Propriety of Posteriors with Improper Priors in Hierarchical Linear Mixed Models

This paper examines necessary and sufficient conditions for the propriety of the posterior distribution in hierarchical linear mixed effects models for a collection of improper prior distributions. In addition to the flat prior for the fixed effects, the collection includes various limiting forms of the invariant gamma distribution for the variance components, including cases considered previously by Datta and Ghosh (1991), and Hobert and Casella (1996). Previous work is extended by considering a family of correlated random effects which include as special cases the intrinsic autoregressive models of Besag, York and Mollié (1991), the Autoregressive (AR) Model of Ord (1975), and the Conditional Autoregressive (CAR) Models of Clayton and Kaldor (1987), which have been found useful in the analysis of spatial effects. Conditions are then presented for the propriety of the posterior distribution for a generalized linear mixed model, where the first stage distribution belongs to an exponential family.