  We consider a particular Cox process from a Bayesian viewpoint and show that
the Bayes estimator of the intensity measure is the so-called P\'olya sum
kernel, which occurred recently in the context of the construction of the
so-called Papangelou processes. More precisely, if the prior, the directing
measure of the Cox process, is a Poisson-Gamma random measure, then the
posterior is again a Poisson-Gamma random measure and the Bayes estimator of
the intensity is the P\'olya sum kernel. Moreover, we extend this result to
doubly stochastic Poisson-Gamma priors and give conditions under which one can
identify the Bayes estimator for the intensity.
