WEBVTT 00:00:01.540 --> 00:00:03.910 The following content is provided under a Creative 00:00:03.910 --> 00:00:05.300 Commons license. 00:00:05.300 --> 00:00:07.510 Your support will help MIT OpenCourseWare 00:00:07.510 --> 00:00:11.600 continue to offer high quality educational resources for free. 00:00:11.600 --> 00:00:14.140 To make a donation or to view additional materials 00:00:14.140 --> 00:00:18.100 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.100 --> 00:00:19.310 at ocw.mit.edu. 00:00:22.672 --> 00:00:24.130 WILLIAM GREEN: So today we're going 00:00:24.130 --> 00:00:31.920 to talk about Bayesian prior estimation and prior estimation 00:00:31.920 --> 00:00:32.500 in general. 00:00:36.830 --> 00:00:44.830 So the last time we were writing down the expressions 00:00:44.830 --> 00:00:52.540 for the probability of observing a mean measurement if you 00:00:52.540 --> 00:00:54.470 know what the model is. 00:00:54.470 --> 00:00:55.730 So let's try to do that again. 00:00:55.730 --> 00:01:04.840 So suppose I have a model that predicts some observable 00:01:04.840 --> 00:01:06.390 and it depends on some knobs, and it 00:01:06.390 --> 00:01:09.850 depends on some parameters. 00:01:09.850 --> 00:01:15.480 And suppose because I have great powers of faith 00:01:15.480 --> 00:01:18.670 that I believe this model is 100% correct with every core 00:01:18.670 --> 00:01:20.440 of my being. 00:01:20.440 --> 00:01:22.620 And also because I have tremendous confidence in all 00:01:22.620 --> 00:01:25.810 the people who built my apparatus, 00:01:25.810 --> 00:01:28.210 and the knobs that I turn actually correspond 00:01:28.210 --> 00:01:31.030 to the real values, and I have tremendous confidence 00:01:31.030 --> 00:01:34.320 in all the literature that reports the parameter values. 00:01:34.320 --> 00:01:37.920 And so I'm absolutely certain that this is the truth. 00:01:37.920 --> 00:01:41.410 So we'll start from a position of absolute certainty, 00:01:41.410 --> 00:01:45.229 and then we'll degrade into doubt as the collector goes on. 00:01:45.229 --> 00:01:47.020 So let's start from the position of someone 00:01:47.020 --> 00:01:51.665 who has absolute faith that this is the truth. 00:01:51.665 --> 00:01:52.900 Is true. 00:01:55.470 --> 00:01:58.090 So I have a model, I really believe this model. 00:01:58.090 --> 00:02:01.330 So for example, I believe that the kilogram 00:02:01.330 --> 00:02:04.810 weight in the SI Institute in Paris 00:02:04.810 --> 00:02:07.510 weighs exactly one kilogram. 00:02:07.510 --> 00:02:09.800 I believe that with every core of my being. 00:02:09.800 --> 00:02:13.390 I'm completely confident that model is correct. 00:02:13.390 --> 00:02:15.921 So there are some things I'm really confident on. 00:02:15.921 --> 00:02:16.420 That's one. 00:02:19.180 --> 00:02:22.410 And maybe guys have some things you really believe, too. 00:02:22.410 --> 00:02:24.160 So let's go with things we really believe. 00:02:28.780 --> 00:02:32.950 So I plan to conduct some experiments that 00:02:32.950 --> 00:02:37.880 measure this observable and are related to this model. 00:02:37.880 --> 00:02:45.865 And so I'm going to do 10 repeats of measuring y. 00:02:52.630 --> 00:02:57.101 So I'm going to get to the kilogram blob that's in Paris, 00:02:57.101 --> 00:02:59.350 and I'm going to stick it on my really expensive scale 00:02:59.350 --> 00:03:00.880 that I really believe is great, and I'm 00:03:00.880 --> 00:03:02.060 going to measure its weight. 00:03:02.060 --> 00:03:03.520 And then I'm going to put it back, and then 00:03:03.520 --> 00:03:05.370 put it back, and put it back, and put it back. 00:03:05.370 --> 00:03:06.850 And I'm going to get another really great scale that I 00:03:06.850 --> 00:03:09.516 really believe is great, and I'm going to measure it there, too. 00:03:09.516 --> 00:03:11.350 So I've got a lot of repeats of measuring 00:03:11.350 --> 00:03:13.270 the weight of this kilogram, and I 00:03:13.270 --> 00:03:15.950 believe it's really a kilogram. 00:03:15.950 --> 00:03:18.340 But the stupid measurements don't say a kilogram. 00:03:18.340 --> 00:03:25.070 They say, you know, 1.0003, 0.99995, all kinds of numbers 00:03:25.070 --> 00:03:27.610 not equal to one kilogram. 00:03:27.610 --> 00:03:33.280 So now I'm going to try to figure out 00:03:33.280 --> 00:03:36.340 what the probability is that it would have measured 00:03:36.340 --> 00:03:38.180 some particular value y. 00:03:38.180 --> 00:03:58.230 So what is the probability that my experimental 00:03:58.230 --> 00:04:01.960 is between some value, say, y and y plus dy. 00:04:04.990 --> 00:04:06.420 So that's a question for you. 00:04:06.420 --> 00:04:07.930 So what's the probability? 00:04:19.700 --> 00:04:20.915 Sorry, what? 00:04:20.915 --> 00:04:22.590 [INAUDIBLE] 00:04:22.590 --> 00:04:29.704 OK, so we think that the probability that y 00:04:29.704 --> 00:04:37.815 is in this interval given that the model is true. 00:04:43.050 --> 00:04:45.420 And I know the theta values perfectly, 00:04:45.420 --> 00:04:49.220 and I know the x values perfectly, 00:04:49.220 --> 00:04:51.590 is equal to some integral of what? 00:04:55.100 --> 00:04:59.820 The bounds integral, probably y to y plus dy, believe that? 00:05:02.520 --> 00:05:03.791 What's the integrand? 00:05:11.660 --> 00:05:13.220 Sorry, what? 00:05:13.220 --> 00:05:16.040 AUDIENCE: The probability [INAUDIBLE] AC function of y. 00:05:16.040 --> 00:05:17.540 WILLIAM GREEN: Right, so what is it? 00:05:20.071 --> 00:05:22.043 AUDIENCE: [INAUDIBLE] 00:05:22.043 --> 00:05:23.760 You wrote it down last time, I think. 00:05:33.260 --> 00:05:36.340 So this [INAUDIBLE] is large? 00:05:36.340 --> 00:05:37.380 Standard normal, right? 00:05:37.380 --> 00:05:42.880 So it should be one over sigma root of 2 pi. 00:06:00.220 --> 00:06:01.090 Does that sound OK? 00:06:06.510 --> 00:06:07.010 I mean here. 00:06:10.794 --> 00:06:12.674 It's probably the same, it's fine. 00:06:20.578 --> 00:06:21.673 Yep. 00:06:21.673 --> 00:06:23.048 AUDIENCE: What does that notation 00:06:23.048 --> 00:06:26.602 mean, if your model is true? 00:06:26.602 --> 00:06:29.060 WILLIAM GREEN: So this means, given that the model is true, 00:06:29.060 --> 00:06:31.434 and I know these data values are exactly certain numbers, 00:06:31.434 --> 00:06:34.040 and the x values are actually certain numbers, what's 00:06:34.040 --> 00:06:35.870 the probability that I would make 00:06:35.870 --> 00:06:40.220 a measurement whose average would fall in this interval? 00:06:40.220 --> 00:06:42.139 So this line means given that this is true, 00:06:42.139 --> 00:06:43.430 what's the probability of that? 00:06:48.070 --> 00:06:50.020 OK, is this right? 00:06:50.020 --> 00:06:52.572 Is this surprising? 00:06:52.572 --> 00:06:54.412 This is OK? 00:06:54.412 --> 00:06:55.370 So this is what I mean. 00:06:55.370 --> 00:06:58.160 So we say that our probability distribution converges 00:06:58.160 --> 00:07:00.462 to a Gaussian distribution, this is what we expect. 00:07:00.462 --> 00:07:02.170 So we expect it to have been large enough 00:07:02.170 --> 00:07:03.003 for this to be true. 00:07:06.130 --> 00:07:07.692 Yeah? 00:07:07.692 --> 00:07:08.650 This is very important. 00:07:08.650 --> 00:07:10.511 This is like the whole course, actually. 00:07:10.511 --> 00:07:12.510 This is the whole section, is this one equation. 00:07:12.510 --> 00:07:15.340 So I just wanted to make sure you really get what this says. 00:07:15.340 --> 00:07:16.930 And if you don't like the integral, 00:07:16.930 --> 00:07:18.721 you can make dy really small, and then it's 00:07:18.721 --> 00:07:19.650 just this times dy. 00:07:23.506 --> 00:07:24.006 OK? 00:07:28.850 --> 00:07:31.050 Actually, this notation's like [INAUDIBLE] I 00:07:31.050 --> 00:07:33.220 think I should do this way. 00:07:33.220 --> 00:07:35.492 I should do this. 00:07:35.492 --> 00:07:36.200 This is a number. 00:07:38.731 --> 00:07:39.980 Let's get rid of the integral. 00:07:39.980 --> 00:07:42.240 Let's make dy really small. 00:07:42.240 --> 00:07:43.774 I'll make it [INAUDIBLE]. 00:07:48.596 --> 00:07:49.220 That all right? 00:07:52.470 --> 00:07:55.510 So this is the probability density that we would observe, 00:07:55.510 --> 00:07:59.820 this is the experimental value y that we observe from the mean, 00:07:59.820 --> 00:08:05.204 and this is the little with of our tiny little interval. 00:08:05.204 --> 00:08:06.350 Is that all right? 00:08:10.570 --> 00:08:11.070 Yes? 00:08:11.070 --> 00:08:14.353 AUDIENCE: So is sigma the [INAUDIBLE] on there? 00:08:14.353 --> 00:08:16.231 WILLIAM GREEN: Ah, what is sigma? 00:08:16.231 --> 00:08:17.230 That's a great question. 00:08:17.230 --> 00:08:18.729 We didn't write down what sigma was. 00:08:18.729 --> 00:08:19.423 What is sigma? 00:08:19.423 --> 00:08:21.235 AUDIENCE: Standard deviation? 00:08:21.235 --> 00:08:23.526 WILLIAM GREEN: It's not the standard deviation exactly. 00:08:23.526 --> 00:08:25.650 Standard deviation of the mean, right? 00:08:25.650 --> 00:08:28.170 So there's two sigmas. 00:08:28.170 --> 00:08:32.049 We have the sigma of y, of the measurements, 00:08:32.049 --> 00:08:42.559 and that's equal to average value of y squared minus. 00:08:49.430 --> 00:08:51.680 So we just figure that for how many experiments we do, 00:08:51.680 --> 00:08:54.030 we just compute the average of y squared, the average y, 00:08:54.030 --> 00:08:55.180 subtract them. 00:08:55.180 --> 00:08:56.340 That's the variance. 00:08:56.340 --> 00:08:58.550 And then sigma that I used in that equation 00:08:58.550 --> 00:09:04.580 there is 1 over n times sigma y. 00:09:04.580 --> 00:09:08.160 And we call this the variation of the mean, 00:09:08.160 --> 00:09:11.987 it's the uncertainty in the mean value of y. 00:09:11.987 --> 00:09:13.445 And the central limits theorem said 00:09:13.445 --> 00:09:15.230 that as long as n gets really large, 00:09:15.230 --> 00:09:18.500 we expect that this should converge to this. 00:09:18.500 --> 00:09:21.690 And we talked last time about how when n get bigger, 00:09:21.690 --> 00:09:24.710 these averages don't really change when it gets big. 00:09:24.710 --> 00:09:25.910 They're just the average. 00:09:28.580 --> 00:09:30.620 But this number declined as n gets big because 00:09:30.620 --> 00:09:34.390 of this one over n formula. 00:09:34.390 --> 00:09:39.170 And to understand that, suppose I measure the weight, 00:09:39.170 --> 00:09:43.080 and I measure, it should be around one kilogram. 00:09:43.080 --> 00:09:46.430 But in fact my measurements are all over here. 00:09:46.430 --> 00:09:50.610 Lots of measurements. 00:09:50.610 --> 00:09:54.020 So they have a variance something like this. 00:09:54.020 --> 00:09:56.060 But if I make a plot of-- 00:09:59.650 --> 00:10:03.370 as I run, I compute the running average. 00:10:03.370 --> 00:10:08.470 So when I run the first two points, 00:10:08.470 --> 00:10:10.960 I get some average value here. 00:10:10.960 --> 00:10:15.370 After I run 27 points more, the average value is here. 00:10:15.370 --> 00:10:18.700 After I run 1,000 repeats, the average value is here. 00:10:18.700 --> 00:10:20.117 It's getting pretty close to this, 00:10:20.117 --> 00:10:21.616 and the uncertainty in this number's 00:10:21.616 --> 00:10:23.350 getting smaller and smaller as I'm doing 00:10:23.350 --> 00:10:25.120 better and better averages. 00:10:25.120 --> 00:10:26.900 The average more and more repeats. 00:10:26.900 --> 00:10:29.054 Does that make sense? 00:10:29.054 --> 00:10:31.590 OK. 00:10:31.590 --> 00:10:37.910 So from this key equation, I can derive a lot of things. 00:10:37.910 --> 00:10:39.510 And it depends what you want to do. 00:10:39.510 --> 00:10:44.220 So one thing people do a lot is it was called model validation. 00:10:50.060 --> 00:10:51.180 And what does this mean? 00:10:51.180 --> 00:10:54.890 It means I have a model, I believe it's true. 00:10:54.890 --> 00:10:57.420 I have some parameters, I believe they're true. 00:10:57.420 --> 00:10:59.420 But there are some foolish skeptics out there 00:10:59.420 --> 00:11:01.400 who don't have the faith that I do. 00:11:01.400 --> 00:11:04.520 And they think that my model's baloney, or my parameter 00:11:04.520 --> 00:11:06.440 values are wrong, or something. 00:11:06.440 --> 00:11:10.520 And so to prove I'm right, I'm going to make some experiments. 00:11:10.520 --> 00:11:12.020 And I'm going to show that I make 00:11:12.020 --> 00:11:16.340 a plot that looks like the experiment and model agree. 00:11:16.340 --> 00:11:18.795 Some of you might have done this in your life, yes? 00:11:18.795 --> 00:11:21.920 Everybody might make a parity plot or something. 00:11:21.920 --> 00:11:24.830 You've seen these things before. 00:11:24.830 --> 00:11:28.919 Now, this is like a confidence builder. 00:11:28.919 --> 00:11:30.710 You're trying to get the skeptics out there 00:11:30.710 --> 00:11:33.560 to believe that there's some evidence 00:11:33.560 --> 00:11:38.120 to back up your faith that this model is perfect. 00:11:38.120 --> 00:11:40.490 And what you really want to know is 00:11:40.490 --> 00:11:45.200 like, if the measurement that I measure, 00:11:45.200 --> 00:11:49.050 the average for my 10,000 repeated measurements, 00:11:49.050 --> 00:11:54.720 I expect that this quantity should be pretty big somehow 00:11:54.720 --> 00:11:56.294 in some way. 00:11:56.294 --> 00:11:57.710 By then quantitatively saying what 00:11:57.710 --> 00:12:02.124 that means exactly what's a good fit, what's a bad fit, 00:12:02.124 --> 00:12:04.040 this is actually kind of a difficult question, 00:12:04.040 --> 00:12:05.510 and we'll come back to this one. 00:12:05.510 --> 00:12:09.470 But that's a very common use of this equation 00:12:09.470 --> 00:12:13.200 is to try to do validation. 00:12:13.200 --> 00:12:15.400 Now because it's kind of complicated, 00:12:15.400 --> 00:12:17.250 most people don't actually do it. 00:12:17.250 --> 00:12:19.530 So instead what they do is they just plot 00:12:19.530 --> 00:12:22.860 some data points, and they plot your model curve. 00:12:22.860 --> 00:12:26.040 And as long as they look good, then you're done. 00:12:26.040 --> 00:12:30.990 So that's the normal way that it's done in the literature 00:12:30.990 --> 00:12:31.810 currently. 00:12:31.810 --> 00:12:33.810 But of course, that's completely unquantitative. 00:12:33.810 --> 00:12:36.690 It doesn't really say whether the model and the data 00:12:36.690 --> 00:12:39.750 really agree, it just means they look sort of like each other. 00:12:39.750 --> 00:12:41.615 So that's like a human qualitative thing. 00:12:41.615 --> 00:12:42.990 Now, if the purpose of validation 00:12:42.990 --> 00:12:48.270 is just to convince humans, then you've done the purpose. 00:12:48.270 --> 00:12:50.670 Now, if your purpose is to try to quantitatively say 00:12:50.670 --> 00:12:51.630 something, then you really have to get 00:12:51.630 --> 00:12:53.610 into this equation, which usually is not done 00:12:53.610 --> 00:12:56.770 but would be the right thing to do for validation. 00:12:56.770 --> 00:13:01.740 Now the alternative view is disproving a model. 00:13:08.636 --> 00:13:14.200 But I just say that there's several ways this can happen. 00:13:14.200 --> 00:13:16.500 You can try to disprove a model, but you might also 00:13:16.500 --> 00:13:20.648 show that the theta values are incorrect. 00:13:27.360 --> 00:13:31.780 Or you might show that the experiment is wrong. 00:13:41.910 --> 00:13:43.410 These are all possibilities, reasons 00:13:43.410 --> 00:13:48.330 why the model and the data might not agree with each other. 00:13:48.330 --> 00:13:52.820 So this equation, it only holds if the model is really true, 00:13:52.820 --> 00:13:55.470 if the parameter values are all perfectly correct, 00:13:55.470 --> 00:14:00.790 if we know exactly what all the knob values are perfectly. 00:14:00.790 --> 00:14:02.700 If any of those things are not true, 00:14:02.700 --> 00:14:05.240 then you should have some discrepancy, 00:14:05.240 --> 00:14:07.270 and there should be a way to show it. 00:14:07.270 --> 00:14:08.820 And really what you're showing is 00:14:08.820 --> 00:14:13.650 that you'd observed some y that is 00:14:13.650 --> 00:14:15.270 very unlikely to be observed. 00:14:15.270 --> 00:14:18.720 So probably observing that y is very extremely unlikely 00:14:18.720 --> 00:14:20.470 if all these other things were true. 00:14:20.470 --> 00:14:24.210 So if all these things are true, and you compute this value, 00:14:24.210 --> 00:14:26.490 and this value's very tiny, then it 00:14:26.490 --> 00:14:29.910 makes you think that it's unlikely that you 00:14:29.910 --> 00:14:31.140 would have observed that. 00:14:31.140 --> 00:14:33.180 And therefore, you might try to use that as an argument to say 00:14:33.180 --> 00:14:34.290 that something must be wrong. 00:14:34.290 --> 00:14:36.840 The model's wrong, parameters are wrong, the knobs are wrong, 00:14:36.840 --> 00:14:39.180 something's wrong. 00:14:39.180 --> 00:14:40.970 My y values are wrong. 00:14:40.970 --> 00:14:42.870 It could be any of those things. 00:14:42.870 --> 00:14:46.599 So this is often the most exciting papers to publish. 00:14:46.599 --> 00:14:49.140 You publish a paper, you take some model that a lot of people 00:14:49.140 --> 00:14:50.030 believe. 00:14:50.030 --> 00:14:52.810 You tell them they're full of baloney, it's completely wrong. 00:14:52.810 --> 00:14:56.940 My great experiment shows you are completely wrong. 00:14:56.940 --> 00:14:59.495 And so you'll see a lot of these in Nature. 00:14:59.495 --> 00:15:00.870 I should warn you, a lot of those 00:15:00.870 --> 00:15:06.780 get retracted later, a very high retraction rate in Nature. 00:15:06.780 --> 00:15:08.250 Because they want to publish papers 00:15:08.250 --> 00:15:11.760 like that that show that the common view is incorrect, 00:15:11.760 --> 00:15:12.900 and sometimes it's true. 00:15:12.900 --> 00:15:14.430 But oftentimes the common view is actually correct, 00:15:14.430 --> 00:15:15.730 and there's something wrong with the experiment, 00:15:15.730 --> 00:15:18.146 or the interpretation, or how they computed this equation, 00:15:18.146 --> 00:15:19.040 or whatever. 00:15:19.040 --> 00:15:21.770 And so actually it turns out the common view is perfectly fine, 00:15:21.770 --> 00:15:26.280 and it's just that the foolish authors went off on a tangent. 00:15:26.280 --> 00:15:28.080 And then they have to six months later 00:15:28.080 --> 00:15:29.790 publish a retraction, by the way, sorry, 00:15:29.790 --> 00:15:31.322 paper was completely wrong. 00:15:31.322 --> 00:15:32.530 And so you see a lot of that. 00:15:36.570 --> 00:15:39.210 So that's a second kind of thing. 00:15:39.210 --> 00:15:42.870 And we'll talk more about that a little bit later, too. 00:15:42.870 --> 00:15:51.920 And then another thing is I'll relax my assumptions. 00:15:51.920 --> 00:15:56.360 So I'll say, well, I'm sure that the model is true, 00:15:56.360 --> 00:15:59.540 and I'm sure that my knob settings are perfect, 00:15:59.540 --> 00:16:01.530 and I know what they are. 00:16:01.530 --> 00:16:04.460 But I'm not really sure about all the parameters. 00:16:04.460 --> 00:16:06.710 And therefore I want to use the experiment 00:16:06.710 --> 00:16:10.134 to try to refine parameter values. 00:16:19.580 --> 00:16:22.910 So I'm trying to take my y's that I measure and somehow 00:16:22.910 --> 00:16:26.630 infer something about the thetas. 00:16:26.630 --> 00:16:29.680 And this is a very common thing to do. 00:16:29.680 --> 00:16:33.570 So in my group we've tried to measure the rate coefficient 00:16:33.570 --> 00:16:35.220 for a reaction. 00:16:35.220 --> 00:16:37.297 We believe there is value of that theta, 00:16:37.297 --> 00:16:39.630 and in fact, we probably have an estimate of what it is. 00:16:39.630 --> 00:16:41.540 But we're not sure of the exact number, 00:16:41.540 --> 00:16:43.790 and we'd like to do an experiment to refine the number 00:16:43.790 --> 00:16:45.600 and get it more accurately determined. 00:16:45.600 --> 00:16:49.150 So that's another useful thing to do. 00:16:49.150 --> 00:16:54.100 And this leads into two somewhat different points of view 00:16:54.100 --> 00:16:55.870 about this. 00:16:55.870 --> 00:16:59.760 One you've probably done already called least squares fitting. 00:16:59.760 --> 00:17:00.820 That's one view. 00:17:00.820 --> 00:17:03.580 And the other is this Bayesian view 00:17:03.580 --> 00:17:06.119 that I'll tell you about next. 00:17:06.119 --> 00:17:10.609 So there's sort of A and B. There's 00:17:10.609 --> 00:17:14.920 one that I'll call Bayesian, and one I'll call least squares. 00:17:18.764 --> 00:17:20.680 They're sort of related to each other, but not 00:17:20.680 --> 00:17:22.544 exactly the same conceptually. 00:17:22.544 --> 00:17:23.710 So I'll try to explain that. 00:17:27.069 --> 00:17:30.190 So the Bayesian view is probabilistic, 00:17:30.190 --> 00:17:38.250 so it's actually pretty straightforward to write down. 00:17:38.250 --> 00:17:49.170 Remember that we wrote that the probability of A and B 00:17:49.170 --> 00:17:52.880 is equal to the probability of A times 00:17:52.880 --> 00:17:58.980 the probability of B given A, and it's also 00:17:58.980 --> 00:18:02.646 equal to the probability of B times the probability 00:18:02.646 --> 00:18:07.650 of A given B. And what we have here is 00:18:07.650 --> 00:18:09.440 one of these conditional probabilities, 00:18:09.440 --> 00:18:11.330 if the thetas have a certain value, 00:18:11.330 --> 00:18:13.290 this is a certain probability. 00:18:13.290 --> 00:18:16.410 So I should be able to use that formula somehow. 00:18:16.410 --> 00:18:21.770 So I can write down that the probability of measuring 00:18:21.770 --> 00:18:35.920 y given theta is equal to the probability of y times 00:18:35.920 --> 00:18:43.280 the probability of theta given y divided 00:18:43.280 --> 00:18:49.960 by the probability of theta. 00:18:49.960 --> 00:18:52.460 So I just took this formula, and I plugged in y's and thetas 00:18:52.460 --> 00:18:54.710 instead of A's and B's. 00:18:54.710 --> 00:18:57.770 So I said, these two are equal to each other. 00:18:57.770 --> 00:18:59.510 Rearranged it so then I can rewrite this. 00:18:59.510 --> 00:19:01.700 This is the way we have it in here, probably 00:19:01.700 --> 00:19:03.800 of measuring y given theta. 00:19:03.800 --> 00:19:05.370 Let's flip it around. 00:19:05.370 --> 00:19:11.000 So probability of theta given that we measured 00:19:11.000 --> 00:19:21.740 y is equal to the probability of theta times 00:19:21.740 --> 00:19:28.430 the probability of observing y if theta was true 00:19:28.430 --> 00:19:31.823 divided by the probability of y. 00:19:36.725 --> 00:19:37.850 Terrible handwriting there. 00:19:41.470 --> 00:19:43.147 That's just algebra. 00:19:43.147 --> 00:19:44.480 So this is what we want to know. 00:19:44.480 --> 00:19:46.646 We want to know, what's the probability distribution 00:19:46.646 --> 00:19:48.090 of the parameter values y? 00:19:48.090 --> 00:19:50.490 Because some of them are uncertain. 00:19:50.490 --> 00:19:52.680 Now, before we started the experiment, 00:19:52.680 --> 00:19:56.220 we had some idea of what the ranges were 00:19:56.220 --> 00:19:59.340 for all the primary values. 00:19:59.340 --> 00:20:01.620 Like I'm trying to measure a rate coefficient. 00:20:01.620 --> 00:20:06.000 I know from experience with other similar reactions, 00:20:06.000 --> 00:20:08.550 from a quantum chemistry calculation, 00:20:08.550 --> 00:20:12.420 from some indirect evidence, from some other more 00:20:12.420 --> 00:20:14.910 complicated experiment, I have some idea 00:20:14.910 --> 00:20:17.274 that this rate coefficient has to be in a certain range. 00:20:17.274 --> 00:20:18.690 Now, it could be pretty uncertain. 00:20:18.690 --> 00:20:21.150 It might be five orders of magnitude uncertain. 00:20:21.150 --> 00:20:23.580 But I know it's not less than 0. 00:20:23.580 --> 00:20:26.730 I know it can't be faster than the diffusion limit, how 00:20:26.730 --> 00:20:28.590 fast things can come together. 00:20:28.590 --> 00:20:30.905 So for sure I know some range, and oftentimes I 00:20:30.905 --> 00:20:32.640 know a much narrower range than that. 00:20:32.640 --> 00:20:35.700 So I have some information about these parameter values 00:20:35.700 --> 00:20:36.709 before I even start. 00:20:36.709 --> 00:20:38.250 Some of the parameters of the model I 00:20:38.250 --> 00:20:39.960 know perfectly, or pretty well. 00:20:39.960 --> 00:20:43.090 So you know maybe there's a Planck's constant, or the heat 00:20:43.090 --> 00:20:45.090 of formation of one of my chemicals or something 00:20:45.090 --> 00:20:46.590 like that shows up in the numbers, 00:20:46.590 --> 00:20:48.860 and I might know that parameter really pretty accurately. 00:20:48.860 --> 00:20:51.026 Whereas the particular rate coefficient I care about 00:20:51.026 --> 00:20:53.940 is the thing I really don't know very well. 00:20:53.940 --> 00:20:57.180 So some of these have tight probability distributions 00:20:57.180 --> 00:21:00.720 ahead of time, and some of them have loose ones. 00:21:00.720 --> 00:21:01.900 And this thing has a name. 00:21:01.900 --> 00:21:02.816 It's called the prior. 00:21:05.670 --> 00:21:09.864 And it's our prior information before we did the experiment. 00:21:09.864 --> 00:21:11.780 And this one, after we've done the experiment, 00:21:11.780 --> 00:21:12.770 we're going to change it. 00:21:12.770 --> 00:21:14.395 So we're going to say previously people 00:21:14.395 --> 00:21:17.474 thought that the parameters all lied in these certain ranges. 00:21:17.474 --> 00:21:18.890 And now I'm going to get a tighter 00:21:18.890 --> 00:21:20.973 range, because I have some additional experimental 00:21:20.973 --> 00:21:22.050 information. 00:21:22.050 --> 00:21:23.430 So this is called the posterior. 00:21:28.650 --> 00:21:31.440 This means before, it means after. 00:21:31.440 --> 00:21:33.330 So this is what I know about parameter values 00:21:33.330 --> 00:21:36.920 before and after the experiment. 00:21:36.920 --> 00:21:41.050 This is the formula that I have over there. 00:21:41.050 --> 00:21:44.624 It's a probability that if the thetas had a certain value, 00:21:44.624 --> 00:21:46.290 probably would have observed what I saw. 00:21:46.290 --> 00:21:46.790 Yeah? 00:21:46.790 --> 00:21:50.234 AUDIENCE: Which one refers to which? 00:21:50.234 --> 00:21:53.420 WILLIAM GREEN: Sorry, this is the prior, 00:21:53.420 --> 00:21:54.616 this is the posterior. 00:21:59.450 --> 00:22:02.115 And those of you who are paying attention to notation 00:22:02.115 --> 00:22:03.740 realize I'm not doing this very nicely. 00:22:03.740 --> 00:22:06.639 Because these are continuous variables, 00:22:06.639 --> 00:22:08.180 and I'm writing capital PRs, and they 00:22:08.180 --> 00:22:10.880 should be not be capital PRs, it should be probably density 00:22:10.880 --> 00:22:12.516 functions instead. 00:22:12.516 --> 00:22:13.640 So let's rewrite it nicely. 00:22:18.140 --> 00:22:23.370 So the probability of theta given y, 00:22:23.370 --> 00:22:26.780 probability density is equal to the probability density 00:22:26.780 --> 00:22:37.422 of theta initially times the probability of y given theta, 00:22:37.422 --> 00:22:41.470 [INAUDIBLE] density divided by the probability density of-- 00:22:44.900 --> 00:22:47.420 all right? 00:22:47.420 --> 00:22:48.920 And what I just basically did was 00:22:48.920 --> 00:22:51.378 this is the correct equation the previous other one was all 00:22:51.378 --> 00:22:54.807 multiplied by d theta dy, it shouldn't be done that way. 00:22:54.807 --> 00:22:55.390 So this is OK? 00:22:59.640 --> 00:23:02.100 Now this is the prior information 00:23:02.100 --> 00:23:04.670 I have about the parameter values. 00:23:04.670 --> 00:23:08.010 I know that they have to fall into some ranges. 00:23:08.010 --> 00:23:11.084 And really all I'm doing is I'm correcting that information. 00:23:11.084 --> 00:23:13.500 I'm improving the information to tighten the distribution. 00:23:13.500 --> 00:23:18.910 So initially I know that my rate constant, here's 00:23:18.910 --> 00:23:20.700 my rate constant. 00:23:20.700 --> 00:23:24.727 I know that it's got to be greater than zero, 00:23:24.727 --> 00:23:27.060 and I don't think it's really down there at zero anyway. 00:23:27.060 --> 00:23:28.351 I think it's somewhere in here. 00:23:28.351 --> 00:23:29.670 I really don't know much. 00:23:29.670 --> 00:23:31.740 And I really don't think it's all up at the diffusion limit, 00:23:31.740 --> 00:23:33.740 and no way it's higher than the diffusion limit. 00:23:33.740 --> 00:23:35.910 So that's my initial information that I 00:23:35.910 --> 00:23:41.340 have about the probability distribution of K. 00:23:41.340 --> 00:23:43.800 So it's the rate coefficient I want to know, 00:23:43.800 --> 00:23:45.480 and I know it's bigger than zero, 00:23:45.480 --> 00:23:47.480 and I know it's less than infinity. 00:23:47.480 --> 00:23:49.500 And actually I know there's some physical limit, 00:23:49.500 --> 00:23:51.291 it can't be higher than something or other. 00:23:53.509 --> 00:23:55.300 And you can do this for any problem, right? 00:23:55.300 --> 00:23:56.990 I give you any parameter, you should 00:23:56.990 --> 00:23:58.909 be able to tell me something about it. 00:23:58.909 --> 00:24:00.950 You might be uncertain by 20 orders of magnitude, 00:24:00.950 --> 00:24:02.866 but at least you have an error bar some width. 00:24:02.866 --> 00:24:04.742 It can't be anything, right? 00:24:04.742 --> 00:24:06.950 A lot of parameters have to be positive, for example. 00:24:06.950 --> 00:24:08.390 You know that. 00:24:08.390 --> 00:24:10.017 And you usually know something. 00:24:10.017 --> 00:24:12.100 you might not think you know anything, but you do, 00:24:12.100 --> 00:24:13.683 you actually do know before you start. 00:24:13.683 --> 00:24:17.110 So you actually know, this is the P of theta to start with. 00:24:19.880 --> 00:24:23.605 And after I've done the experiment, 00:24:23.605 --> 00:24:25.980 hopefully you're going to know more information about it. 00:24:25.980 --> 00:24:29.610 I might know that this quantity here 00:24:29.610 --> 00:24:33.120 is going to be like a Gaussian distribution. 00:24:33.120 --> 00:24:35.980 It might have a kind of goofball dependence on theta. 00:24:35.980 --> 00:24:38.110 I should comment that. 00:24:38.110 --> 00:24:42.740 Notice how theta appears inside F. 00:24:42.740 --> 00:24:44.680 So theta's up in the exponent. 00:24:44.680 --> 00:24:49.520 It's sort of inside a Gaussian, but it's like processed by F 00:24:49.520 --> 00:24:53.240 and so the observable might have a pretty goofball dependence 00:24:53.240 --> 00:24:55.240 on this rate coefficient. 00:24:55.240 --> 00:24:58.640 So this thing could be some weird thing. 00:24:58.640 --> 00:25:03.549 But for sure, when I change theta so this changes a lot, 00:25:03.549 --> 00:25:05.340 it's going to make a pretty big difference. 00:25:05.340 --> 00:25:08.700 Because up inside the exponent of a Gaussian, 00:25:08.700 --> 00:25:11.940 so it's going to drop off a lot somewhere. 00:25:11.940 --> 00:25:15.370 So I should get something that looks something like this 00:25:15.370 --> 00:25:18.210 maybe for my experiment. 00:25:18.210 --> 00:25:24.060 So this one is P of K initially, the prior. 00:25:24.060 --> 00:25:28.010 This one is P of yk. 00:25:31.910 --> 00:25:34.010 And what this equation says is I want 00:25:34.010 --> 00:25:37.330 to multiply those two together. 00:25:37.330 --> 00:25:39.240 And so I'm going to multiply this times this, 00:25:39.240 --> 00:25:50.180 and I'm going to get some new thing that's 00:25:50.180 --> 00:25:55.432 something like that when I multiply this time that. 00:25:55.432 --> 00:25:58.340 Is that OK? 00:25:58.340 --> 00:26:03.090 And so that's my new numerator of this equation. 00:26:03.090 --> 00:26:05.226 Now this denominator doesn't make too much sense. 00:26:05.226 --> 00:26:06.600 This says, what's the probability 00:26:06.600 --> 00:26:13.035 that I measured the mean I measured, given nothing? 00:26:13.035 --> 00:26:14.910 So this is sort of like the prior probability 00:26:14.910 --> 00:26:16.750 that I would have measured it or something. 00:26:16.750 --> 00:26:17.833 I don't know what this is. 00:26:17.833 --> 00:26:22.100 So instead what people do, is they say, forget this. 00:26:22.100 --> 00:26:27.234 But instead, let's multiply this by a constant that's 00:26:27.234 --> 00:26:29.400 going to normalize it to make it probability density 00:26:29.400 --> 00:26:31.370 so that it integrates to one. 00:26:34.370 --> 00:26:36.950 So that's the way Bayes' theorem is used. 00:26:36.950 --> 00:26:38.540 This is called Bayesian analysis. 00:26:42.140 --> 00:26:43.850 And so what it's telling you is how 00:26:43.850 --> 00:26:49.220 to take your experimental information as expressed 00:26:49.220 --> 00:26:59.010 in this formula and use all your previous information 00:26:59.010 --> 00:27:03.030 about the parameters, put them all together, 00:27:03.030 --> 00:27:06.490 now we have a cumulative information about everything. 00:27:06.490 --> 00:27:09.760 So we have some parameters that came into our problem 00:27:09.760 --> 00:27:12.370 into my experiment, but from previous work, 00:27:12.370 --> 00:27:14.630 I also knew something about those parameters. 00:27:14.630 --> 00:27:16.330 Now I put it all together and I get 00:27:16.330 --> 00:27:19.180 a new value of probability distribution 00:27:19.180 --> 00:27:21.020 of those parameters. 00:27:21.020 --> 00:27:23.530 And if my expert was really good, 00:27:23.530 --> 00:27:27.237 it would make this really tight [WHOOSHING SOUND].. 00:27:27.237 --> 00:27:29.070 And then when I multiply these two together, 00:27:29.070 --> 00:27:32.470 it's going to make this really sharp, 00:27:32.470 --> 00:27:35.380 and we have a really good value of k. 00:27:35.380 --> 00:27:37.250 So that's like the ideal case if I 00:27:37.250 --> 00:27:39.440 have a really great, well-designed experiment 00:27:39.440 --> 00:27:44.090 executed perfectly with great precision, then I can do this. 00:27:44.090 --> 00:27:46.214 More generally, when I don't think about it, 00:27:46.214 --> 00:27:47.630 I get some distribution like this. 00:27:50.210 --> 00:27:52.830 I still learn something compared to what I had before, 00:27:52.830 --> 00:27:54.440 but it might not be much. 00:27:54.440 --> 00:27:57.410 So now I can end up with some distribution that's 00:27:57.410 --> 00:27:59.630 a little tighter than before. 00:28:04.130 --> 00:28:07.190 So is this OK so far? 00:28:07.190 --> 00:28:11.040 All right, now this is super simple. 00:28:11.040 --> 00:28:13.200 I didn't have to solve anything, all 00:28:13.200 --> 00:28:17.120 I had to do was multiply two distributions together. 00:28:17.120 --> 00:28:20.810 So in some respects, this is what you should always do. 00:28:20.810 --> 00:28:22.962 All you do is you take your experiment, 00:28:22.962 --> 00:28:25.170 you multiply the probability distribution corresponds 00:28:25.170 --> 00:28:27.334 to your experiment times the prior, 00:28:27.334 --> 00:28:29.750 and you get some posterior, and that's why new information 00:28:29.750 --> 00:28:31.340 about the distribution. 00:28:31.340 --> 00:28:33.870 And if I have a distribution like this, 00:28:33.870 --> 00:28:37.370 suppose this is my new distribution here, 00:28:37.370 --> 00:28:41.240 I can still get it central value, that's my mean value, k. 00:28:41.240 --> 00:28:44.010 I can get an estimate of the range of k. 00:28:44.010 --> 00:28:45.780 So I end up with a k plus or minus 00:28:45.780 --> 00:28:50.870 dk maybe, from just looking at the plot. 00:28:50.870 --> 00:28:53.670 In fact, I never even have to evaluate what this constant is 00:28:53.670 --> 00:28:54.540 in order to do this. 00:28:54.540 --> 00:28:57.830 I can just go look at the plot, see where the peak is, 00:28:57.830 --> 00:29:01.030 figure out the width, and I can report now 00:29:01.030 --> 00:29:03.460 because in my experiment, k plus or minus 00:29:03.460 --> 00:29:05.590 dk is more precisely determined than it was before. 00:29:09.150 --> 00:29:12.840 Now, a practical challenge with this 00:29:12.840 --> 00:29:15.980 is that theta is usually a lot of parameters. 00:29:15.980 --> 00:29:19.700 And I only drew the plot here in one dimension, 00:29:19.700 --> 00:29:23.280 but really it's a multi-dimensional plot. 00:29:23.280 --> 00:29:27.205 So really what looked like, suppose I had two parameters. 00:29:27.205 --> 00:29:29.580 I had my k I care about, and I have some other parameter, 00:29:29.580 --> 00:29:34.870 theta 2, that also it shows up in my model. 00:29:34.870 --> 00:29:39.170 And say, before I started, I knew 00:29:39.170 --> 00:29:43.570 theta 2 fell in this kind of range, 00:29:43.570 --> 00:29:47.950 and I knew k fell in this kind of range. 00:29:47.950 --> 00:29:51.880 So really before I started, if I think what it looks like, 00:29:51.880 --> 00:29:57.440 I really had sort of a blobby rectangular contour 00:29:57.440 --> 00:30:02.460 plot, where I think it's more likely that the k 00:30:02.460 --> 00:30:05.506 value and the theta 2 value are somewhere in this range. 00:30:05.506 --> 00:30:08.130 And the most likely one is maybe somewhere in the middle there. 00:30:08.130 --> 00:30:09.780 But I really didn't know much. 00:30:09.780 --> 00:30:12.720 So it could be anywhere in this whole blob. 00:30:12.720 --> 00:30:15.960 Now, when I do the experiment, the experimental value 00:30:15.960 --> 00:30:19.240 depends of both k and theta 2. 00:30:19.240 --> 00:30:22.510 And commonly what'll happen is that the distribution 00:30:22.510 --> 00:30:25.190 from the experiment-- 00:30:25.190 --> 00:30:28.718 need color chalk here. 00:30:28.718 --> 00:30:31.100 Let's get rid of these guys. 00:30:31.100 --> 00:30:33.932 So this is my probably distribution, there's my prior. 00:30:33.932 --> 00:30:37.850 If I do the experiment, maybe I'll have something like this. 00:30:37.850 --> 00:30:40.510 That the experiment says that the guys 00:30:40.510 --> 00:30:47.094 have to be somewhere in the contour plot like this. 00:30:47.094 --> 00:30:48.510 Because I can get pretty good fits 00:30:48.510 --> 00:30:50.080 of the data with different values of k 00:30:50.080 --> 00:30:52.038 as long as I compensate with the value theta 2. 00:30:54.780 --> 00:30:58.370 Now I multiply these two dimensional functions. 00:30:58.370 --> 00:31:02.410 The original is a blob function, and this 00:31:02.410 --> 00:31:05.020 is a stretched out blob. 00:31:05.020 --> 00:31:08.140 And I multiply a stretched out blob times a fat blob, 00:31:08.140 --> 00:31:09.970 I get some stretched out blob that 00:31:09.970 --> 00:31:12.170 looks something like the intersection of these guys. 00:31:12.170 --> 00:31:14.587 And so I end up with some kind of blob like that. 00:31:14.587 --> 00:31:15.670 I'll draw it really thick. 00:31:15.670 --> 00:31:21.940 So this is my posterior, some kind of blob like this. 00:31:21.940 --> 00:31:25.060 So now I know a little bit more about these two parameters 00:31:25.060 --> 00:31:29.370 than I did before I started because of my experiment. 00:31:29.370 --> 00:31:30.417 Is this OK? 00:31:30.417 --> 00:31:32.500 I really can't say I know what the real value of k 00:31:32.500 --> 00:31:34.005 is, or the value of theta 2. 00:31:34.005 --> 00:31:36.439 But I know that combinations of k 00:31:36.439 --> 00:31:38.730 and theta 2 that are sort of in this range, all of them 00:31:38.730 --> 00:31:40.438 will give me pretty good fits to my data, 00:31:40.438 --> 00:31:43.500 and also be consistent with all the previous information I have 00:31:43.500 --> 00:31:46.500 about those parameter values. 00:31:46.500 --> 00:31:48.880 Is that all right? 00:31:48.880 --> 00:31:51.394 Now, I drew it with two parameters. 00:31:51.394 --> 00:31:53.560 In a lot of models we have, we have five parameters, 00:31:53.560 --> 00:31:55.643 six parameters, seven parameters, nine parameters, 00:31:55.643 --> 00:31:56.540 14 parameters. 00:31:56.540 --> 00:31:58.360 We have a lot of parameters. 00:31:58.360 --> 00:32:01.660 And so then we try to make this plot, even how 00:32:01.660 --> 00:32:05.784 to display the plot is going to be a little problematic. 00:32:05.784 --> 00:32:06.700 But it's there, right? 00:32:06.700 --> 00:32:10.630 And somehow, we still narrowed down the hypervolume 00:32:10.630 --> 00:32:12.970 in the parameter space from whatever 00:32:12.970 --> 00:32:15.580 it was to begin with to now we know 00:32:15.580 --> 00:32:16.850 something a little bit better. 00:32:16.850 --> 00:32:19.389 We have a narrower range of the parameters that 00:32:19.389 --> 00:32:21.680 would be consistent with all the information available, 00:32:21.680 --> 00:32:24.400 including my new experiment. 00:32:24.400 --> 00:32:26.560 And then the next guy does his experiment, 00:32:26.560 --> 00:32:29.350 and he does an experiment that shows that these guys have 00:32:29.350 --> 00:32:32.320 to be somewhere in this range in order to be 00:32:32.320 --> 00:32:34.570 consistent with his experiment. 00:32:34.570 --> 00:32:37.660 And so now I can narrow down the range 00:32:37.660 --> 00:32:39.490 to be something like that. 00:32:39.490 --> 00:32:41.240 And the next person does their experiment, 00:32:41.240 --> 00:32:42.540 and they get something else, and something else, 00:32:42.540 --> 00:32:43.360 and something else. 00:32:43.360 --> 00:32:46.030 And eventually by 2050, we have a pretty nice determination 00:32:46.030 --> 00:32:48.970 of the parameter values. 00:32:48.970 --> 00:32:51.760 So that's the advance of science, 00:32:51.760 --> 00:32:55.212 as drawn in chalk by Professor Green at the board. 00:32:59.890 --> 00:33:02.020 So this is a very important way to think 00:33:02.020 --> 00:33:04.740 about it, is what you're doing when you do experiments, 00:33:04.740 --> 00:33:09.730 is you're generally restricting the range of parameter space 00:33:09.730 --> 00:33:12.030 that's still consistent with everything. 00:33:12.030 --> 00:33:13.870 And when we mean consistent, we mean 00:33:13.870 --> 00:33:16.286 that the probability that you would have observed what you 00:33:16.286 --> 00:33:18.174 did observe is reasonably high. 00:33:18.174 --> 00:33:20.590 We'll still have to come back to quantitatively figure out 00:33:20.590 --> 00:33:21.796 what reasonably high means. 00:33:25.600 --> 00:33:29.440 Now, when you did this before when you were kids, 00:33:29.440 --> 00:33:31.450 nobody mentioned the word Bayes, or Bayesian, 00:33:31.450 --> 00:33:34.630 or conditional probabilities, right? 00:33:34.630 --> 00:33:38.077 So they just said, oh, just do a least squares fit. 00:33:38.077 --> 00:33:39.410 How many of you did that before? 00:33:42.930 --> 00:33:45.660 So somebody told me before, forget this stuff, 00:33:45.660 --> 00:33:47.790 we're going to never even mention this stuff. 00:33:47.790 --> 00:33:49.581 We're just going to do a least squares fit. 00:33:56.300 --> 00:33:59.060 Now, where did the least squares fit idea came from? 00:33:59.060 --> 00:34:01.670 It came from looking at this formula and saying, 00:34:01.670 --> 00:34:06.020 you know, this is the deviations between the experiment 00:34:06.020 --> 00:34:11.719 and the model prediction, and I weigh them somehow, 00:34:11.719 --> 00:34:13.061 and I have the square. 00:34:13.061 --> 00:34:14.810 And that's the thing I want to make small. 00:34:14.810 --> 00:34:20.239 If I have a high probability that what I observed really 00:34:20.239 --> 00:34:23.659 happened, or the probably I'm going to observe this, 00:34:23.659 --> 00:34:25.550 it's got to be that these guys have to be 00:34:25.550 --> 00:34:26.580 reasonably close to each other. 00:34:26.580 --> 00:34:27.610 They're really different. 00:34:27.610 --> 00:34:29.485 And it's going to be very small, because it's 00:34:29.485 --> 00:34:30.770 inside an exponential. 00:34:30.770 --> 00:34:32.659 And if those guys are really different, 00:34:32.659 --> 00:34:35.170 and the squared thing is really large, 00:34:35.170 --> 00:34:37.210 then the probability is incredibly small 00:34:37.210 --> 00:34:39.290 that I would have observed that. 00:34:39.290 --> 00:34:42.679 So we think that this thing should be small. 00:34:42.679 --> 00:34:47.600 And in fact, if I want to get the very best fit I can get, 00:34:47.600 --> 00:34:50.840 which means like the probability was the highest of what 00:34:50.840 --> 00:34:54.460 I observed in the real observation or something, 00:34:54.460 --> 00:34:56.492 then if I'm free to adjust one of these thetas, 00:34:56.492 --> 00:34:57.950 I can adjust the theta, try to make 00:34:57.950 --> 00:35:01.057 this thing like equal to zero, or small as I can. 00:35:01.057 --> 00:35:03.390 So that's where the concept of least squares comes from. 00:35:07.290 --> 00:35:11.360 Now, when you're doing least squares, 00:35:11.360 --> 00:35:14.236 you almost always have multiple parameters, 00:35:14.236 --> 00:35:16.610 and therefore you're going to have to have multiple data. 00:35:16.610 --> 00:35:20.234 And they can't just be a repeat of one number. 00:35:20.234 --> 00:35:22.400 Can't be your data, it's not sufficient to determine 00:35:22.400 --> 00:35:23.870 the parameters. 00:35:23.870 --> 00:35:25.710 So normally when you do an experiment, 00:35:25.710 --> 00:35:27.700 you have to change the knobs. 00:35:27.700 --> 00:35:30.660 We have to make measurements in a couple different conditions. 00:35:30.660 --> 00:35:31.970 Like for example, kinetics. 00:35:31.970 --> 00:35:35.120 You often want the Arrhenius A factor and the EA. 00:35:35.120 --> 00:35:37.092 And so I got to run the experiment 00:35:37.092 --> 00:35:39.050 in more than one temperature or I'm never going 00:35:39.050 --> 00:35:39.960 to be able to figure that out. 00:35:39.960 --> 00:35:42.080 So I have to change the temperature in my reactor. 00:35:42.080 --> 00:35:43.829 Make some measurements at one temperature, 00:35:43.829 --> 00:35:46.310 and make some measurements at a different temperature. 00:35:46.310 --> 00:35:49.040 And for almost everything in life that you want to measure, 00:35:49.040 --> 00:35:50.540 you're going to have to do this. 00:35:50.540 --> 00:35:52.940 You vary the concentration of your enzyme 00:35:52.940 --> 00:35:55.796 if you want to see how the enzyme kinetics depends 00:35:55.796 --> 00:35:57.170 on something. you can't just keep 00:35:57.170 --> 00:35:59.630 running exactly the same condition over and over. 00:35:59.630 --> 00:36:01.657 You'll get that number really precise, 00:36:01.657 --> 00:36:03.740 but it's not enough information to really fill out 00:36:03.740 --> 00:36:05.280 the parameters in your model. 00:36:05.280 --> 00:36:08.870 So you're going to have to run several different experiments 00:36:08.870 --> 00:36:10.640 with different knob settings. 00:36:10.640 --> 00:36:13.920 Also, normally we don't just measure one quantity, one 00:36:13.920 --> 00:36:15.170 observable in each experiment. 00:36:15.170 --> 00:36:17.790 We usually try to measure as many things as we can. 00:36:17.790 --> 00:36:19.527 So we actually have several observables 00:36:19.527 --> 00:36:21.860 at each knob setting, and we have several knob settings, 00:36:21.860 --> 00:36:23.152 so we have quite a lot of data. 00:36:23.152 --> 00:36:25.485 And each one of those is repeated a whole bunch of times 00:36:25.485 --> 00:36:28.400 so that we're confident that we can use this Gaussian formula. 00:36:28.400 --> 00:36:39.120 And so what we really have is the pi-th observable measured 00:36:39.120 --> 00:36:47.150 at the l-th knob position. 00:36:47.150 --> 00:36:48.740 Well, I'm sorry, l's not good either, 00:36:48.740 --> 00:36:51.684 it's used in your notes for something else. 00:36:51.684 --> 00:36:54.460 M, there you go. 00:36:54.460 --> 00:36:56.000 The m-th knob position. 00:36:56.000 --> 00:36:59.380 Now, normally you have several knobs, so that's a factor. 00:36:59.380 --> 00:37:03.720 And we have a lot of observables we can make at each position. 00:37:03.720 --> 00:37:08.340 So this thing is a measurement. 00:37:08.340 --> 00:37:12.650 And we repeated this multiple times so I can get the average. 00:37:12.650 --> 00:37:20.960 And we're also going to have a corresponding sigma I M, which 00:37:20.960 --> 00:37:25.230 is the variance of the mean. 00:37:25.230 --> 00:37:27.350 So it's variance, that's going to be divided 00:37:27.350 --> 00:37:30.020 by the square root of the number of repeats 00:37:30.020 --> 00:37:33.560 for that particular experiment and that particular observable. 00:37:33.560 --> 00:37:36.260 So this is your incoming data set, 00:37:36.260 --> 00:37:44.750 and you also have your model which predicts y model, 00:37:44.750 --> 00:37:54.878 it predicts the observable i for the sequel to fi of xk theta. 00:37:59.559 --> 00:38:01.100 So if you have certain knob settings, 00:38:01.100 --> 00:38:04.500 like certain temperature, and you have your parameter values, 00:38:04.500 --> 00:38:07.610 then you can calculate what the model thinks 00:38:07.610 --> 00:38:09.895 should be the observable value, and then you 00:38:09.895 --> 00:38:14.570 can actually measure it and measure its variance. 00:38:14.570 --> 00:38:15.960 so that's the normal situation. 00:38:15.960 --> 00:38:19.250 And now you want to figure out, are there some values 00:38:19.250 --> 00:38:24.789 of the theta that make the model and the data agree? 00:38:24.789 --> 00:38:26.580 And that's the least squares fitting thing. 00:38:26.580 --> 00:38:32.093 So what we can define as a new quantity, 00:38:32.093 --> 00:38:43.170 weight of the residual vector EJ, which is defined to be jk, 00:38:43.170 --> 00:38:45.220 to be consistent with Joe Scott's notes. 00:38:45.220 --> 00:38:47.822 AUDIENCE: Is k the same as m? 00:38:47.822 --> 00:38:49.280 WILLIAM GREEN: M is in oppositions, 00:38:49.280 --> 00:38:50.823 I'll tell you what k is in a second. 00:39:00.000 --> 00:39:02.890 m, sorry. 00:39:02.890 --> 00:39:03.780 Too many indices. 00:39:17.390 --> 00:39:21.240 OK, so this is the residual between the model position. 00:39:21.240 --> 00:39:25.500 And now-- oh man, I'm sorry, [INAUDIBLE].. 00:39:25.500 --> 00:39:31.730 K is an index over i and m. 00:39:31.730 --> 00:39:34.127 So k is just going to list all the data you got. 00:39:34.127 --> 00:39:36.210 Some of the data came from the same knob settings, 00:39:36.210 --> 00:39:37.835 some came from different knob settings. 00:39:37.835 --> 00:39:38.372 Yeah? 00:39:38.372 --> 00:39:42.660 AUDIENCE: So is x the m the y model i [INAUDIBLE]?? 00:39:42.660 --> 00:39:43.910 WILLIAM GREEN: Thank you, yes. 00:39:49.110 --> 00:39:57.480 y model i, I guess this is now k. 00:40:06.280 --> 00:40:08.005 And so k is one of these indices that 00:40:08.005 --> 00:40:10.630 carry-- you can bind two indices together and put them together 00:40:10.630 --> 00:40:12.940 just like you did in your PD problems. 00:40:12.940 --> 00:40:13.440 All right. 00:40:17.020 --> 00:40:20.541 Now, I wrote down this sigma. 00:40:20.541 --> 00:40:22.540 But actually if you're measuring multiple things 00:40:22.540 --> 00:40:24.070 at the same experiment, you should 00:40:24.070 --> 00:40:26.500 expect them to be correlated. 00:40:26.500 --> 00:40:29.010 So really what we should worry about 00:40:29.010 --> 00:40:34.930 is the c, the covariance matrix, that we defined last time. 00:40:34.930 --> 00:40:38.750 So you should also compute that thing. 00:40:38.750 --> 00:40:45.170 And so what you should expect is the probability density 00:40:45.170 --> 00:40:51.800 that we would measure any particular residuals 00:40:51.800 --> 00:40:53.900 if the model is true. 00:40:53.900 --> 00:40:56.842 And if we have these certain parameters, theta, 00:40:56.842 --> 00:41:02.715 this should be equal to 2 pi negative k over 2 00:41:02.715 --> 00:41:10.200 the determinant of c negative 1.2 exponential 00:41:10.200 --> 00:41:22.180 of negative 1/2, epsilon transpose c inverse epsilon. 00:41:22.180 --> 00:41:26.671 So this is the multi measurement version of the same equation 00:41:26.671 --> 00:41:27.170 here. 00:41:31.420 --> 00:41:35.410 So this is the quantity that we think 00:41:35.410 --> 00:41:40.150 should be small if we have good parameter values 00:41:40.150 --> 00:41:42.019 and we did a good experiment. 00:41:42.019 --> 00:41:43.810 Actually, even when we did bad experiments, 00:41:43.810 --> 00:41:46.210 still should be small if we have good parameter values. 00:41:48.990 --> 00:41:51.360 And that's because the c's, if we did a bad experiment, 00:41:51.360 --> 00:41:53.330 we'll have a high variance or something, 00:41:53.330 --> 00:41:55.780 then we should see the c's will give us weightings 00:41:55.780 --> 00:41:57.420 that will reflect that. 00:41:57.420 --> 00:41:58.580 Yeah? 00:41:58.580 --> 00:42:00.980 AUDIENCE: [INAUDIBLE] 00:42:00.980 --> 00:42:02.900 WILLIAM GREEN: Is that-- 00:42:02.900 --> 00:42:07.700 AUDIENCE: So you have the next [INAUDIBLE] K [INAUDIBLE].. 00:42:07.700 --> 00:42:13.410 WILLIAM GREEN: Oh I'm sorry, this is the capital K, 00:42:13.410 --> 00:42:17.810 this is the number of data points. 00:42:17.810 --> 00:42:24.748 So little k is equal to 1 to capital K. 00:42:24.748 --> 00:42:28.140 AUDIENCE: So does capital K count for both experiments? 00:42:28.140 --> 00:42:32.330 WILLIAM GREEN: It's the number of distinct data values 00:42:32.330 --> 00:42:34.740 after you've already averaged over repeats. 00:42:34.740 --> 00:42:39.960 So you do m experiments, at each experiment you measure capital 00:42:39.960 --> 00:42:42.270 I observables. 00:42:42.270 --> 00:42:45.860 So it's like m times I, so K. If you measured everything 00:42:45.860 --> 00:42:50.010 in every experiment, it's equal to I times m. 00:42:59.060 --> 00:43:04.130 Now there's two ways that people approach this in literature. 00:43:04.130 --> 00:43:05.720 The fancy way is you say, you know, 00:43:05.720 --> 00:43:09.490 this covariance matrix comes in in a pretty important way 00:43:09.490 --> 00:43:11.870 into this probability distribution function. 00:43:11.870 --> 00:43:14.270 And so maybe I need to worry a lot about whether I really 00:43:14.270 --> 00:43:16.500 know the covariance matrix. 00:43:16.500 --> 00:43:22.440 And my uncertainty in the mean drops pretty fast 00:43:22.440 --> 00:43:26.020 as I do averaging, but I'm not so confident 00:43:26.020 --> 00:43:29.350 that my answer in the covariance matrix was small. 00:43:29.350 --> 00:43:31.980 So what people do sometimes is they'll 00:43:31.980 --> 00:43:40.940 try to vary both c and theta, and try to get a best fit where 00:43:40.940 --> 00:43:41.750 they're varying c. 00:43:41.750 --> 00:43:43.458 But then they have additional constraints 00:43:43.458 --> 00:43:46.190 on c that c has to satisfy the equations you gave last time 00:43:46.190 --> 00:43:49.384 about how you calculate the covariance matrix from data. 00:43:49.384 --> 00:43:51.050 And so I was saying, well, I want this c 00:43:51.050 --> 00:43:54.050 to personify these equations pretty well, 00:43:54.050 --> 00:44:02.930 but true covariance of the world of the system 00:44:02.930 --> 00:44:04.370 is not the same as what I actually 00:44:04.370 --> 00:44:08.460 measure by just measuring, say, five repeats of an experiment. 00:44:08.460 --> 00:44:10.722 And so I might want to vary the c. 00:44:10.722 --> 00:44:14.120 You try to vary the c, turns out to be kind of complicated math, 00:44:14.120 --> 00:44:15.770 so not many people do it. 00:44:15.770 --> 00:44:17.646 Even though conceptually it makes some sense, 00:44:17.646 --> 00:44:19.686 you should worry about the fact you're not really 00:44:19.686 --> 00:44:20.810 sure about the covariance. 00:44:20.810 --> 00:44:22.950 So what a lot of people do is they say, 00:44:22.950 --> 00:44:25.460 let's just use the c that's computed from the formulas 00:44:25.460 --> 00:44:27.180 I gave you last time experimentally. 00:44:27.180 --> 00:44:34.190 So just say, let's just take c experimental, put them in here. 00:44:34.190 --> 00:44:36.600 And now this is a constant. 00:44:36.600 --> 00:44:40.440 And now the only thing that varies in this problem 00:44:40.440 --> 00:44:43.290 is thetas which come into the epsilons. 00:44:43.290 --> 00:44:47.290 Because the epsilons depend on theta. 00:44:47.290 --> 00:44:49.980 And so in that case, I can just try 00:44:49.980 --> 00:44:53.390 to maximize this probability. 00:44:53.390 --> 00:44:55.990 And what that happens to do is to minimize 00:44:55.990 --> 00:44:59.330 this quantity in the exponent. 00:44:59.330 --> 00:45:02.650 And so all I need to do is say, for example, 00:45:02.650 --> 00:45:11.500 theta best is equal to arg [INAUDIBLE] theta epsilon 00:45:11.500 --> 00:45:16.737 of theta c epsilon. 00:45:22.550 --> 00:45:24.550 And so this is the least squares fitting problem 00:45:24.550 --> 00:45:26.320 that you guys have probably done before. 00:45:26.320 --> 00:45:27.695 And probably what you did was you 00:45:27.695 --> 00:45:30.580 assumed I had perfectly uncorrelated data, 00:45:30.580 --> 00:45:32.170 and all my errors were the same. 00:45:32.170 --> 00:45:35.380 And so c, which is the identity matrix, and I took it out. 00:45:35.380 --> 00:45:36.521 Probably did that before? 00:45:36.521 --> 00:45:39.290 Yeah, OK. 00:45:39.290 --> 00:45:42.710 That's pretty dangerous to do, I'd say. 00:45:42.710 --> 00:45:45.400 What people do a lot, which is a little bit less dangerous, 00:45:45.400 --> 00:45:47.480 is at least say, well, you know, when 00:45:47.480 --> 00:45:53.540 I measure the concentration of species x by GC, 00:45:53.540 --> 00:45:57.020 I have an error bar of plus or minus 5%. 00:45:57.020 --> 00:46:00.080 And when I measure the temperature 00:46:00.080 --> 00:46:02.525 with my thermocouple, I have an error bar 00:46:02.525 --> 00:46:04.790 of plus or minus 2 degrees. 00:46:04.790 --> 00:46:08.360 And so the variances of these guys should be a lot different, 00:46:08.360 --> 00:46:11.180 temperature and GC signal. 00:46:11.180 --> 00:46:14.895 And therefore I definitely need to weight my deviation somehow. 00:46:14.895 --> 00:46:16.520 And it's really what you do is you keep 00:46:16.520 --> 00:46:19.551 the diagonal entries of this. 00:46:19.551 --> 00:46:20.300 That's often done. 00:46:20.300 --> 00:46:23.570 And we just forget the fact that they might be covariant. 00:46:23.570 --> 00:46:25.040 But if you've done the experiments, 00:46:25.040 --> 00:46:26.310 you actually do have enough information 00:46:26.310 --> 00:46:27.840 to compute this thing anyway, so you might as well just 00:46:27.840 --> 00:46:29.068 use the experimental value. 00:46:33.860 --> 00:46:35.340 So this is the least squares thing. 00:46:35.340 --> 00:46:38.502 And let's think, what the heck is this doing? 00:46:38.502 --> 00:46:40.960 We're saying, all of a sudden we grabbed all the parameters 00:46:40.960 --> 00:46:42.626 in the model, which might include things 00:46:42.626 --> 00:46:45.940 like the molecular weight of hydrogen or something. 00:46:45.940 --> 00:46:49.120 And we can find the very best values 00:46:49.120 --> 00:46:52.445 that would make our data match the experiment as best as 00:46:52.445 --> 00:46:52.945 possible. 00:46:55.630 --> 00:46:57.820 And in some sense, that's great, we 00:46:57.820 --> 00:47:00.310 know the best values and parameters for our experiment. 00:47:00.310 --> 00:47:02.170 But of course, if we vary the molecular weight of hydrogen, 00:47:02.170 --> 00:47:04.330 it's going to screw up somebody else's experiment. 00:47:04.330 --> 00:47:05.830 Because somebody else did some other experiment 00:47:05.830 --> 00:47:07.913 that depended on the molecular weight of hydrogen, 00:47:07.913 --> 00:47:09.980 and they had to get some other value 00:47:09.980 --> 00:47:12.400 to match their experiment. 00:47:12.400 --> 00:47:15.820 So in these parameter set, anything 00:47:15.820 --> 00:47:17.380 I do to vary those parameters, I got 00:47:17.380 --> 00:47:20.209 to watch out that maybe some of those parameters 00:47:20.209 --> 00:47:22.500 are involved with somebody else's model and [INAUDIBLE] 00:47:22.500 --> 00:47:24.010 some other experiments. 00:47:24.010 --> 00:47:27.230 And I'm not really free to vary them all freely. 00:47:27.230 --> 00:47:28.850 So this is the idea from the Bayesian 00:47:28.850 --> 00:47:32.020 of having the prior intimation is 00:47:32.020 --> 00:47:35.500 so you know some of the ranges on these thetas already, 00:47:35.500 --> 00:47:38.454 and some of them you might know really sharp distributions, 00:47:38.454 --> 00:47:40.870 like the molecular weight of hydrogen. You might know that 00:47:40.870 --> 00:47:43.070 to a lot of decimal places. 00:47:43.070 --> 00:47:46.810 And so when people do this, normally you 00:47:46.810 --> 00:47:49.120 don't vary all of the thetas. 00:47:49.120 --> 00:47:51.450 Usually what you do is you select a set of thetas 00:47:51.450 --> 00:47:55.020 that you feel free to vary because they're so uncertain, 00:47:55.020 --> 00:47:58.380 and other thetas that you think, oh, I better not touch them. 00:47:58.380 --> 00:48:01.716 Because if I adjust them, I may go 00:48:01.716 --> 00:48:03.840 to crazy values that are inconsistent with somebody 00:48:03.840 --> 00:48:05.275 else's experiment. 00:48:05.275 --> 00:48:07.150 So a lot of times like the molecular weights, 00:48:07.150 --> 00:48:08.080 you would not touch them. 00:48:08.080 --> 00:48:09.704 You would just say, I got to just stick 00:48:09.704 --> 00:48:12.640 to the recommended values and the tables. 00:48:12.640 --> 00:48:14.890 I'm not free to vary the molecular weight of hydrogen, 00:48:14.890 --> 00:48:16.640 even though if I did it would make my data 00:48:16.640 --> 00:48:18.460 match my experiment better. 00:48:18.460 --> 00:48:20.390 Makes my model and the experiment 00:48:20.390 --> 00:48:22.520 match more precisely. 00:48:22.520 --> 00:48:25.450 So deciding which parameters to vary in this 00:48:25.450 --> 00:48:27.490 is a really crucial thing. 00:48:30.610 --> 00:48:35.800 And that's a lot of the art of doing 00:48:35.800 --> 00:48:39.430 this has to do with that issue. 00:48:39.430 --> 00:48:42.130 Also, you don't have to keep the thetas 00:48:42.130 --> 00:48:43.310 in the form you have them. 00:48:43.310 --> 00:48:44.560 You could do a transformation. 00:48:44.560 --> 00:48:49.420 So you could change to, say, W's that's equal to, say, 00:48:49.420 --> 00:48:52.540 some matrix times the thetas, and I could express 00:48:52.540 --> 00:48:54.760 the equation in terms of the W's. 00:48:54.760 --> 00:48:57.640 So I could transform my original representational parameters 00:48:57.640 --> 00:48:59.670 as some other parameters. 00:48:59.670 --> 00:49:03.530 And often times, your experiment might 00:49:03.530 --> 00:49:06.230 be really good at determining some of these W's, even 00:49:06.230 --> 00:49:09.800 if it might be incapable of determining any of the thetas. 00:49:09.800 --> 00:49:13.830 So you often might know some linear combination 00:49:13.830 --> 00:49:15.830 of parameters, or maybe not linear combinations, 00:49:15.830 --> 00:49:18.590 some non-linear combination of parameters 00:49:18.590 --> 00:49:22.790 might actually be determinable very well from your experiment, 00:49:22.790 --> 00:49:24.950 even though you can't determine things separately. 00:49:24.950 --> 00:49:27.290 And this gets into the idea of dimensionless numbers. 00:49:27.290 --> 00:49:30.680 So your experiment might depend on some dimensionless number 00:49:30.680 --> 00:49:32.241 very sensitively. 00:49:32.241 --> 00:49:34.490 And you can be quite confident from your external data 00:49:34.490 --> 00:49:36.657 what the value of that dimensionless number must be. 00:49:36.657 --> 00:49:38.656 But if you look inside the dimensionless number, 00:49:38.656 --> 00:49:40.350 it depends on a lot of different things. 00:49:40.350 --> 00:49:41.990 And you might not have any information about them 00:49:41.990 --> 00:49:42.770 separately. 00:49:42.770 --> 00:49:44.600 All you know is about your experiment just 00:49:44.600 --> 00:49:48.410 tells you the value of that one parameter very accurately. 00:49:48.410 --> 00:49:51.080 So this is another big part of the art 00:49:51.080 --> 00:49:54.240 of doing the model versus data is setting up 00:49:54.240 --> 00:49:58.050 your model in terms of parameters that you really 00:49:58.050 --> 00:50:00.900 can determine, and getting out all the ones you can't 00:50:00.900 --> 00:50:02.370 determine and fixing them. 00:50:02.370 --> 00:50:04.469 So we're really going to generally change do 00:50:04.469 --> 00:50:05.260 this kind of thing. 00:50:05.260 --> 00:50:13.050 But we're going to say that some thetas are fixed, 00:50:13.050 --> 00:50:21.780 and also we might change to a different representation, 00:50:21.780 --> 00:50:26.780 change to W's instead. 00:50:26.780 --> 00:50:27.950 Yeah? 00:50:27.950 --> 00:50:30.800 AUDIENCE: Can you explain where this transform-- 00:50:30.800 --> 00:50:32.470 I don't really know what's up with-- 00:50:32.470 --> 00:50:34.220 WILLIAM GREEN: Yeah, let's get an example. 00:50:34.220 --> 00:50:39.640 Suppose I was doing a reactor that had A equilibrium of B. 00:50:39.640 --> 00:50:41.520 And I was really interested in kf, 00:50:41.520 --> 00:50:44.720 the forward rate for A going to B. I'm a kineticist, 00:50:44.720 --> 00:50:48.100 I love to know A goes to B. However, 00:50:48.100 --> 00:50:50.210 if I setup the experiment wrong, it 00:50:50.210 --> 00:50:53.140 might be that this readction ran all the way to equilibrium. 00:50:53.140 --> 00:50:54.890 And what I see in the products is actually 00:50:54.890 --> 00:50:57.620 just the equilibrium ratio of A to B. 00:50:57.620 --> 00:51:03.630 So what I'm measuring might be something that's dependent 00:51:03.630 --> 00:51:06.110 really on kf over kr, and that might be the quantity 00:51:06.110 --> 00:51:07.136 I can really determine. 00:51:07.136 --> 00:51:08.635 Because that's equilibrium constant. 00:51:11.160 --> 00:51:13.030 If I didn't think about it, I could just 00:51:13.030 --> 00:51:16.950 try to have the model fitting procedure, just optimize 00:51:16.950 --> 00:51:18.760 to find the very best value of kf. 00:51:18.760 --> 00:51:21.700 And in that situation, it might have a lot of trouble, 00:51:21.700 --> 00:51:24.130 because it might be quite indeterminant what 00:51:24.130 --> 00:51:27.074 the kf is, because really all that matters is the ratio. 00:51:27.074 --> 00:51:28.490 Also I think about this some more, 00:51:28.490 --> 00:51:33.080 suppose I run at short times, and I 00:51:33.080 --> 00:51:34.580 measure the time dependence. 00:51:34.580 --> 00:51:37.230 What I'm really measuring is kf plus kr. 00:51:37.230 --> 00:51:39.490 Do you remember we did the analysis of A 00:51:39.490 --> 00:51:43.930 goes to B, one of the early homework problems? 00:51:43.930 --> 00:51:48.680 The time cost was actually kf plus kr, not kf separately. 00:51:48.680 --> 00:51:51.210 And so if I measure the exponential decay 00:51:51.210 --> 00:51:53.507 time constant, I'm really determining kf plus kr, 00:51:53.507 --> 00:51:55.340 I might be able to determine that very well. 00:51:55.340 --> 00:51:57.170 Actually, in my lab, I can do a great job for this. 00:51:57.170 --> 00:51:58.670 I have an instrument that can measure 00:51:58.670 --> 00:52:00.378 the time constant of the exponential of k 00:52:00.378 --> 00:52:02.540 really precisely, but it's determining the sum. 00:52:02.540 --> 00:52:05.170 It's not determining either one of them separately. 00:52:05.170 --> 00:52:07.045 And I might have to do a separate experiment, 00:52:07.045 --> 00:52:09.200 say a thermo-experiment to get the ratio. 00:52:09.200 --> 00:52:12.000 And then from the two I can put them together and get the two 00:52:12.000 --> 00:52:13.914 values distinctly. 00:52:13.914 --> 00:52:15.330 So this will be an example of this 00:52:15.330 --> 00:52:24.760 would be a W. My W is kf plus kr, the matrix would be 1001, 00:52:24.760 --> 00:52:26.410 something like that. 00:52:26.410 --> 00:52:29.220 1-1, something like that. 00:52:29.220 --> 00:52:32.860 Where I add these two guys, kf, kr. 00:52:32.860 --> 00:52:36.280 These are my two parameters, 1 plus 1. 00:52:36.280 --> 00:52:39.690 And I can determine W1 now very accurately. 00:52:39.690 --> 00:52:44.820 sorry, this is m, this is W. 00:52:44.820 --> 00:52:50.570 So now in terms of W, this has two parameters now, W1 and W2. 00:52:50.570 --> 00:52:52.790 I can't determine W2 from my experiment, 00:52:52.790 --> 00:52:54.497 but I can determine W1 really well. 00:52:54.497 --> 00:52:56.330 So then when I do the least squares fitting, 00:52:56.330 --> 00:52:58.010 I should vary W1. 00:52:58.010 --> 00:53:00.350 I can fix it it for my experimental data, 00:53:00.350 --> 00:53:02.400 and just leave W2 fixed at some value. 00:53:02.400 --> 00:53:05.470 I can't do anything about it. 00:53:05.470 --> 00:53:06.291 That all right? 00:53:09.060 --> 00:53:11.630 Now, do you get the difference in these two points of view? 00:53:11.630 --> 00:53:16.650 This is like, two completely different ways 00:53:16.650 --> 00:53:18.702 to look at the problem. 00:53:18.702 --> 00:53:20.660 You can think about it as, these parameters are 00:53:20.660 --> 00:53:23.150 free for me to vary, and I just have 00:53:23.150 --> 00:53:25.975 to be careful to select the ones I'm really free to vary. 00:53:25.975 --> 00:53:28.100 And that's the least squares fitting point of view. 00:53:28.100 --> 00:53:32.421 Or I could say, I'm not really determining anything 00:53:32.421 --> 00:53:34.670 in particular, all I'm doing is taking the whole range 00:53:34.670 --> 00:53:36.545 of uncertainty that we have about parameters, 00:53:36.545 --> 00:53:41.080 and by my experiment, I narrowed it down in the Bayesian view. 00:53:41.080 --> 00:53:42.790 So it's the two different points of view. 00:53:42.790 --> 00:53:46.040 To do this one, I need to make sure I have enough data 00:53:46.040 --> 00:53:48.204 to determine something. 00:53:48.204 --> 00:53:49.620 So I have to have enough determine 00:53:49.620 --> 00:53:52.490 some parameter, at least one, otherwise 00:53:52.490 --> 00:53:54.212 there's no point in doing this. 00:53:54.212 --> 00:53:56.420 This one I can do even if I can't determine anything, 00:53:56.420 --> 00:54:00.230 because I could still narrow down the range of parameters. 00:54:00.230 --> 00:54:04.490 But this might be harder to report in a table. 00:54:04.490 --> 00:54:08.120 Because all I have at the end is a new probably density function 00:54:08.120 --> 00:54:11.130 of multiple parameters. 00:54:11.130 --> 00:54:11.820 All right? 00:54:11.820 --> 00:54:13.470 OK, we're done. 00:54:13.470 --> 00:54:16.220 See you guys on Friday.