条件信息测度下Fano不等式的优控理论推广
Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory
摘要：
Fano不等式是信息论中最基本、最普遍、最重要的工具之一。利用优控制理论，将Fano不等式推广到包含Shannon和Rényi的信息测度的广义信息测度上.当专门为这些措施，它恢复和推广的经典不等式。推导的关键是构造一个适当的条件分布，在可数无限字母表上诱导出所需的边际分布。的建设是基于无限维版本的伯克霍夫定理证明Révész [数学学报。1962，3，188{ \textendash }198]，并且保持期望的边缘分布的约束类似于概率论中的耦合。利用我们关于Shannon和Rényi信息测度的Fano型不等式，我们还研究了当错误概率为零时Shannon和Rényi的模糊序列的渐近行为.这种渐近行为通过Fano不等式提供了渐近均分性质（AEP）的一个新的表征。
英文摘要：
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.
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