Exact Bayesian Gaussian Cox Processes Using Random Integral

TL;DR

This paper introduces an exact Bayesian inference method for Gaussian Cox processes that models the intensity and cumulative intensity jointly, avoiding likelihood approximations and enabling precise posterior inference.

Contribution

It presents a novel nonparametric Bayesian approach with an exact MCMC sampler for Gaussian Cox processes, bypassing the need for likelihood approximations.

Findings

Effective on simulated data

Demonstrated on real-world temporal and spatial data

Applicable to aggregated count data at multiple resolutions

Abstract

A Gaussian Cox process is a popular model for point process data, in which the intensity function is a transformation of a Gaussian process. Posterior inference of this intensity function involves an intractable integral (i.e., the cumulative intensity function) in the likelihood resulting in doubly intractable posterior distribution. Here, we propose a nonparametric Bayesian approach for estimating the intensity function of an inhomogeneous Poisson process without reliance on large data augmentation or approximations of the likelihood function. We propose to jointly model the intensity and the cumulative intensity function as a transformed Gaussian process, allowing us to directly bypass the need of approximating the cumulative intensity function in the likelihood. We propose an exact MCMC sampler for posterior inference and evaluate its performance on simulated data. We demonstrate the utility of our method in three real-world scenarios including temporal and spatial event data, as well as aggregated time count data collected at multiple resolutions. Finally, we discuss extensions of our proposed method to other point processes.