Unique Information and Secret Key Agreement
Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
Author to whom correspondence should be addressed.
Received: 1 November 2018 / Revised: 12 December 2018 / Accepted: 16 December 2018 / Published: 24 December 2018
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable
has on a target variable Y
, relative to the other sources. For two sources, influence breaks down into the information that both
redundantly share with Y
uniquely shares with Y
uniquely shares with Y
, and finally what
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 information. 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. Specifically, the use of a consistent PID quantified using a secret key agreement rate naturally induces a directional interpretation of the PID. We further reveal a surprising connection between third-order connected information, two-way secret key agreement rate, and synergy. We also consider difficulties which arise with a popular PID measure in light of the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.
This is an open access article distributed under the Creative Commons Attribution License
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MDPI and ACS Style
James, R.G.; Emenheiser, J.; Crutchfield, J.P. Unique Information and Secret Key Agreement. Entropy 2019, 21, 12.
James RG, Emenheiser J, Crutchfield JP. Unique Information and Secret Key Agreement. Entropy. 2019; 21(1):12.
James, Ryan G.; Emenheiser, Jeffrey; Crutchfield, James P. 2019. "Unique Information and Secret Key Agreement." Entropy 21, no. 1: 12.
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