Unique Information and Secret Key Agreement

Abstract

The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable Xi has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X0 and X1 redundantly share with Y, what X0 uniquely shares with Y, what X1 uniquely shares with Y, and finally what X0 and X1 synergistically share with Y. Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique informations. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PID's meaning---interpretations not present in PID's definition but that, we argue, need to be explicit. These reveal an inconsistency between third-order connected information, two-way secret key agreement rate, and synergy. Similar difficulties arise with a popular PID measure in light the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.

arXiv comment9 pages, 3 figures, 4 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/pid_skar.htm. arXiv admin note: text overlap with arXiv:1808.08606

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Cited by in corpus (10)

• A Novel Approach to the Partial Information Decomposition
• An operational information decomposition via synergistic disclosure
• Decomposing past and future: Integrated information decomposition based on shared probability mass exclusions
• Generalised Measures of Multivariate Information Content
• Quantifying high-order interdependencies on individual patterns via the local O-information: theory and applications to music analysis
• Modes of Information Flow in Collective Cohesion
• Pooling Probability Distributions and the Partial Information Decomposition
• A Measure of Synergy based on Union Information
• MAXENT3D_PID: An Estimator for the Maximum-entropy Trivariate Partial Information Decomposition
• Quantifying Influence and Information Transfer in a Modified Vicsek Model with Non-reciprocal Interactions