From Babel to Boole: The Logical Organization of Information Decompositions

TL;DR

This paper introduces a logical framework for understanding the structure of Partial Information Decomposition (PID), revealing a unified pattern that encompasses existing concepts and proposing a new notion called 'vulnerable information.'

Contribution

It applies a mereological, logic-based approach to unify and extend the conceptual landscape of PID, including the introduction of novel base-concepts like vulnerable information.

Findings

Unified logical pattern for PID base-concepts

Revealed a new concept: vulnerable information

Provided a formal framework for analyzing information decomposition

Abstract

The conventional approach to the general Partial Information Decomposition (PID) problem has been redundancy-based: specifying a measure of redundant information between collections of source variables induces a PID via Moebius-Inversion over the so called redundancy lattice. Despite the prevalence of this method, there has been ongoing interest in examining the problem through the lens of different base-concepts of information, such as synergy, unique information, or union information. Yet, a comprehensive understanding of the logical organization of these different based-concepts and their associated PIDs remains elusive. In this work, we apply the mereological formulation of PID that we introduced in a recent paper to shed light on this problem. Within the mereological approach base-concepts can be expressed in terms of conditions phrased in formal logic on the specific parthood relations between the PID components and the different mutual information terms. We set forth a general pattern of these logical conditions of which all PID base-concepts in the literature are special cases and that also reveals novel base-concepts, in particular a concept we call "vulnerable information".