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 $\wedge$.
@article{1176993801,
author = {Glotzl, E. and Wakolbinger, A.},
title = {Bayes Estimators and Ergodic Decomposability with an Application to Cox Processes},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 872-876},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993801}
}
Glotzl, E.; Wakolbinger, A. Bayes Estimators and Ergodic Decomposability with an Application to Cox Processes. Ann. Probab., Tome 10 (1982) no. 4, pp. 872-876. http://gdmltest.u-ga.fr/item/1176993801/