A nonparametric Bayesian analysis of independent and identically distributed observations of covariate-driven Poisson processes

2025-09-02

AI Summary

This paper addresses the challenge of anisotropic smoothing in intensity function estimation for inhomogeneous Poisson processes, arising from disparate physical units and scales of covariates. We propose a hierarchical Gaussian process prior with multiple bandwidths to automatically adapt smoothness across distinct covariate directions. Theoretically, we establish that the posterior distribution contracts at the optimal minimax rate. Computationally, we design a dimension-robust Metropolis-within-Gibbs MCMC algorithm enabling efficient nonparametric Bayesian inference. Numerical experiments demonstrate superior performance over competing methods, and real-data application to Canadian wildfire occurrences reveals nuanced, multivariate environmental drivers of spatial event intensity. Our core contribution lies in embedding anisotropic adaptivity within a nonparametric Bayesian framework—achieving both theoretical rigor and practical interpretability.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that $n$ independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as $n oinfty$, and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

Problem

• Modeling anisotropic intensity functions in Poisson processes
• Investigating covariate influence on point-generating mechanisms
• Achieving adaptive Bayesian inference for nonparametric regression

Innovation

• Nonparametric Bayesian approach for Poisson processes
• Multi-bandwidth Gaussian processes for anisotropic intensity
• Metropolis-within-Gibbs MCMC with dimension-robust sampling