Gaussian Process Methods for Covariate-Based Intensity Estimation

TL;DR

This paper develops a Bayesian nonparametric approach using Gaussian processes to estimate the intensity function of covariate-driven point processes, achieving optimal convergence rates under common statistical assumptions.

Contribution

It extends Gaussian process methods to covariate-based intensity estimation, demonstrating minimax optimal posterior contraction rates for a broad class of priors including Matérn processes.

Findings

Achieves minimax optimal posterior contraction rates.

Applicable to popular Gaussian priors like Matérn processes.

Works under increasing domain asymptotics with stationary ergodic covariates.

Abstract

We study nonparametric Bayesian inference for the intensity function of a covariate-driven point process. We extend recent results from the literature, showing that a wide class of Gaussian priors, combined with flexible link functions, achieve minimax optimal posterior contraction rates. Our result includes widespread prior choices such as the popular Mat\'ern processes, with the standard exponential (and sigmoid) link, and implies that the resulting methodologically attractive procedures optimally solve the statistical problem at hand, in the increasing domain asymptotics and under the common assumption in spatial statistics that the covariates are stationary and ergodic.