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تسجيل مستخدم جديدOn $f$-Divergences: Integral Representations, Local Behavior, and Inequalities
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Igal Sason
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This paper is focused on $f$-divergences, consisting of three main contributions. The first one introduces integral representations of a general $f$-divergence by means of the relative information spectrum. The second part provides a new approach for the derivation of $f$-divergence inequalities, and it exemplifies their utility in the setup of Bayesian binary hypothesis testing. The last part of this paper further studies the local behavior of $f$-divergences.
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This paper is focused on derivations of data-processing and majorization inequalities for $f$-divergences, and their applications in information theory and statistics. For the accessibility of the material, the main results are first introduced witho
ut proofs, followed by exemplifications of the theorems with further related analytical results, interpretations, and information-theoretic applications. One application refers to the performance analysis of list decoding with either fixed or variable list sizes; some earlier bounds on the list decoding error probability are reproduced in a unified way, and new bounds are obtained and exemplified numerically. Another application is related to a study of the quality of approximating a probability mass function, induced by the leaves of a Tunstall tree, by an equiprobable distribution. The compression rates of finite-length Tunstall codes are further analyzed for asserting their closeness to the Shannon entropy of a memoryless and stationary discrete source. Almost all the analysis is relegated to the appendices, which form a major part of this manuscript.
This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the be
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This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas f
or quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive noninteger real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, $n$, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over $n$ dimensions, as needed in the direct calculation of the desired moment.
We consider a sub-class of the $f$-divergences satisfying a stronger convexity property, which we refer to as strongly convex, or $kappa$-convex divergences. We derive new and old relationships, based on convexity arguments, between popular $f$-divergences.
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