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arXiv:1807.05103 · Allerton Conference on Communication, Control, and Computing

Unique Informations and Deficiencies

Pradeep Kr. Banerjee, Eckehard Olbrich, Jurgen Jost, Johannes Rauh

Published Oct 1, 2018• 20 citations
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Abstract

Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its input or output. We relate the proposed quantities to an existing measure of unique information which we call the minimum-synergy unique information. We give an operational interpretation of the latter in terms of an upper bound on the one-way secret key rate and discuss the role of the unique informations in the context of nonnegative mutual information decompositions into unique, redundant and synergistic components.

DOI
10.1109/allerton.2018.8635984

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Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its input or output. We relate the proposed quantities to an existing measure of unique information which we call the minimum-synergy unique information. We give an operational interpretation of the latter in terms of an upper bound on the one-way secret key rate and discuss the role of the unique informations in the context of nonnegative mutual information decompositions into unique, redundant and synergistic components.
The problem of constructing information measures with a well-defined interpretation is of fundamental significance in information theory. A good definition of an information measure entails certain desirable properties while also providing answers to operational problems. In this work, we investigate the properties of the unique information, an information measure that quantifies a deviation from the Blackwell order. Beyond providing an accessible introduction to the topic from a channel ordering perspective, we present a novel resource-theoretic characterization of unique information in a cryptographic task related to secret key agreement. Our operational view of unique information entails rich physical intuition that leads to new insights into secret key agreement in the context of non-negative decompositions of the mutual information into redundant and synergistic contributions. Through this lens, we illuminate new directions for research in partial information decompositions and information-theoretic cryptography.
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable X i has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X 0 and X 1 redundantly share with Y, what X 0 uniquely shares with Y, what X 1 uniquely shares with Y, and finally what X 0 and X 1 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.
In analogy to the zero-error variant of the channel capacity, the zero-error information between two random variables is defined. We show that our definition is natural in the sense that the representation of the channel capacity with respect to mutual information carries over to the zero-error variants of the quantities. It is shown that the new notion, together with two operators introduced in the same context, namely the common random variable of two random variables and the dependent part of a random variable with respect to another, is useful for giving characterizations of the possibility of realizing cryptographic tasks - such as bit commitment, coin tossing, or oblivious transfer - from correlated pieces of information.
This paper is concerned with secret-key agreement by public discussion: two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X , Y , and Z, respectively, with joint distribution PXY Z . The secret key rate S(X ; Y jjZ) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between X and Y when given Z, denoted by I(X ; Y#Z), which is an upper bound on S(X ; Y jjZ). The special scenarios where X , Y , and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or where Z is generated by sending X and Y over erasure channels, are analyzed. In the first scenario it can be shown that the secret key rate is str...