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: . At level-1 of the model where are known constants (e.g.,group sample sizes),and . At level-2, where: are known q×1 covariate vectors. The unknown varameters are the J group means, 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 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 , having specified vague prior distributions for A, and σ²This enables exact Bayesian inference for all model parameters. The algorithm is implemented as an SPlus program,TLNise, available from www.swarthmore.edu/NatSci/peversol/tlnise.htm.

Keywords:

• Constrained gamma distribution
• Inverse gamma distribution
• Random effects model
• Rejection sampling
• TLNise