derive 8.25 and 8.26

Various approaches to inference for random effects have been proposed. The simplest, which we describe first, is to take an empirical Bayes approach. From a Bayesian standpoint, there is no distinction inferentially between fixed and random effects (the distinction is in the priors that are assigned). Consequently, inference is simply based on the posterior distribution p(b; | y). Consider the LMM yi = x;B+ z;bite;, and assume b; and e; are independent with bi | D ~;id Nati ( 0, D) and e; | Of wind Nn, (0, o I), so that a = [o, D]. We begin by considering the simple, albeit unrealistic, situation, in which / and o are known. Letting y* = yi - c;B, we have p(bi | yi, B, a) x p(yi | bi, B, a) x T(bi | ax) x exp -202 (y - z;b;)"(y; - zib;) - 56:D-1b. which we recognize as a multiple linear regression with a zero-centered normal prior on the parameters b; (this model is closely linked to that used in ridge regression, see Sect. 10.5.1). Using a standard derivation, (5.7), b; | yi, B, a ~ Ng+1 [E(b; | yi, B, Q), var(b; | yi, B, ax) ] with mean and variance E[b; | yi, B, a] = #2 + D-1 (yi - x; B) = D= V '(y; - x;B) (8.25)\f8.4 Derive (8.25) and (8.26). [Hint: The identities Z Zi + -1 = DEV 02 D-' + = D - D= V-1 zD are useful. These follow from 8.12 Exercises 419 ( E + F)RE = I -(E+F) 'F (G + EFE) '= G-1-G E(EG E+F-1)-'EG-1, respectively.]