Posterior Distribution of Hierarchical Models Using CAR(1) Distributions

We examine properties of the CAR1 model, which is commonly used to represent regional eeects in Bayesian analyses of mortality rates. We consider a Bayesian hierarchical linear mixed model where the xed eeects have a v ague prior such a s a constant prior and the random eeect follows a class of CAR1 models including those whose joint prior distribution of the regional eeects is improper. We give suucient conditions for the existence of the posterior distribution of the xed and random eeects and variance components. We then prove the necessity of the conditions and give a one-way analysis of variance example where the posterior may o r m a y not exist. Finally, w e 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.