Propriety of posteriors with improper priors in hierarchical linear mixed models

This paper examines necessary and sufficient conditions for the propriety o f the posterior distribution in hierarchical linear mixed effects models fo r a collection of improper prior distributions. In addition to the flat pri or for the fixed effects, the collection includes various limiting forms of the invariant gamma distribution for the variance components, including ca ses considered previously by Datta and Ghosh (1991), and Hobert and Casella (1996). Previous work is extended by considering a family of correlated ra ndom effects which include as special cases the intrinsic autoregressive mo dels of Besag, York and Mollie (1991), the Autoregressive (BR) Model of Ord (1975), and the Conditional Autoregressive (CAR) Models of Clayton and Kal dor (1987), which have been found useful in the analysis of spatial effects . Conditions are then presented for the propriety of the posterior distribu tion for a generalized linear mixed model, where the first stage distributi on belongs to an exponential family.