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Advances in intelligent and soft computing

Information Entropy of Discrete Quasi-random Variables and Its Properties

Yang Yang, Chao Wang, Ming-Hu Ha

Published Jan 1, 2012• 1 citation
Publisher
DOI
10.1007/978-3-642-28592-9_17

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This work presents a new methodology for obtaining information theoretic necessary conditions directly from general operational requirements. This methodology is based on the construction of a discrete random variable that, when conditioned upon, ensures information stability of quasi-images. The induced information stability allows a more direct way to develop information theoretic necessary conditions from operational requirements beyond using Fano's inequality. That is, while Fano's inequality uses the probability of error to establish an upper bound on the entropy of a random variable given its estimator, the proposed new methodology can be applied to arbitrary operational requirements to obtain corresponding conditions on information theoretic quantities. To demonstrate its power, this new methodology is employed, to derive new necessary conditions for keyed authentication over a discrete memoryless channels and to establish the capacity region subject to finite leakage and finite error of the wiretap channel under two different secrecy metrics. These examples establish the usefulness of the proposed methodology.
The paper deals with linear information inequalities valid for entropy functions induced by discrete random variables. Specifically, the so-calledconditional Ingleton inequalitiesare in the center of interest: these are valid under conditional independence assumptions on the inducing random variables. We discuss five inequalities of this particular type, four of which has appeared earlier in the literature. Besides the proof of the new fifth inequality, simpler proofs of (some of) former inequalities are presented. These five information inequalities are used to characterize all conditional independence structures induced by four discrete random variables.
We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.