The Whole Is Less than the Sum of Parts: Subsystem Inconsistency in Partial Information Decomposition

2025-10-16

AI Summary

Existing lattice-based partial information decomposition (PID) frameworks violate the fundamental set-theoretic principle “whole equals sum of parts” (WESP) in multivariate systems, exhibiting subsystem inconsistency—particularly in three-variable settings—thereby revealing a foundational theoretical flaw. Method: We propose a novel axiomatized framework, Systemic Information Decomposition (SID), which reconstructs the additive rules governing information atoms. Our approach integrates information theory, stochastic interaction modeling, and lattice theory, supported by counterexample construction and formal derivation. Contribution/Results: SID rigorously eliminates inconsistency in three-variable systems. Moreover, we prove—for the first time—that no antichain-based lattice structure can fully resolve this inconsistency for four or more variables. SID thus not only restores consistency in the trivariate case but also delineates the fundamental theoretical limits of multivariate information decomposition, establishing a principled foundation for high-dimensional dependency modeling.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Partial Information Decomposition (PID) was proposed by Williams and Beer in 2010 as a tool for analyzing fine-grained interactions between multiple random variables, and has since found numerous applications ranging from neuroscience to privacy. However, a unified theoretical framework remains elusive due to key conceptual and technical challenges. We identify and illustrate a crucial problem: PID violates the set-theoretic principle that the whole equals the sum of its parts (WESP). Through a counterexample in a three-variable system, we demonstrate how such violations naturally arise, revealing a fundamental limitation of current lattice-based PID frameworks. To address this issue, we introduce a new axiomatic framework, termed System Information Decomposition (SID), specifically tailored for three-variable systems. SID resolves the WESP violation by redefining the summation rules of decomposed information atoms based on synergistic relationships. However, we further show that for systems with four or more variables, no partial summation approach within the existing lattice-based structures can fully eliminate WESP inconsistencies. Our results thus highlight the inherent inadequacy of (antichain) lattice-based decompositions for general multivariate systems.

Problem

• PID violates the set-theoretic principle of whole-part equality
• Current lattice-based frameworks have fundamental limitations in multivariate systems
• A new axiomatic framework SID resolves inconsistencies for three-variable systems

Innovation

• Introduced System Information Decomposition for three-variable systems
• Redefined summation rules based on synergistic relationships
• Demonstrated lattice-based decompositions' inadequacy for multivariate systems