Gaussian Process Methods for Covariate-Based Intensity Estimation
Patric Dolmeta, Matteo Giordano
TL;DR
The paper addresses nonparametric Bayesian estimation of the covariate-driven intensity of a Cox process in an increasing-domain setting with ergodic covariates. It develops an $n$-dependent log-Gaussian prior on the log-intensity built from Gaussian processes with Sobolev RKHS and combines it with an exponential (or sigmoid) link to enforce positivity, showing minimax-optimal posterior contraction rates of order $n^{-\beta/(2\beta+d)}$ for ground truths with regularity $\beta$. The findings extend prior work to a broader class of Gaussian priors, including Matérn processes, and weaken link-function assumptions, while proving contraction in both the $L^1$ norm and a covariate-dependent empirical metric, with implications for predictive accuracy of the intensity. The work offers a theoretically grounded, practically appealing Bayesian approach for covariate-informed intensity estimation in spatial statistics, and it highlights directions for adaptivity, computation, and extensions to infill asymptotics.
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 Matérn 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.