Novel Inconsistency Results for Partial Information Decomposition

Abstract

Partial Information Decomposition (PID) seeks to disentangle how information about a target variable is distributed across multiple sources, separating redundant, unique, and synergistic contributions. Despite extensive theoretical development and applications across diverse fields, the search for a unique, universally accepted solution remains elusive, with numerous competing proposals offering different decompositions. A promising but underutilized strategy for making progress is to establish inconsistency results, proofs that certain combinations of intuitively appealing axioms cannot be simultaneously satisfied. Such results clarify the landscape of possibilities and force us to recognize where fundamental choices must be made. In this work, we leverage the recently developed mereological approach to PID to establish novel inconsistency results with far-reaching implications. Our main theorem demonstrates that three cornerstone properties of classical information theory, namely non-negativity, the chain rule, and invariance under invertible transformations, become mutually incompatible when extended to the PID setting. This result reveals that any PID framework must sacrifice at least one property that seems fundamental to information theory itself. Additionally, we strengthen the classical result of Rauh et al., which showed that non-negativity, the identity property, and the Williams and Beer axioms cannot coexist.