1 00:00:00,000 --> 00:00:05,013 We previously defined the notion of Bayesian estimation in which we have a 2 00:00:05,013 --> 00:00:10,627 prior over the parameters and we continue to maintain a posterior over the parameters 3 00:00:10,627 --> 00:00:16,108 as we accumulate the data. What we haven't discussed though, is how one might use a 4 00:00:16,108 --> 00:00:21,322 model in which we have a distribution over the parameters. That is, how do we take 5 00:00:21,322 --> 00:00:27,240 such a model and use it to make predictions about new instances? [sound] 6 00:00:31,160 --> 00:00:36,994 So, now let's think about how we use a Dirichlet distribution, once we have it. 7 00:00:36,994 --> 00:00:42,367 So assume that we have a param-, a, a model, where our parameter Theta is 8 00:00:42,367 --> 00:00:49,589 distributed Dirichlet with some set of hyper-parameters. Now, if we're trying to 9 00:00:49,589 --> 00:00:55,330 make a prediction over the value of the variable x that depends on the parameter 10 00:00:55,330 --> 00:01:00,999 theta, well, we're just, this is just now a probabilistic inference problem, and 11 00:01:00,999 --> 00:01:08,144 so the probability of x. Is simply the probability of x given theta, marginaliz- 12 00:01:08,144 --> 00:01:15,125 times the prior over theta marginalizing in this case corresponding to an 13 00:01:15,125 --> 00:01:23,619 integration Over the value of the theta. And then give this integral over here. So 14 00:01:23,619 --> 00:01:28,803 if we plug through the integral, what we're going to get is the following form. 15 00:01:28,803 --> 00:01:33,855 And I'm not going to go through the integration by parts that's required to 16 00:01:33,855 --> 00:01:38,640 actually show this. But it's really a straightforward consequence of the 17 00:01:38,640 --> 00:01:44,224 properties of integrals of polynomials. In this case, we have that the probability 18 00:01:44,224 --> 00:01:51,608 that X takes a particular value XI, Is one, is one over Z times the integral over 19 00:01:51,608 --> 00:02:00,678 all the parameters theta of theta I which is the, the probability given the 20 00:02:00,678 --> 00:02:09,208 parameterization theta that X takes the value of little xi times this thing over here 21 00:02:09,208 --> 00:02:15,979 which is the prior. And we multiply the two together, integrate out over the 22 00:02:15,979 --> 00:02:22,806 parameter vector theta, which in this case K dimensional parameter vector, 23 00:02:22,806 --> 00:02:29,379 and it turns out that you end up with alpha I over the sum of all J's. Alpha J, 24 00:02:29,379 --> 00:02:35,953 a quantity typically known as little alpha. And so we end up with a case where, 25 00:02:36,206 --> 00:02:45,553 the prediction over the next instance represents the fraction of the instances 26 00:02:45,553 --> 00:02:55,500 that we've seen as represented in the hyper parameters of the Dirichlet. Where. 27 00:02:55,760 --> 00:03:02,791 We have X, little XI. But if alpha I represents the number of instances that 28 00:03:02,791 --> 00:03:08,404 we've seen where X, where the variable has the value little xi. The prediction 29 00:03:08,404 --> 00:03:13,532 very naturally represents, is, is, is simply the fractions of instances with 30 00:03:13,532 --> 00:03:19,145 that property. And so once again we see that there is a natural intuition for the 31 00:03:19,145 --> 00:03:27,699 hyper parameter as representing a notion of counts. Now let's put these two results 32 00:03:27,699 --> 00:03:32,209 together and think about Bayesian prediction as a function of, as the number 33 00:03:32,209 --> 00:03:36,422 of data instances that we have [inaudible]. So here we have a parameter 34 00:03:36,422 --> 00:03:41,110 theta, which initially was distributed as a Dirichlet, as some sort of a hyper 35 00:03:41,110 --> 00:03:45,738 parameter, and lets imagine that we've seen data instances X1 up to XM. And now 36 00:03:45,738 --> 00:03:50,307 we have the M plus first data instance and we want to make a prediction 37 00:03:50,307 --> 00:03:57,165 about that. So, the problem that we're trying to solve is now the probability of 38 00:03:57,165 --> 00:04:04,056 the m plus first data instance given the m first, the m instances that we've seen 39 00:04:04,056 --> 00:04:10,863 previously. And so we can once again plug that into a probabilistic inference 40 00:04:10,863 --> 00:04:17,073 equation. So this is going to be the probability of the m plus first data 41 00:04:17,073 --> 00:04:22,968 instance, given. Everything including theta. Times the probability of theta given 42 00:04:22,968 --> 00:04:28,551 x one up to x m. So we've introduced the variable theta into this probability. And 43 00:04:28,551 --> 00:04:35,850 we're marginalizing out over the variable theta. Well one thing that immediately 44 00:04:35,850 --> 00:04:41,827 follows is because of the structure of the probabilistic graphical model here, we have 45 00:04:41,827 --> 00:04:47,031 that x m + one is conditionally independent of all of these previous x's 46 00:04:47,031 --> 00:04:52,587 given theta. So we can cancel these from the right hand side of the 47 00:04:52,587 --> 00:04:57,720 conditioning bar, which gives us over here probability of x, M + one given 48 00:04:57,720 --> 00:05:03,891 theta and here we have the probability of theta Given X one 49 00:05:03,891 --> 00:05:12,665 up to X M. And so now let's think about the blue equation, the blue expression over 50 00:05:12,665 --> 00:05:23,726 here, which is just the posterior. Over theta, given D. Which are X1 up to XM. And we 51 00:05:23,726 --> 00:05:29,297 already seen what that looks like. That as we show just on the previous slide 52 00:05:29,297 --> 00:05:34,940 is simply a Dirichlet whose hyper parameter are alpha one plus M1 up to alpha K plus MK. 53 00:05:35,780 --> 00:05:41,630 And so now we're making a prediction of a single random variable from a Dirichlet 54 00:05:41,632 --> 00:05:46,545 that has a certain set of hyper parameters. And that was the thing we 55 00:05:46,545 --> 00:05:52,180 showed on the slide just before that, which is simply the fraction of the alp-, 56 00:05:52,180 --> 00:05:57,383 the fraction of the hyper parameter corresponding to the outcome xi as a 57 00:05:57,383 --> 00:06:02,729 fraction of all of, the sum of all the hyper parameters, where again just to 58 00:06:02,729 --> 00:06:08,798 introduce notation, alpha is equal to the sum of the alpha I and m to the sum of the 59 00:06:08,798 --> 00:06:17,937 M-Is. Now notice what happens here. This parameter alpha that we just defined, 60 00:06:17,937 --> 00:06:24,756 which is the sum over all of the alpha I's that I have . It?s a parameter known as 61 00:06:24,756 --> 00:06:30,986 the equivalent sample size. And this represents the number of, if you will 62 00:06:30,986 --> 00:06:37,890 imaginary samples that I would have seen prior to receiving the new data, X1 to Xn. 63 00:06:37,890 --> 00:06:44,728 Now look what happens if we multiply alpha by a constant. So say we double all of our 64 00:06:44,728 --> 00:06:51,648 alpha I's, then we have we're going to let the m I's affect our estimate a lot less 65 00:06:51,648 --> 00:06:58,085 than for smaller values of alpha. So the larger the alpha, the more confidence we 66 00:06:58,085 --> 00:07:06,421 have in our prior and the less we let our data move us away from that prior. Let's 67 00:07:06,421 --> 00:07:12,085 look at an example of the influence that this might have. So let's go back to 68 00:07:12,085 --> 00:07:17,454 binomial data or a Bernoulli random variable and let's take the simplest 69 00:07:17,454 --> 00:07:23,412 example where a prior is uniform for theta in zero, one and we've previously seen 70 00:07:23,412 --> 00:07:31,628 that corresponds to a Dirichlet with hyper parameters one, one. So this is our, 71 00:07:31,628 --> 00:07:38,083 so this is a general-purpose Dirichlet distribution. In this case the hyper 72 00:07:38,083 --> 00:07:45,409 parameters are one, one and let's imagine that we get a five data instances of which 73 00:07:45,409 --> 00:07:54,123 we have four ones. And one zero. And if you actually. Think about the differences 74 00:07:54,123 --> 00:07:59,612 between what the Bayesian estimate gives you for the sixth. Next coin toss. 75 00:07:59,612 --> 00:08:05,988 Relative in, in when doing maximum likelihood estimation versus the Bayesian estimation. 76 00:08:05,988 --> 00:08:12,217 For maximum likelihood estimation we have four heads, four tails maximum likelihood estimate 77 00:08:12,217 --> 00:08:17,703 is 4/5ths. So that's going to be the prediction for the sixth instance. The 78 00:08:17,703 --> 00:08:23,042 Bayesian prediction, on the other hand, remember is going to do the hyper 79 00:08:23,042 --> 00:08:28,974 parameter alpha one plus M1 divided by alpha plus M which in this case is going 80 00:08:28,974 --> 00:08:34,987 to be one plus four divided by two plus. Find a high enough score to give us five 81 00:08:34,987 --> 00:08:43,479 over seven. So let's look more qualitatively at the effects of these 82 00:08:43,479 --> 00:08:49,572 predictions on a next instance after seeing certain amounts of data. And for 83 00:08:49,572 --> 00:08:55,820 the moment we're going to assume that the ratio between the number of 1s and the 84 00:08:55,820 --> 00:09:02,299 number of 0s is fixed so that we have one. One for every four zero's and that's the 85 00:09:02,299 --> 00:09:08,556 data that we are getting. And now let's see what happens as the function of the sample 86 00:09:08,556 --> 00:09:14,298 size. So as we get more and more data all of which satisfies this, a particular 87 00:09:14,298 --> 00:09:21,775 ratio. So here we are playing around with a different strength, or equivalent sample 88 00:09:21,775 --> 00:09:28,810 size. But we're fixing the ratio of alpha one to alpha zero to represent, in this 89 00:09:28,810 --> 00:09:34,878 case, the 50 percent level. So our prior is a uniform prior but of greater and 90 00:09:34,878 --> 00:09:41,737 greater or changing strength. And so this little green line down at the bottom 91 00:09:41,737 --> 00:09:49,096 represents a low alpha. Because we can see that the data gets pulled, our, posterior, 92 00:09:49,096 --> 00:09:55,442 so sorry. The line is drawing the posterior over the parameter or rather 93 00:09:55,442 --> 00:10:01,865 equivalency, the prediction of the next data instance, over time. And you can see 94 00:10:01,865 --> 00:10:08,524 here that alpha is low and that means that for even for fairly small amounts of data, 95 00:10:08,759 --> 00:10:14,869 , say twenty data points, we're already very close to the data estimates. On the 96 00:10:14,869 --> 00:10:23,108 other hand, this bluish line over here, We can see that the alpha is high. And that 97 00:10:23,108 --> 00:10:29,662 means it takes more time for the data to pull us, to the empirical, fraction, of 98 00:10:29,662 --> 00:10:36,050 heads versus tails Now lets look at varying the other parameter, we're going 99 00:10:36,050 --> 00:10:42,089 to now fix the equivalent sample size. And we are just going to start out with priors 100 00:10:42,089 --> 00:10:47,703 and we can see that now we get pulled down to the 0.2 value that we see in the 101 00:10:47,703 --> 00:10:53,245 empirical data, and , the further away from it we start though it takes us a 102 00:10:53,245 --> 00:10:59,143 little bit longer to actually get pulled out the data estimate. But in all cases we 103 00:10:59,143 --> 00:11:06,782 eventually get convergence to the value in the actual data set. And from a pragmatic 104 00:11:06,782 --> 00:11:14,839 perspective it turns out that. Bayesian estimates provide us with a smoothness, 105 00:11:14,839 --> 00:11:22,695 where the random fluctuations in the data don't, Don't cause quite as much random 106 00:11:22,695 --> 00:11:29,070 jumping around as they do, for example, in maximum likelihood estimates. So, if what we have 107 00:11:29,070 --> 00:11:35,246 here is the actual value of the coin toss. At different points in the process you can 108 00:11:35,246 --> 00:11:39,976 see the blue line, this light blue line, that corresponds to maximize data 109 00:11:39,976 --> 00:11:44,900 estimation, basically bops around the [inaudible], especially in the low data 110 00:11:44,900 --> 00:11:50,278 regime. Where as the, where as the ones that use a prior, the estimates that use a 111 00:11:50,278 --> 00:11:55,317 prior, are considerably smoother, and less subject to random noise. In summary, 112 00:11:55,317 --> 00:12:01,035 Bayesian prediction combines, two types of, you might call them sufficient 113 00:12:01,035 --> 00:12:06,746 statistics. There's the sufficient statistics from the real data. But there's 114 00:12:06,746 --> 00:12:12,948 also sufficient statistics from the imaginary samples that contribute to the 115 00:12:12,948 --> 00:12:18,850 Dirichlet distribution, these alpha hyper parameters. And the Bayesian prediction 116 00:12:18,850 --> 00:12:25,201 effectively makes the prediction about the new data instance by combining both of 117 00:12:25,201 --> 00:12:30,223 these. Now, as the amount of data increases that is at the asymptotic limit of 118 00:12:30,223 --> 00:12:35,128 many data instances, the term that corresponds to the real data samples is 119 00:12:35,128 --> 00:12:40,365 going to dominate. And, therefore, the prior is going to become vanishingly small 120 00:12:40,365 --> 00:12:45,138 in terms of the contribution that it makes. So at the limit, the Bayesian 121 00:12:45,138 --> 00:12:51,634 prediction is the same as maximum likelihood estimation. But. Initially in 122 00:12:51,634 --> 00:12:57,382 the early stages of estimation before we have a lot of data that the priors 123 00:12:57,382 --> 00:13:02,340 actually make a fairly significant difference. And we've seen that 124 00:13:02,340 --> 00:13:07,317 Dirichlet hyper parameters basically determine both. Our prior beliefs 125 00:13:07,317 --> 00:13:12,256 initially before we have a lot of data as well as the strengths of these beliefs 126 00:13:12,256 --> 00:13:17,194 that is how long it takes for the data to outweigh the prior and move us towards 127 00:13:17,194 --> 00:13:22,316 what we see in the empirical distribution. But importantly even as we've see here in 128 00:13:22,316 --> 00:13:27,315 the very simple example [inaudible] see what's wrong when you talk about learning 129 00:13:27,315 --> 00:13:31,644 [inaudible] networks it turns out that this Bayesian learning paradigm is 130 00:13:31,644 --> 00:13:35,851 considerably more robust in the sparse data regime in terms of its 131 00:13:36,217 --> 00:13:37,620 generalization ability.