Posterior Contraction Rates for Gaussian Cox Processes with Non-identically Distributed Data

TL;DR

This paper establishes posterior contraction rates for non-parametric Bayesian inference on non-homogeneous Poisson processes with non-identically distributed data, focusing on Sigmoidal and Quadratic Gaussian Cox Process models, extending theoretical understanding in this area.

Contribution

It provides the first analysis of posterior contraction rates for non-identically distributed data in Gaussian Cox Process models, including the quadratic transformation case.

Findings

Derived contraction rates for Sigmoidal and Quadratic Gaussian Cox Processes.

Extended theoretical analysis to non-i.i.d. data in Poisson process models.

Provided finite-sample and hyperparameter-specific results.

Abstract

This paper considers the posterior contraction of non-parametric Bayesian inference on non-homogeneous Poisson processes. We consider the quality of inference on a rate function $\lambda$, given non-identically distributed realisations, whose rates are transformations of $\lambda$. Such data arises frequently in practice due, for instance, to the challenges of making observations with limited resources or the effects of weather on detectability of events. We derive contraction rates for the posterior estimates arising from the Sigmoidal Gaussian Cox Process and Quadratic Gaussian Cox Process models. These are popular models where $\lambda$ is modelled as a logistic and quadratic transformation of a Gaussian Process respectively. Our work extends beyond existing analyses in several regards. Firstly, we consider non-identically distributed data, previously unstudied in the Poisson process setting. Secondly, we consider the Quadratic Gaussian Cox Process model, of which there was previously little theoretical understanding. Thirdly, we provide rates on the shrinkage of both the width of balls around the true $\lambda$ in which the posterior mass is concentrated and on the shrinkage of posterior mass outside these balls - usually only the former is explicitly given. Finally, our results hold for certain finite numbers of observations, rather than only asymptotically, and we relate particular choices of hyperparameter/prior to these results.