Structural Impossibility of Antichain-Lattice Partial Information Decomposition

TL;DR

This paper demonstrates fundamental limitations in using antichain-lattice structures for multivariate partial information decomposition, showing that such representations cannot universally recover mutual information components.

Contribution

It introduces System Information Decomposition for a special case and proves the impossibility of a universal rule within antichain-lattice frameworks for general cases.

Findings

Antichain-lattice structures cannot universally recover mutual information.

System Information Decomposition provides a consistent entropy decomposition for three variables.

Different systems can have identical atoms but different mutual information.

Abstract

Partial Information Decomposition (PID) represents multivariate mutual information via antichain-lattice that aims to specify which source groups can recover which informational components of a target. For three or more sources, widely desired PID axioms become mutually incompatible. This is often treated as an axiomatic tuning issue. This paper argues that the obstruction is representational, rooted in the antichain indexing itself, so that purely axiomatic adjustments within an antichain-lattice structure cannot resolve it in general. We first introduce System Information Decomposition (SID) for the special target-free three-variable setting, obtaining a self-consistent entropy decomposition with an operational redundancy definition. More fundamentally, we then show that for general multivariate PID, there is no universal rule that recovers the decomposed mutual information from the antichain-indexed information atoms. In particular, two systems can share identical atoms regardless of any axioms while having different mutual information. These results reveal the limits of antichain-lattice and motivate relation-based foundations for multivariate information measures.