:44:17.270 align:middle line:84% Take expectations of both sides, and you get this 00:44:17.270 --> 00:44:21.360 align:middle line:84% inequality between expectations overall. 00:44:21.360 --> 00:44:29.130 align:middle line:84% And this last inequality tells me that the person who's using 00:44:29.130 --> 00:44:34.430 align:middle line:84% this estimator who produces estimates according to this 00:44:34.430 --> 00:44:45.090 align:middle line:84% machine will have a mean squared estimation error 00:44:45.090 --> 00:44:48.580 align:middle line:84% that's less than or equal to the estimation error that's 00:44:48.580 --> 00:44:51.290 align:middle line:90% produced by the other person. 00:44:51.290 --> 00:44:54.710 align:middle line:84% In a few words, the conditional expectation 00:44:54.710 --> 00:44:58.500 align:middle line:84% estimator is the optimal estimator. 00:44:58.500 --> 00:45:01.765 align:middle line:84% It's the ultimate estimating machine. 00:45:01.765 --> 00:45:04.430 align:middle line:90% 00:45:04.430 --> 00:45:08.720 align:middle line:84% That's how you should solve estimation problems and report 00:45:08.720 --> 00:45:10.240 align:middle line:90% a single value. 00:45:10.240 --> 00:45:14.510 align:middle line:84% If you're forced to report a single value and if you're 00:45:14.510 --> 00:45:18.060 align:middle line:84% interested in estimation errors. 00:45:18.060 --> 00:45:24.620 align:middle line:84% OK, while we could have told you that story, of course, a 00:45:24.620 --> 00:45:29.500 align:middle line:84% month or two ago, this is really about interpretation -- 00:45:29.500 --> 00:45:32.550 align:middle line:84% about realizing that conditional expectations have 00:45:32.550 --> 00:45:35.160 align:middle line:90% a very nice property. 00:45:35.160 --> 00:45:38.180 align:middle line:84% But other than that, any probabilistic skills that come 00:45:38.180 --> 00:45:41.180 align:middle line:84% into this business are just the probabilistic skills of 00:45:41.180 --> 00:45:44.330 align:middle line:84% being able to calculate conditional expectations, 00:45:44.330 --> 00:45:46.750 align:middle line:84% which you already know how to do. 00:45:46.750 --> 00:45:51.380 align:middle line:84% So conclusion, all of optimal Bayesian estimation just means 00:45:51.380 --> 00:45:54.655 align:middle line:84% calculating and reporting conditional expectations. 00:45:54.655 --> 00:45:58.380 align:middle line:84% Well, if the world were that simple, then statisticians 00:45:58.380 --> 00:46:02.670 align:middle line:84% wouldn't be able to find jobs if life is that simple. 00:46:02.670 --> 00:46:05.690 align:middle line:84% So real life is not that simple. 00:46:05.690 --> 00:46:07.540 align:middle line:90% There are complications. 00:46:07.540 --> 00:46:10.050 align:middle line:84% And that perhaps makes their life a little more 00:46:10.050 --> 00:46:11.300 align:middle line:90% interesting. 00:46:11.300 --> 00:46:22.010 align:middle line:90% 00:46:22.010 --> 00:46:25.500 align:middle line:84% OK, one complication is that we would deal with the vectors 00:46:25.500 --> 00:46:28.580 align:middle line:84% instead of just single random variables. 00:46:28.580 --> 00:46:31.830 align:middle line:84% I use the notation here as if X was a 00:46:31.830 --> 00:46:33.500 align:middle line:90% single random variable. 00:46:33.500 --> 00:46:37.710 align:middle line:84% In real life, you get several data. 00:46:37.710 --> 00:46:39.520 align:middle line:90% Does our story change? 00:46:39.520 --> 00:46:41.950 align:middle line:90% Not really, same argument-- 00:46:41.950 --> 00:46:44.410 align:middle line:84% given all the data that you have observed, you should 00:46:44.410 --> 00:46:47.660 align:middle line:84% still report the conditional expectation of Theta. 00:46:47.660 --> 00:46:51.260 align:middle line:84% But what kind of work does it take in order to report this 00:46:51.260 --> 00:46:53.080 align:middle line:90% conditional expectation? 00:46:53.080 --> 00:46:57.030 align:middle line:84% One issue is that you need to cook up a plausible prior 00:46:57.030 --> 00:46:58.810 align:middle line:90% distribution for Theta. 00:46:58.810 --> 00:46:59.960 align:middle line:90% How do you do that? 00:46:59.960 --> 00:47:03.570 align:middle line:84% In a given application , this is a bit of a judgment call, 00:47:03.570 --> 00:47:05.970 align:middle line:84% what prior would you be working with. 00:47:05.970 --> 00:47:08.840 align:middle line:84% And there's a certain skill there of not 00:47:08.840 --> 00:47:12.100 align:middle line:90% making silly choices. 00:47:12.100 --> 00:47:16.690 align:middle line:84% A more pragmatic, practical issue is that this is a 00:47:16.690 --> 00:47:21.180 align:middle line:84% formula that's extremely nice and compact and simple that 00:47:21.180 --> 00:47:24.560 align:middle line:84% you can write with minimal ink. 00:47:24.560 --> 00:47:29.180 align:middle line:84% But the behind it there could be hidden a huge amount of 00:47:29.180 --> 00:47:31.520 align:middle line:90% calculation. 00:47:31.520 --> 00:47:34.820 align:middle line:84% So doing any sort of calculations that involve 00:47:34.820 --> 00:47:39.640 align:middle line:84% multiple random variables really involves calculating 00:47:39.640 --> 00:47:42.240 align:middle line:90% multi-dimensional integrals. 00:47:42.240 --> 00:47:46.230 align:middle line:84% And the multi-dimensional integrals are hard to compute. 00:47:46.230 --> 00:47:50.830 align:middle line:84% So implementing actually this calculating machine here may 00:47:50.830 --> 00:47:54.340 align:middle line:84% not be easy, might be complicated computationally. 00:47:54.340 --> 00:47:58.250 align:middle line:84% It's also complicated in terms of not being able to derive 00:47:58.250 --> 00:47:59.890 align:middle line:90% intuition about it. 00:47:59.890 --> 00:48:03.680 align:middle line:84% So perhaps you might want to have a simpler version, a 00:48:03.680 --> 00:48:07.940 align:middle line:84% simpler alternative to this formula that's easier to work 00:48:07.940 --> 00:48:10.950 align:middle line:90% with and easier to calculate. 00:48:10.950 --> 00:48:13.440 align:middle line:84% We will be talking about one such simpler 00:48:13.440 --> 00:48:15.540 align:middle line:90% alternative next time. 00:48:15.540 --> 00:48:18.570 align:middle line:84% So again, to conclude, at the high level, Bayesian 00:48:18.570 --> 00:48:22.330 align:middle line:84% estimation is very, very simple, given that you have 00:48:22.330 --> 00:48:24.180 align:middle line:84% mastered everything that has happened in 00:48:24.180 --> 00:48:26.370 align:middle line:90% this course so far. 00:48:26.370 --> 00:48:29.860 align:middle line:84% There are certain practical issues and it's also good to 00:48:29.860 --> 00:48:33.590 align:middle line:84% be familiar with the concepts and the issues that in 00:48:33.590 --> 00:48:36.620 align:middle line:84% general, you would prefer to report that complete posterior 00:48:36.620 --> 00:48:37.360 align:middle line:90% distribution. 00:48:37.360 --> 00:48:40.890 align:middle line:84% But if you're forced to report a point estimate, then there's 00:48:40.890 --> 00:48:43.130 align:middle line:84% a number of reasonable ways to do it. 00:48:43.130 --> 00:48:45.690 align:middle line:84% And perhaps the most reasonable one is to just the 00:48:45.690 --> 00:48:48.220 align:middle line:84% report the conditional expectation itself. 00:48:48.220 --> 00:48:49.470 align:middle line:90%