Posterior distribution of hierarchical models using CAR(1) distributions

We examine properties of the conditional autoregressive model, or CAR(1) model, which is commonly used to represent regional effects in Bayesian analyses of mortality rates. We consider a Bayesian hierarchical linear mixed model where the fixed effects have a vague prior such as a constant prior and the random effect follows a class of CAR(1) models including those whose joint prior distribution of the regional effects is improper. We give sufficient conditions for the existence of the posterior distribution of the fixed and random effects and variance components. We then prove the necessity of the conditions and give a one-way analysis of variance example where the posterior may or may not exist. Finally, we extend the result to the generalised linear mixed model, which includes as a special case the Poisson log-linear model commonly used in disease mapping.