Bayesian Modeling and Decision Theory for Non-Homogeneous Poisson Point Processes

• Activity Number: 168 - Bayesian Models for Gaussian and Point Processes
• Type: Contributed
• Date/Time: Monday, July 31, 2017 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Jiaxun Chen* and Athanasios Micheas and Scott H. Holan
• Companies: University of Missouri-Columbia and Univ of Missouri- Columbia and University of Missouri
• Keywords: Bayesian Decision Theory ; Exponential Family ; Kullback-Leibler loss function ; Mixture models ; Point process ; Poisson process

Abstract

We present a flexible hierarchical Bayesian model and a first attempt towards developing a comprehensive Bayesian decision theoretic framework for point process theory. We investigate more closely the most commonly used point process model, that of a 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 a Kullback-Leibler loss function and compare it with the usual estimator, the posterior mean under square error loss. The methodology is exemplified through simulations and applications to estimate the intensity surface of homicide incidents in 2015 in the city of Chicago.