Find the approximate 90% HPD credible set, using the normal approximation to the posterior, in the situations of (a) Exercise 23; (b) Exercise 27; (c) Exercise 9. In the situation of Exercise 33,...

Find the approximate 90% HPD credible set, using the normal approximation to the posterior, in the situations of (a) Exercise 23; (b) Exercise 27; (c) Exercise 9.

In the situation of Exercise 33, suppose that the size of a credible set, C, for a is measured by

(a) If
  find the S-optimal 90% credible set for a.

(b) If n = 10 and S²
= 2, find the S-optimal 90% credible set for a.

(c) Show that the interval C = (as, bs) has the same size under S( C) for any s.

(d) Show that, if S( C) measures the size of a set for a, then

should be used to measure the size of a set for a²