The mathematical landscape of partial information decomposition: A comprehensive review of properties and measures

TL;DR

This paper provides a comprehensive review of the mathematical properties and measures of Partial Information Decomposition (PID), clarifying the relationships and differences among various approaches to guide future theoretical and empirical work.

Contribution

It systematically analyzes existing PID measures, maps their properties and interdependencies, and offers a unified framework to advance the understanding and application of PID.

Findings

Mapped relationships and incompatibilities among PID properties

Identified conditions under which PID measures satisfy key properties

Provided a unified perspective to guide future PID research

Abstract

Partial Information Decomposition (PID) has become one of the most prominent information-theoretic frameworks for describing the structure and quality of information in complex systems. Despite its widespread utility, there exists no unique solution constraining precisely how a PID should be constructed, leading to a multiverse of different formalisms with different mathematical commitments. In this work, we provide a comprehensive overview of the mathematical landscape of PID. By integrating existing PID measures into a common language, we systematically examine all major approaches to the PID framework that have emerged so far, determining for each measure whether or not each known property holds. In addition, we derive a web of all known theorems mapping the relationships and incompatibilities between these properties, before also revealing some novel interdependency results. In doing so, we chart a brief history of the framework, promote a unified perspective for its discussions, and offer a path towards both theoretical refinement and informed empirical applications for the future of this powerful method.