7.2.2 For the model discussed in Example 7.1.2, determine the posterior distribution of the third quartile y = / + 0020.75. Determine the posterior mode and the posterior expectation of w .EXAMPLE 7.1.2 Location Normal Model Suppose that (X1, . . ., Xn) is a sample from an N(u, o() distribution, where u e R is unknown and o is known. The likelihood function is then given by n L (u | x1, . . . , Xn) = exp (x - 1) 2 20 Suppose we take the prior distribution of / to be an N(Mo, to) for some specified choice of Mo and to. The posterior density of / is then proportional toexp ( 14 - 10)2 exp " (x - 14) 2 1 = exp ( 12 - 2 410 + 10) - n 20% = exp 1- 2 + x exp Ho 2to 20 7 N OxP - ( # + :2 ) ( " -( *+ n + yexp 3082+ 2 n X x exp (7.1.2) We immediately recognize this, as a function of , as being proportional to the density of an distribution. Notice that the posterior mean is a weighted average of the prior mean Mo and the sample mean x, with weights1 n n n + 2 2 and + TO respectively. This implies that the posterior mean lies between the prior mean and the sample mean. Furthermore, the posterior variance is smaller than the variance of the sample mean. So if the information expressed by the prior is accurate, inferences about / based on the posterior will be more accurate than those based on the sample mean alone. Note that the more diffuse the prior is - namely, the larger to 2 is - the less influence the prior has. For example, when n = 20 and o = 1, To = 1, then the ratio of the posterior variance to the sample mean variance is 20/21 ~ 0.95. So there has been a 5% improvement due to the use of prior information.For example, suppose that 0 = 1, /0 = 0, to = 2, and that for n = 10, we observe x = 1.2. Then the prior is an A(0, 2) distribution, while the posterior is an ~(+49) "(2+ 1912).(+ 9) = N(1.1429, 95238 x 10-) distribution. These densities are plotted in Figure 7.1.4. Notice that the posterior is quite concentrated compared to the prior, so we have learned a lot from the data. I 1.2 + 1.0+ 0.8+ 0.6+ 0.4+ 0.2 -5 Figure 7.1.4: Plot of the N(0, 2) prior (dashed line) and the N (1.1429, 9.523 8 x 10-2) posterior (solid line) in Example 7.1.2