Shrinkage priors for nonparametric Bayesian prediction of nonhomogeneous Poisson processes

TL;DR

This paper introduces a novel class of improper priors for nonparametric Bayesian estimation of nonhomogeneous Poisson processes, demonstrating their theoretical validity and practical usefulness for intensity function prediction.

Contribution

It proposes a new class of improper priors for nonparametric Bayesian inference in Poisson processes and establishes their theoretical properties and applicability.

Findings

Improper priors lead to admissible Bayesian estimators under Kullback-Leibler loss.

Theoretical results extend finite-dimensional properties to infinite-dimensional models.

Numerical methods for inference with these priors are developed.

Abstract

We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and prediction based on the improper priors are shown to be admissible under the Kullback--Leibler loss. Numerical methods for Bayesian inference based on the priors are investigated.