Introduction to Bayesian Data Analysis

1. 00:00What we've done so far is that we fit a very simple linear model.
2. 00:04Ignoring some of the details of the data.
3. 00:07The non independence in the data and treating it as independent.
4. 00:10A simple model.
5. 00:12Looking at the effect of attentional load centered attentional load on people's size.
6. 00:17I just showed you the code for the model.
7. 00:19Now I'm going to show you what you get out of this model and how to interpret what's in this model.
8. 00:25So first of all, one easy way to summarize the output from the model is to use this built in plot function.
9. 00:32So the plot function knows that you're looking at a stand object.
10. 00:36So the fitted model is a stand object underlying lee and it uses that information to print out the appropriate plot function
11. 00:44as you know in art there's also this plot function for standard based graphics but this plot function is working with the
12. 00:51rMS output.
13. 00:53So what this this is the fit pupil data that I had fit earlier and the things to look at here is that there are
14. 00:59three parameters indeed that we had set up is the intercept, here's the slope and I know that it's the slope because the
15. 01:07name of the predictor c load, center load is listed here and here's the sigma parameter.
16. 01:13As you can see the sigma parameter is positive.
17. 01:16The posterior distributions ranging from in the positive positive range.
18. 01:21And this comes from the fact that I have
19. 01:28enough data that the posterior distribution is heavily influenced by the likelihood.
20. 01:35So what's going to happen is that the mean of this sigma and the variability of this sigma is going to be not very different
21. 01:42from the the mean that I would get from a standard linear model in the frequently setting.
22. 01:48Okay, what is more interesting for me is to look at the distribution, the posterior distribution of the slope parameter because
23. 01:57the slope parameter is now telling me what the range of variability is going to be in the pupil size when I increase attentional
24. 02:05load by one unit.
25. 02:07So what the intercept is telling me the average pupil size?
26. 02:12Why?
27. 02:13Because I centered the predictor.
28. 02:15Once you center the predictor, the intercept has the interpretation that it represents the grand mean.
29. 02:21So the mean pupil size independent of any attentional load.
30. 02:25You see the advantage of centering now.
31. 02:27And so what the slope now means is that when you increase attention attentional load by one unit, the increase in pupil size
32. 02:37is represented by this posterior distribution.
33. 02:39So, roughly speaking, The pupil size would increase by about 10-60 units.
34. 02:47With every unit increase in attentional load.
35. 02:49That's what the model predicts.
36. 02:51Whether this is the truth or not is a separate matter.
37. 02:54We're just trying to draw inferences from the data we happen to have.
38. 02:57So what we learned from the data is that this is the effect of attention loan on pupil size.
39. 03:05So what you see on the right hand side are the four chains.
40. 03:08The
41. 03:09m c sampler is now trying to get samples from these distributions.
42. 03:16And what you're seeing is that these four chains are sitting on top of each other.
43. 03:21These are called fat hairy caterpillars.
44. 03:23And this is a good sign because it shows that the chains are landing on top of each other.
45. 03:28And what that means is that each of the chains for each parameter is sampling from the same distribution.
46. 03:35If you didn't have these these chains sitting on top of each other, that would indicate some problems with convergence.
47. 03:42And there are examples of non convergence in the textbook.
48. 03:45And of course we explain how to deal with that in the textbook.
49. 03:48But right now, I'm only showing you well behaved models that perform nicely.
50. 03:52So you can see what happens in the usual case.
51. 03:56Okay,
52. 03:59so that's one way to summarize the output with a figure like this.
53. 04:03Usually what we do in a paper is that we don't show the chains.
54. 04:07One doesn't generally show the chains.
55. 04:09One just shows the posterior distributions of the parameters and perhaps also a table that summarizes the results.
56. 04:16So what are the important things that you can summarize for the reader when you're writing a paper, you can some the mean
57. 04:21of the posterior for the intercept and the 95% credible interval.
58. 04:26This is the lower and upper bound of the 95% credible interval.
59. 04:29These are diagnostics about convergence.
60. 04:32That right now we don't need to worry about because we're going to fit models only those models that converge right now.
61. 04:39So this is okay.
62. 04:41You can look at the details in the textbook Later.
63. 04:43But what's important in this output is that you get to see what the model tells you about the plausible values of each of
64. 04:53the parameters.
65. 04:54For example, the average pupil size would be about 701 within 95% probability of lying between 661 and 740.
66. 05:05That's what the data tell you.
67. 05:06That's not necessarily the truth.
68. 05:07But that's what we learned from these data.
69. 05:10And so one of the great advantages of the bayesian framework is that you can talk about your uncertainty about the parameter
70. 05:17Not just you know, like in the frequentist world we have, is it significant?
71. 05:22Is it not significant?
72. 05:23That is a much less interesting issue.
73. 05:25Much more interesting is what is the range of variability of that parameter.
74. 05:29Similarly, the thing that we are really interested in scientifically is the effect of attentional load
75. 05:35on people's sides.
76. 05:36And what this output this second line is telling me is that for every unit increase in attentional load there's a 34 unit
77. 05:45increase predicted In um pupil size.
78. 05:49And my uncertainty about that is between 11 and 57 units here.
79. 05:55So this is telling me how uncertain I am about this prediction.
80. 05:59So this is a very important piece of information because if you have sparse data then your uncertainty will be much larger.
81. 06:06So you've learned a lot less from that particular data set.
82. 06:10If you happen to have a lot more data, you will get pretty tight
83. 06:13these are called credible intervals, you get tighter credible intervals which are much more informative about your problem
84. 06:19because they're constraining what the plausible values are given the data.
85. 06:23And finally, if you have the sigma parameter here with the posterior mean and the 95% credible interval here.
86. 06:30So that's an easy way to summarize, you know, the output, we are writing out we wrote our own function that produces the
87. 06:37short output.
88. 06:38It's in the code that will be provided with this course.
89. 06:41So you can look at that later.
90. 06:44So one other interesting thing you can do with this model now is that you can study what the model would predict for future
91. 06:52data.
92. 06:53Remember that in this, in this experiment design we have either an attentional load of 0 1 2 3 4
93. 07:02right, 0 to 4, that was all there was right, so five levels of attention load.
94. 07:08So what we can now do is that having fit the model and we got posterior distributions for each of the parameters, we will
95. 07:15just generate new data repeatedly using the posterior distributions of the parameters that we have.
96. 07:23So, what we are getting from the model now is future data simulated datasets that represent what the model predicts future
97. 07:32data would look like.
98. 07:33This is a very useful thing to do because it gives you a good understanding of how realistic your model is relative to the
99. 07:39data that you have.
100. 07:41So if you get predicted data that is completely divergent or very far away from your observed data, that's an indication
101. 07:48that there's something wrong in your model specification.
102. 07:50Need to go back and fix it.
103. 07:51There are many examples of such wrong model specifications in the textbook.
104. 07:56But right now, I just want to show you how this would work.
105. 07:58The code for all this is of course available to you.
106. 08:01And so you can look at it later.
107. 08:03So I don't want to distract you with the code right now.
108. 08:05What I want to show you is that the observed data is this density plot this black one here.
109. 08:10These are the actual data points.
110. 08:12And the blue lines that you're seeing.
111. 08:14These are the simulated datasets from the model given the posterior distributions.
112. 08:19So the posterior distributions are informed by the data now and the prior of course.
113. 08:23And so we're getting posterior predictive distributions of the data.
114. 08:28Future data sets given the posterior distribution of the parameter, this is for load zero and you can go and now check for
115. 08:35every possible load level.
116. 08:38How am I doing this?
117. 08:39I'm putting in different load levels into the, into the model.
118. 08:43And I'm just generating data now because I have posteriors for the parameters so I can just plug in different load levels
119. 08:49and I'm going to get predictions from the model.
120. 08:51So for load zero load one, you can see that there are some really wild data here,
121. 08:57There's some really sharp distributions in the simulated datasets, but most of them are generally spread out, which is quite
122. 09:03realistic.
123. 09:03They look quite a lot like the observed data here.
124. 09:06Also we have these occasional outliers, These strange extreme, extremely tight distributions, but most of them are spread
125. 09:12out again for load to load three, it's the same story and load for as well.
126. 09:18There might be something going on in the data.
127. 09:21There might be actually a finite mixture happening here.
128. 09:25There could be two separate distributions that we might be looking at but we are ignoring all that.
129. 09:30We're just treating this data as if it's coming from a simple linear model.
130. 09:34That's what we often do.
131. 09:35We simplify our assumptions even though there might be a more sophisticated model that would be better justified by the by
132. 09:42the data that you have.
133. 09:43This is something that we discuss in more advanced chapters.
134. 09:46But for now this is as a start, this is a good enough model and we will of course improve on this in future iterations
135. 09:56So that's a simple example of what you can do with regression modeling.
136. 10:01You can add one predictor.
137. 10:03Or you can add more than one predictor.
138. 10:05You can have many more predictors.
139. 10:06Of course there are complications associated with that.
140. 10:09But for now let's think about the situation where you add just one predictor and you can use that predictor to figure out
141. 10:16the effect of that variable on your dependent variable.
142. 10:19That's a very standard statistical modeling technique that we use quite frequently in many different areas of science.
143. 10:27Okay, so in the next lecture, what I'm going to do is I'm going to present another model now as an example.
144. 10:35But this time instead of using the normal likelihood, I'm going to use the log normal likelihood.
145. 10:41The log normal has some small complications and subtleties that I want to illustrate with this example.
146. 10:48So that's what we will look at next.

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