Bayesian Inference and Model Assessment for Spatial Point Patterns Using Posterior Predictive Samples

Abstract

Spatial point pattern data describes locations of events observed over a given domain, with the number of and locations of these events being random. Historically, data analysis for spatial point patterns has focused on rejecting complete spatial randomness and then on fitting a richer model specification. From a Bayesian standpoint, the literature is growing but primarily considers versions of Poisson processes, focusing on specifications for the intensity. However, the Bayesian literature on, e.g., clustering or inhibition processes is limited, primarily attending to model fitting. There is little attention given to full inference and scant with regard to model adequacy or model comparison.

The contribution here is full Bayesian analysis, implemented through generation of posterior point patterns using composition. Model features, hence broad inference, can be explored through functions of these samples. The approach is general, applicable to any generative model for spatial point patterns.

The approach is also useful in considering model criticism and model selection both in-sample and, when possible, out-of-sample. Here, we adapt or extend familiar tools. In particular, for model criticism, we consider Bayesian residuals, realized and predictive, along with empirical coverage and prior predictive checks through Monte Carlo tests. For model choice, we propose strategies using predictive mean square error, empirical coverage, and ranked probability scores. For simplicity, we illustrate these methods with standard models such as Poisson processes, log-Gaussian Cox processes, and Gibbs processes. The utility of our approach is demonstrated using a simulation study and two real datasets.