C:/Users/Laura/Documents/Education/UMN/2011_Spring/Writing/Thesis Paper 3/manuscript/HHC_for_Biostatistics_supplement.dvi

We begin by recalling the core Bayesian hierarchical modeling result of Lindley and Smith (1972), henceforth abbreviated L&S. For n × p1-dimensional response vector z, p1-vector of parameters θ, known n× p1 design matrix A1, and known n× n covariance matrix C1, let the likelihood be z ∼ N(A1θ, C1). Then for second-level p2-vector of parameters μ, known design and covariance matrices A2 and C2, let the prior be θ ∼ N(A2μ, C2). L & S showed that the marginal distribution is z ∼ N(A1A2μ, C1 +A1C2A ′ 1) and the posterior is θ|z ∼ N(Dd, D) where D −1 = A1C −1 1 A1 + C 2 and d = A ′ 1C −1 1 z+C −1 2 A2μ. If we assume all covariance parameters (and α) are known, we To whom correspondence should be addressed.