Information Divergences and Likelihood Ratios for Poisson Processes and Point Patterns

核心概念

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.

摘要

The key highlights and insights from the content are:

The article develops a general theoretical framework for studying information divergences and likelihood ratios of Poisson processes and point patterns on abstract measurable spaces, without any topological assumptions.

It introduces the concept of Tsallis divergences of sigma-finite measures, which provides a unified way to characterize various information-theoretic quantities such as Rényi divergences, Kullback-Leibler divergences, and Hellinger distances.

The article derives explicit analytical formulas for the likelihood ratios of Poisson point patterns in terms of their intensity measures. This includes a general formula that holds for Poisson point patterns with sigma-finite intensity measures.

Using the Tsallis divergence framework, the article provides simple characterizations of absolute continuity and mutual singularity of Poisson point pattern distributions in terms of their intensity measures.

The article shows that there exists a common dominating Poisson point pattern distribution if and only if the Hellinger distance between the intensity measures is finite. This extends previous results on the existence of a common dominating measure.

The general results are applied to analyze information-theoretic quantities for various types of Poisson processes, compound Poisson processes, and marked Poisson point patterns.

統計資料

The article does not contain any explicit numerical data or statistics. The key results are analytical formulas and characterizations involving measures, densities, and information-theoretic quantities.

引述

"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 analytical toolbox is based on Tsallis divergences of sigma-finite measures on abstract measurable spaces. The treatment is purely information-theoretic and free of any topological assumptions."

從以下內容提煉的關鍵洞見

Information divergences and likelihood ratios of Poisson processes and point patterns

by Lass... 於 arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00294.pdf

深入探究

What are some potential applications of the information-theoretic analysis of Poisson processes and point patterns developed in this article

The information-theoretic analysis of Poisson processes and point patterns developed in this article has various potential applications across different fields.

Machine Learning: These techniques can be applied in machine learning for modeling and analyzing spatial data, such as in image recognition, object detection, and natural language processing.

Biomedical Research: In biomedical research, the analysis of point patterns can be used to study the spatial distribution of cells, tissues, or diseases, providing insights into biological processes and health outcomes.

Environmental Studies: Understanding the distribution of species in ecology or the spatial patterns of natural phenomena can benefit from this analysis, aiding in conservation efforts and environmental management.

Network Analysis: Point patterns can also represent network nodes or events in a network, making this analysis useful in studying network structures, connectivity, and dynamics.

Epidemiology: Analyzing the spatial distribution of diseases or health-related events can help in epidemiological studies, outbreak investigations, and public health interventions.

How could the techniques and results be extended to study other types of stochastic processes or random measures beyond Poisson models

The techniques and results developed for Poisson processes and point patterns can be extended to study other types of stochastic processes or random measures beyond Poisson models.

Markov Processes: The framework can be adapted to analyze Markov processes, where the transition probabilities between states play a similar role to the intensity measures in Poisson processes.

Gaussian Processes: Extending the analysis to Gaussian processes can provide insights into spatial correlations, uncertainty quantification, and regression modeling in various applications.

Renewal Processes: By considering renewal processes, one can study the interarrival times of events and their distributions, which are fundamental in reliability analysis and queuing theory.

Lévy Processes: The framework can be applied to analyze Lévy processes, which have jumps of random sizes at random times, offering a versatile model for various phenomena in finance, physics, and biology.

Are there any connections or implications of the Tsallis divergence framework introduced here to other areas of information theory or statistical physics

The Tsallis divergence framework introduced in this article has connections and implications to other areas of information theory and statistical physics.

Generalized Information Theory: Tsallis divergence is a generalization of the Kullback–Leibler divergence and R´enyi divergence, offering a broader perspective on information measures and their applications in diverse fields.

Complex Systems: In the study of complex systems, Tsallis divergence can quantify the complexity and non-extensivity of systems, providing a tool to analyze the behavior of systems with long-range interactions and memory effects.

Statistical Mechanics: In statistical physics, Tsallis divergence is used to characterize the non-extensive properties of systems far from equilibrium, offering a framework to study phase transitions, critical phenomena, and self-organization in complex systems.