Rate-optimal Bayesian intensity smoothing for inhomogeneous Poisson processes

TL;DR

This paper introduces a Bayesian nonparametric method using B-spline priors for estimating the intensity function of inhomogeneous Poisson processes, achieving rate-optimality and adaptivity to unknown regularity.

Contribution

It develops a new theoretical framework for contraction rates of posteriors in intensity estimation and applies it to construct a practical, adaptive Bayesian smoothing method.

Findings

Method is computationally feasible

Achieves rate-optimal convergence

Automatically adapts to unknown regularity

Abstract

We apply nonparametric Bayesian methods to study the problem of estimating the intensity function of an inhomogeneous Poisson process. We exhibit a prior on intensities which both leads to a computationally feasible method and enjoys desirable theoretical optimality properties. The prior we use is based on B-spline expansions with free knots, adapted from well-established methods used in regression, for instance. We illustrate its practical use in the Poisson process setting by analyzing count data coming from a call centre. Theoretically we derive a new general theorem on contraction rates for posteriors in the setting of intensity function estimation. Practical choices that have to be made in the construction of our concrete prior, such as choosing the priors on the number and the locations of the spline knots, are based on these theoretical findings. The results assert that when properly constructed, our approach yields a rate-optimal procedure that automatically adapts to the regularity of the unknown intensity function.