Improper and proper posteriors with improper priors in a hierarchical model with a beta-binomial likelihood
In a multistage hierarchical model statisticians often use improper prior distributions in the last stage of the model. The posterior distribution of the parameters, however, is not always proper when the prior is improper. The purpose of the paper is to derive necessary and sufficient conditions for the posterior distribution of the parameters to be proper in a hierarchical model with a beta-binomial likelihood if the prior belongs to a large family of improper distributions. Conditions are given also for the posterior cross-moments of the parameters to be finite.