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, R\'enyi 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.

翻译：本文建立了一个分析框架，用于研究一般可测空间上泊松过程与点模式相关的信息散度及似然比。主要结果包括：以强度测度表示的泊松点模式分布的Kullback-Leibler散度、Rényi散度、Hellinger距离及似然比的显式解析公式。该一般性结果导出了非齐次泊松过程、复合泊松过程以及空间与标记泊松点模式的类似公式。其他结果包括绝对连续性、相互奇异性及公共控制测度存在的简明刻画。本分析工具箱基于抽象可测空间上σ有限测度的Tsallis散度。处理方式纯粹基于信息论，无需拓扑假设。