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Information divergences and likelihood ratios of Poisson processes and point patterns

Lasse Leskelä

Abstract

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.

Information divergences and likelihood ratios of Poisson processes and point patterns

Abstract

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.
Paper Structure (42 sections, 38 theorems, 133 equations)

This paper contains 42 sections, 38 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Outline
  4. Preliminaries
  5. Measures
  6. Point patterns
  7. Poisson PPs
  8. Rényi divergences
  9. Tsallis divergences of sigma-finite measures
  10. Definition
  11. Properties
  12. Absolute continuity and mutual singularity
  13. Hellinger distances of sigma-finite measures
  14. Disintegration of Tsallis divergences
  15. Likelihood ratios of Poisson PPs
  16. ...and 27 more sections

Key Result

Theorem 3.1

$\alpha \mapsto T_\alpha(\lambda \| \mu)$ is a well-defined nondecreasing function from $\mathbb{R}_+$ into $[0,\infty]$ that is continuous on the interval $\{\alpha \colon T_\alpha(\lambda \| \mu) < \infty\}$, and admits a representation where $p_s$ refers to the Poisson distribution $k \mapsto e^{-s k} \frac{s^k}{k!}$ on the nonnegative integers with mean $s$. Furthermore, the value of $T_\alph

Theorems & Definitions (87)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • ...and 77 more