Introduction to Bayesian Data Analysis

1. 00:00Okay, so let's look at the normal distribution again.
2. 00:06But this time we're going to look at some of the functions in R that allow us to ask interesting and useful questions about
3. 00:13this particular example of a continuous random variable.
4. 00:18So what we have seen so far, I'm just reminding you what we've done so far.
5. 00:22We can figure out in theory at least.
6. 00:24We can figure out the area under the curve between a particular range of values in the normal distribution and that gives
7. 00:30us the probability of observing those particular range of values.
8. 00:35So that's what we need the CDF for.
9. 00:37And I told you earlier that we've got two parts to the probability density function associated with the normal the normalizing
10. 00:46constant and the kernel.
11. 00:48And we can work out the normalizing constant once we have a kernel, the relevance of this will of course become clear later.
12. 00:55But I just want you to understand that there are always these two parts and one of them is a constant,
13. 01:02that you can figure out at least in easy cases.
14. 01:06Okay, so let's look at some of the functions that are available in R that allow us to use the normal distribution
15. 01:16in various ways.
16. 01:17That is really, really helpful when doing statistical modeling.
17. 01:20Okay, so if you remember in the Bernoulli case for example, we had these functions are dbern, pbern and qbern functions.
18. 01:29So if you have forgotten what these are, go back and review the materials so that you're clear on what these functions
19. 01:35do because what we're going to do now, we are going to use this D P Q R family of functions in the
20. 01:42continuous space.
21. 01:43And of course there are important differences now between the discrete and continuous case, as I will show you, first of
22. 01:49all, you can generate random data from a normal distribution using the rnorm function.
23. 01:55So there's an example here.
24. 01:57I'm generating five data points, randomly generated data points from the standard normal.
25. 02:02Just as an example, I could have chosen any value from you for the mean and the standard deviation and I would get appropriate
26. 02:09you know, samples from that particular distribution.
27. 02:12And of course, if I ran this command multiple times, I would get different numbers each time.
28. 02:17Now, I also want to point out that in R the default values for the mean and standard deviation of zero and one respectively
29. 02:24so, I could have actually dropped that if I want to generate data from a standard normal, I can just drop the mean and standard
30. 02:32deviation specification because that's the default value anyway.
31. 02:36So I'm getting different numbers of course, but that's because I'm generating new random data each time.
32. 02:43Okay, this tool is extremely important for us in statistical modeling, when we want to understand the properties of an experiment
33. 02:51design will be understanding those properties by randomly generating new data from a particular statistical model.
34. 02:58To understand the properties of that model, that's why this function is so crucial in our training statisticians.
35. 03:08So the next example I want to show you is the use of the pnorm function.
36. 03:14This is of course the cumulative distribution function of the normal distribution.
37. 03:18And here I can ask questions like, what's the probability of observing a value like two or something less than that,
38. 03:25that I could write like this and then I would compute that with the pnorm two.
39. 03:29So this gives me the probability of observing two or something less than that.
40. 03:33Incidentally, I could have just written what's the probability of observing some value
41. 03:37like two,
42. 03:38exactly two or something less than that.
43. 03:40Here I've written strictly less than two, but they will give you the same probabilities.
44. 03:44Why?
45. 03:45Because the probability of exactly getting two is zero.
46. 03:48So sometimes in textbooks you might see this written with a less than or equal sign, but that's just the same
47. 03:55thing.
48. 03:56It's not going to change anything.
49. 03:57So that's my cumulative probability of two or less than two.
50. 04:00And I could even ask a question like what's the probability of observing a value like two or something larger than two?
51. 04:09And to do that there's a specification inside this pnorm function which says lower dot tail equal
52. 04:17to false.
53. 04:18What that's saying is that don't look to the left of two in the distribution rather look to the right of two.
54. 04:26So the lower tail equal to false means that you look at the probability to the right of the number that you're talking about
55. 04:32So that's what this useful functionality is in all the d p q r functions,
56. 04:36they are I think in every function.
57. 04:39Okay, so this is useful way that you can compute probabilities for particular values and you can see how you
58. 04:48might compute the probability of observing a value between two and minus two.
59. 04:52You calculate the cumulative probability of two or less than two that covers the entire area to the left to and you subtract
60. 04:59from that.
61. 05:00The probability of observing -2 or something less than 2.
62. 05:03So it will be a subtraction.
63. 05:05These are exercises that you will do later on to get a handle on how this pnorm function or this P family of functions
64. 05:11works.
65. 05:13Alright.
66. 05:13And another important function is the Q norm function which is the inverse of the CDU.
67. 05:19So you can ask questions like what is the quantile Q,
68. 05:24such that the area under the curve to the left of it is .977.
69. 05:30And what this function does is it gives you that quantile,
70. 05:35in this case it's actually two.
71. 05:36That's exactly what I did here.
72. 05:38I plugged in two,
73. 05:40and I got a probability of 0.977.
74. 05:44Now I'm plugging into the qnorm function this probability and getting back two,
75. 05:49well it's approximately two.
76. 05:53So these are very important functions that allow you to ask useful questions about a particular distribution that you're
77. 06:00working with and that's why I'm talking about this at such length because they are very, very useful for understanding different
78. 06:07aspects of a distribution.
79. 06:09These are the questions you can ask from a distribution.
80. 06:11And so finally, I come to the most important thing that I want to talk about in the normal distribution, which is the d norm
81. 06:17function.
82. 06:18So, if you remember in the D band function and the d binom function, we actually got the probability of a particular outcome
83. 06:26in those discrete random variable cases.
84. 06:29And as I told you before, several times now, in the continuous case, you cannot ask or you can of course.
85. 06:35But the probability of getting a particular point value is always zero.
86. 06:40So the D norm function, unlike the D band and D binom functions does not give you the probability of a particular outcome
87. 06:49but it does give you a non zero number.
88. 06:51What is that number?
89. 06:53That number is the density of that particular value.
90. 06:58And what that really means is that it's telling you the result of computing that function.
91. 07:04You plug into the normal density function, you plug in a particular number like two.
92. 07:09That's what I'm doing here.
93. 07:10This is f(x) here by the way.
94. 07:12There's dnorm function
95. 07:13is that probability density function f(x) and it's returning the Y value,
96. 07:18the y axis value
97. 07:20for that particular number.
98. 07:21And that value is the density of the normal distribution.
99. 07:24It is not the probability.
100. 07:26Please keep this very clear in your minds that when we're talking about continuous random variables, we're talking about
101. 07:33the density of a particular point.
102. 07:35We're not talking about the probability because the probability is always zero.
103. 07:43So this is just a summary to remind you of all the functionality that's available for continuous random variable in this
104. 07:49case the normal.
105. 07:51You can generate random data.
106. 07:53This is just random data on the X axis that I just produced.
107. 07:55If I re ran this command it would give me different data,
108. 08:00I would get different numbers each time.
109. 08:01I can compute the area under the curve between let's say plus one and minus one.
110. 08:06And that would give me this area under the curve.
111. 08:08Using this pnorm function, I'm subtracting minus one.
112. 08:12The cumulative probability of minus one or less than that from the cumulative probability of one or something less than that
113. 08:18and I get this area under the curve this is a very useful function of great practical significance to us later on.
114. 08:25The dnorm function forgiven value on the X axis is going to tell me the point on the curve.
115. 08:31The Y value of the probability density function and this value here is non zero, of course, but it's a density.
116. 08:40It's not a probability.
117. 08:43And the qnorm function gives you for any given probability, it tells you what the quantile is.
118. 08:48Such that the under the curve to the left of that quantile is this probability here.
119. 08:54So it's the inverse of the community distribution function.
120. 08:59So this is the example with a normal 01.
121. 09:02But of course you could play with this now with any normal distribution,
122. 09:06with any mean or any standard deviation.
123. 09:09It's a very useful tool for understanding the probability is under the curve.
124. 09:13So, here's an example of a normal distribution with mean 500 standard deviation 100.
125. 09:17500 is the midpoint here.
126. 09:19So this is where the maximum is.
127. 09:21And the spread of this distribution is determined by the standard deviation.
128. 09:25If I made this 1000, then this standard deviation would become much broader and the X axis range would go out much further
129. 09:32to cover, you know, the 95% of the area under the curve between this range and this range.
130. 09:39So the standard evasion determines the spread and the mean determines the center point of this distribution and it's a symmetric
131. 09:45distribution.
132. 09:46The normal distribution is symmetric.
133. 09:49Here's what a cdf looks like in the continuous space.
134. 09:52I showed you the CDF in the discrete case, I think in the binomial earlier in the continuous case we've got continuous values
135. 09:59on the X axis and the probability is rising up to one.
136. 10:03You see that the maximum value is one.
137. 10:05So this is the cumulative probability of observing some number like this or something less than that.
138. 10:10So of course you could go off all the way to infinity.
139. 10:12This thing is going to asymptote at one.
140. 10:14It's never going to change now.
141. 10:16So that's what the CDF looks like for the normal distribution right now, the inverse of the CDF
142. 10:24just flips the axis on this function.
143. 10:27So this becomes the X axis and this becomes the Y axis.
144. 10:31So this is the inverse here
145. 10:33I plug in a probability and I get back the quantile, you know, from the cumulative distribution function for that particular
146. 10:40probability.
147. 10:40So that's basically the setup here.
148. 10:44These are the important functions of the normal distribution.
149. 10:47And one important thing I want to point out is that any other distribution that we'll be using in this course or in the textbook
150. 10:55later on in the further chapters, the later chapters of the book, we will always be working with these types of functions
151. 11:03in order to understand that distribution.
152. 11:05And we will always be wanting to understand the properties of a particular distribution that we're working with.
153. 11:11And these kinds of the D P Q R family of functions is extremely useful for helping us to understand what the distribution
154. 11:19looks like and what the probabilities are for particular ranges of values.
155. 11:24This will become extremely important, bayesian data analysis in particular when we're trying to derive prior distributions
156. 11:31by thinking about what the plausible values might be.
157. 11:34So of course I will talk about that later, but this is the preparation we need for doing all this bayesian modeling later on.
158. 11:43So what have we done so far?
159. 11:44We've looked at random variables, especially the discrete and random cases with examples of each,
160. 11:53at least two examples from the discrete case and one from the continuous case.
161. 11:58You will see more examples of other distributions later on of course.
162. 12:01But these are the canonical examples that you can use to generalize, you know, two other distributions. Because the story
163. 12:08is not never gonna change now.
164. 12:11All that will change is the f(x),
165. 12:14that's what will change now.
166. 12:16And you should be very clear about this.
167. 12:18The use of the DPQR family of functions because you will always need them whenever you're thinking about a particular
168. 12:23distribution.
169. 12:24Okay, so that's it about the normal distribution.
170. 12:28Next I'm going to talk about maximum likelihood estimation.

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