Improper and proper posteriors with improper priors in a Poisson-gamma hierarchical model

In Bayesian analysis, Markov chain Monte Carlo techniques have become so easy to use that it is possible to erroneously generate observations from a posterior distribution that is improper. In this paper we discuss the Poisson-gamma hierarchical model. A flexible prior distribution is discussed, one which allows the user to choose improper priors. Necessary and sufficient conditions are given for the posterior distribution to be proper and for the posterior moments to exist. An example using data on brain lesions for multiple sclerosis patients is presented to demonstrate the difficulty in diagnosing whether the posterior is proper.