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# In Example 7.2.15, prove that the posterior predictive distribution for Xn+1 is as stated….

In Example 7.2.15, prove that the posterior predictive distribution for Xn+1 is as stated.

Example 7.2.15

Suppose that  distribution, where  known, and we use the prior given in Example 7.1.2.Suppose we want to predict a future observation Xn+1, but this time Xn+1 is from the

distribution. So, in this case, the future observation is not independent of the observeddata, but it is independent of the parameter. A simple calculation)shows that (7.2.11) is the posterior predictive distribution of t and so we would predictt by x, as this is both the posterior mode and mean

We can also construct a Y -prediction region C(s) for a future value t from the  prediction region for t satisfies where Q(. / s) is the posterior predictive measure for t One approach to constructing C(s) is to apply the HPD concept to q(t/ s). We illustrate this via several examples.

Example 7.1.2

Suppose that  is a sample from an  distribution The likelihood function is then given by

Suppose we take the prior distribution of  for some specified  The posterior density of µ is then proportional to

We immediately recognize this, as a function of µ, as being proportional to the densityof an

distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights

distribution.Notice that the posterior mean is a weighted average of the prior mean µ0 and thesample mean x, with weights

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 µ 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  the less influence theprior has. For example, when  then the ratio of the

posterior variance to the sample mean variance iS  So there has been a5% improvement due to the use of prior information For example, suppose that  and that for n = 10 we

observe x= 1.2 Then the prior is an N(0, 2) distribution, while the posterior is an