5.7 Referring to Example 6.6: (a) Using the prior distribution for (b) given in (5.6.27), show...

5.7 Referring to Example 6.6:

(a) Using the prior distribution for γ

(b) given in (5.6.27), show that the mode of the posterior distribution π(b|x) is bˆ = (px¯ + α − 1)/(pa + β − 1), and hence the empirical Bayes estimator based on this bˆ does not equal the hierarchical Bayes estimator (5.6.29).

(b) Show that if we estimate b/(b+1) using its posterior expectation E[b/(b+1)|x], then the resulting empirical Bayes estimator is equal to the hierarchical Bayes estimator.