5.5

(a) Analogous to Problem 1.7.9, establish that for any random variable X, Y , and Z, cov(X, Y ) = E[cov(X, Y )|Z] + cov[E(X|Z), E(Y |Z)].

(b) For the hierarchy Xi|θi ∼ f (x|θi), i = 1,... , p, independent, i|λ ∼ π(θi|λ), i = 1,... , p, independent,

 ∼ γ (λ), show that cov(i, j |x) = cov[E(i|x, λ), E(j |x, λ)].

(c) If E(i|x, λ) = g(xi) + h(λ), i = 1,... , p, where g(·) and h(·) are known, then cov(i, j |x) = var[E(i|x, λ)].
[Part

(c) points to what can be considered a limitation in the applicability of some hierarchical models, that they imply a positive correlation structure in the posterior distribution.]