The P\'olya sum kernel and Bayes estimation

TL;DR

This paper demonstrates that the Pólya sum kernel serves as the Bayes estimator for the intensity measure of a Cox process with Poisson-Gamma priors, linking Bayesian inference with recent stochastic process constructions.

Contribution

It establishes the Pólya sum kernel as the Bayes estimator in Cox processes with Poisson-Gamma priors and extends the result to doubly stochastic models.

Findings

Pólya sum kernel is the Bayes estimator for Cox processes with Poisson-Gamma priors.

Posterior distribution remains Poisson-Gamma, facilitating Bayesian updates.

Conditions identified for Bayes estimator in more complex stochastic models.

Abstract

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.