A Sequential Monte Carlo Method for Bayesian Poisson Process Trees

ABSTRACT Statistical analysis of spatial point patterns often needs to deal with heterogeneous structures in intensity function estimation with possibly high‐dimensional covariate space. Likelihood based inference for spatial point process models needs to address the challenges in evaluating the multidimensional integral of random intensity function. To address these challenges, this paper proposes a Bayesian Poisson process tree (PPT) model to allow exact inference without numerical approximation of this random integral even with high‐dimensional covariate space. In particular, the proposed PPT exploits a stochastic tree prior to model the intensity function as a piecewise smooth function that is adaptively inferred at different spatial resolutions. The paper also shows that the resulting marginal likelihood can be computed in a bottom‐up recursion. To facilitate Bayesian inference in higher dimension, the paper develops a top‐down sequential Monte Carlo (SMC) method to allow efficient tree learning. The proposed SMC method is compared to a standard reversible‐jump Markov chain Monte Carlo (RJ‐MCMC) method and shows improved global exploration of tree space over RJ‐MCMC. The proposed model is demonstrated with various simulation studies and a real application in a multidimensional domain. The method is implemented in the R package poistree .