Bayesian Analysis of Sigmoidal Gaussian Cox Processes via Data Augmentation

TL;DR

This paper develops a Bayesian data augmentation framework for sigmoidal Gaussian Cox processes, clarifying inconsistencies in existing methods and enabling multitype inference with uncertainty quantification on ecological data.

Contribution

It establishes a rigorous link between thinning mechanisms and joint densities, resolving contradictions in data augmentation for Gaussian Cox processes.

Findings

Resolved inconsistencies in data augmentation methods.

Extended model to multitype point processes.

Applied Bayesian inference to ecological data with intertype dependence.

Abstract

Many models for point process data are defined through a thinning procedure where locations of a base process (often Poisson) are either kept (observed) or discarded (thinned). In this paper, we go back to the fundamentals of the distribution theory for point processes to establish a link between the base thinning mechanism and the joint density of thinned and observed locations in any of such models. In practice, the marginal model of observed points is often intractable, but thinned locations can be instantiated from their conditional distribution and typical data augmentation schemes can be employed to circumvent this problem. Such approaches have been employed in the recent literature, but some inconsistencies have been introduced across the different publications. We concentrate on an example: the so-called sigmoidal Gaussian Cox process. We apply our approach to resolve contradicting viewpoints in the data augmentation step of the inference procedures therein. We also provide a multitype extension to this process and conduct Bayesian inference on data consisting of positions of two different species of trees in Lansing Woods, Michigan. The emphasis is put on intertype dependence modeling with Bayesian uncertainty quantification.