Information Divergences and Likelihood Ratios of Poisson Processes and Point Patterns

This article develops an analytical framework for studying information divergences and likelihood ratios associated with Poisson processes and point patterns on general measurable spaces. The main results include explicit analytical formulas for Kullback-Leibler divergences, Rényi divergences, Hellinger distances, and likelihood ratios of the laws of Poisson point patterns in terms of their intensity measures. The general results yield similar formulas for inhomogeneous Poisson processes, compound Poisson processes, as well as spatial and marked Poisson point patterns. Additional results include simple characterisations of absolute continuity, mutual singularity, and the existence of common dominating measures. The analytical toolbox is based on Tsallis divergences of sigma-finite measures on abstract measurable spaces. The treatment is purely information-theoretic and free of topological assumptions.