Bayes Estimators and Ergodic Decomposability with an Application to Cox Processes

Abstract

A prior distribution $\wedge$ on a set of parameters $I$ is said to be ergodically decomposable if $\wedge$ - a. all probability measures $(\pi_i)_{i\in I}$ are mutually singular in some strong sense. Criteria are established for $\wedge$ to be ergodically decomposable in terms of the posterior distribution and the Bayes estimator, which, for $I$ the locally finite measures on a Polish space and $\pi_i$ the Poisson process with intensity $i$, is just the Papangelou kernel of the Cox process directed by $\wedge$.