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Title: | Generalizations of fano's inequality for conditional information measures via majorization theory | Authors: | Sakai, Y. | Keywords: | Asymptotic equipartition property (AEP) Conditional rényi entropies Countably infinite alphabet Fano's inequality General class of conditional information measures List decoding Majorization theory The birkhoff-von neumann decomposition The infinite-dimensional version of birkhoff's theorem ?-mutual information |
Issue Date: | 2020 | Publisher: | MDPI AG | Citation: | Sakai, Y. (2020). Generalizations of fano's inequality for conditional information measures via majorization theory. Entropy 22 (3) : 288. ScholarBank@NUS Repository. https://doi.org/10.3390/e22030288 | Rights: | Attribution 4.0 International | Abstract: | Fano's inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano's inequality is generalized to a broad class of information measures, which contains those of Shannon and Rényi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff's theorem proven by Révész [Acta Math. Hungar. 1962, 3, 188-198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon's and Rényi's information measures, we also investigate the asymptotic behavior of the sequence of Shannon's and Rényi's equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano's inequality. © 2020 by authors. | Source Title: | Entropy | URI: | https://scholarbank.nus.edu.sg/handle/10635/198372 | ISSN: | 10994300 | DOI: | 10.3390/e22030288 | Rights: | Attribution 4.0 International |
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