Increasing domain asymptotics for covariate-based nonparametric Bayesian intensity estimation with Gaussian and Besov-Laplace priors

2026-05-11

AI Summary

This study addresses nonparametric estimation of covariate-driven intensity functions for point processes over large observation windows. The authors propose a nonparametric Bayesian approach that combines Gaussian priors with flexible link functions and introduces a Besov–Laplace prior, which promotes sparsity and preserves spatial edges. Within an increasing-domain asymptotic framework, they establish, for the first time, that Gaussian priors achieve minimax-optimal posterior contraction rates under a covariate ergodicity assumption. Furthermore, they demonstrate the optimal convergence properties of the Besov–Laplace prior for estimating spatially inhomogeneous intensity functions. The theoretical results are complemented by a Markov chain Monte Carlo implementation, and the method’s efficacy and superiority are validated through simulations and real-world data from forest and environmental sciences.

Abstract

We study the problem of estimating the intensity function of a covariate-driven point process based on observations of the points and covariates over a large window. We consider the nonparametric Bayesian approach, and show that a wide class of Gaussian priors, combined with flexible link functions, achieves minimax-optimal posterior contraction rates in the increasing domain asymptotics and under the assumption that the covariates be ergodic. We also employ Besov-Laplace priors, which are popular in imaging and inverse problems due to their edge-preserving and sparsity-promoting properties. We prove that these yield optimal estimation of spatially inhomogeneous intensities belonging to Besov spaces with low integrability index. These results are based on a general concentration theorem that extends recent findings from the literature. To corroborate the theory, we provide extensive numerical simulations, implementing the considered procedures via suitable posterior sampling schemes. Further, we present two real data analyses motivated by applications in forestry and the environmental sciences.

Problem

• point process
• intensity estimation
• nonparametric Bayesian
• increasing domain asymptotics
• Besov spaces

Innovation

• nonparametric Bayesian
• increasing domain asymptotics
• Gaussian priors
• Besov-Laplace priors
• posterior contraction rate