Exact Bayesian inference for normal hierarchical models

Abstract

This paper provides an algorithm for generating independent draws from the exact joint posterior distribution of the parameters of a univariate Normal hierarchical model. Suppose one observes data on J groups: {Y1, . . . ,YJ, S1}. At level-1 of the model, Yj|θj, σ2 ∼ N(θj, σ2/nj), where n1, . . . , nJ are known constants (e.g., group sample sizes), and S1|σ2 ∼ σ2χ2(df). At level-2, θj|γ, A ∼ N(W′jγ,A), where W1, . . . , WJ are known q × 1 covariate vectors. The unknown parameters are the J group means, θ1, . . . , θJ, the q × 1 level-2 regression coefficient γ, and the level-1 and level-2 variance components, σ2 and A. Given the two variance components, the conditional posterior distributions of the θj's and of γ are closed-form Normals, assuming a q-dimensional Normal or Uniform prior on γ. The algorithm of this paper yields independent samples from f(A, σ2|Y1, . . . , YJ, S1), having specified vague prior distributions for A, and σ2. This enables exact Bayesian inference for all model parameters. The algorithm is implemented as an SPlus program, TLNise, available from http://www.swarthmore.edu/NatSci/peversol/tlnise.htm.