Verify the statement in Subsection 4.4.4, that the optimal Bayes choice of a(x) is the posterior probability of C(x). In the situation of Exercise 4. the journal editors decide to investigate the...

Verify the statement in Subsection 4.4.4, that the optimal Bayes choice of a(x) is the posterior probability of C(x).

In the situation of Exercise 4. the journal editors decide to investigate the problem. They survey all of the recent contributors to the journal. and determine the total number of experiments. N. that were conducted by the contributors. Let x denote the number of such experiments that were reported in the journal (i.e . the experiments where HI was accepted by a size ao test). Again assume all experiments had fixed

(a) Show that a reasonable empirical Bayes estimate of
  is (defining

(truncated at zero and one to ensure it is a probability).

(b) If
  what is the estimate of the proportion of correct articles in the journal?