Bayesian modeling and decision theory for non-homogeneous Poisson point processes

Abstract

We present a flexible hierarchical Bayesian model and develop a comprehensive Bayesian decision theoretic framework for point process theory. We closely investigate the commonly used point process model for independent events, the Poisson process, using a finite mixture of exponential family components to model the intensity function. We employ a Bayesian hierarchical framework for parameter estimation and illustrate the Bayesian computations involved. We demonstrate the effectiveness of the Bayes rule under the Kullback–Leibler and Hellinger loss functions and compare them with the usual estimator, the posterior mean under squared error loss. The methodology is exemplified through simulations and a motivating application involving estimation of the intensity surface of homicide incidents in Chicago during 2015.