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

2026-03-03

AI Summary

This study addresses the fragmentation in Partial Information Decomposition (PID) caused by the absence of a unified mathematical foundation, which has led to multiple competing formalisms that are difficult to compare. By integrating information theory, axiomatic analysis, and formal logic, this work systematically synthesizes existing PID frameworks and constructs— for the first time—a comprehensive mapping between PID measures and their theoretical properties. The research rigorously evaluates how mainstream approaches satisfy key desiderata, uncovers logical dependencies and incompatibilities among these properties, and establishes a network of theorems that clarifies the mathematical characteristics of each method. This contribution not only resolves longstanding ambiguities in the field but also provides a coherent pathway toward theoretical unification and robust empirical application of PID.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Partial Information Decomposition
• information theory
• mathematical properties
• framework unification
• measure comparison

Innovation

• Partial Information Decomposition
• information theory
• mathematical framework
• property interdependencies
• unified formalism