Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory

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{\textendash}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.

44 pages, 3 figures

References in corpus (4)

• Majorization and Rényi Entropy Inequalities via Sperner Theory
• On Data-Processing and Majorization Inequalities for $f$-Divergences with Applications
• On the error probability of stochastic decision and stochastic decoding
• Sharp Bounds on Arimoto's Conditional Rényi Entropies Between Two Distinct Orders

Cited by in corpus (2)

• From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
• A tight uniform continuity bound for the Arimoto-Rényi conditional entropy and its extension to classical-quantum states