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UCSC-SOE-19-10: Nonparametric Bayesian modeling for Poisson process intensities via Bernstein polynomials

Chunyi Zhao and Athanasios Kottas
11/16/2019 03:02 PM
We develop a general modeling approach for Poisson processes over time or space, using Bernstein polynomials as basis functions to represent the point process intensity function. The model construction implies a Bernstein-Dirichlet process prior for the Poisson process density, thus supporting flexible inference for point process functionals. A key feature of the methodology is that it balances such model flexibility with computational efficiency in implementation of posterior inference. This feature becomes particularly important in handling practically relevant problems where spatial point patterns are recorded over irregular domains. Indeed, the structured Bernstein mixture model for two-dimensional Poisson processes can be adapted to provide flexible, computationally tractable inference for spatial intensities over irregular domains. We address the choice of the number of Bernstein polynomial basis functions, and develop methods for prior specification and posterior simulation for full inference about functionals of the point process. Further, we discuss the computational approach to deal with specific shapes for irregular domain in space. The methodology is illustrated with both synthetic and real data sets.

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