1. Some general results for Normal linear hierarchical regression models (see Lindley and Smith (1972) for more details).

Let y be an N-vector and 1, 2 and 3 be parameter vectors of length k1, k2 and k3, respectively. Let X, W and Z be known N ðk1, k1ðk2 and k2ðk3 matrices and C1, C2 and C3 be k1 ð k1, k2 ð k2 and k3 ð k3 known positive definite matrices. Assume yj1; 2; 3 ¾ N.X1;C1/
1j2; 3 ¾ N.W2;C2/
and 2j3 ¾ N.Z3;C3/
Throughout this exercise, we will treat 3 as known (e.g. as a vector of prior hyperparameters selected by the researcher).

(a) Show that yj2; 3 ¾ N.XW2;C1 C XC2X0 /
and, thus, that this Normal linear regression model with hierarchical prior can be written as a different Normal linear regression model with nonhierarchical prior.

(b) Derive p.1jy; 3/. What happens to this density as C1 3 ! 0k3ðk3?

(c) Derive p.2jy; 3/.