广义非对称发散测度与不等式

对称发散测度间的不等式及其加细

Inequalities Among Symmetric Divergence Measures and Their Refinement

在信息论和统计学的文献中有三种经典的散度度量，即Jeffryes-Kullback-Leiber的J-散度，Sibson-Burbea-Rao的Jensen-Shannon散度和Taneja的算术-几何平均散度。它们之间有着有趣的关系，并且基于对数表达式。像Hellinger歧视，对称卡方发散和三角歧视的分歧措施不是基于对数表达式。这六个发散度量相对于概率分布是对称的。本文研究了这些对称散度测度之间的一些有趣的不等式。对这些不等式也进行了改进。对Dragomir等人的一些不等式也作了改进。

广义对称发散测度与不等式

Generalized Symmetric Divergence Measures and Inequalities

在信息论和统计学的文献中已知有三种经典的发散度量。它们是Jeffryes-Kullback-Leiber \cite{jef} \cite{kul} \textit{J-divergence}。Sibson-Burbea-Rao \cite{sib} \cite{bur1, bur2} \textit{Jensen-Shannon divegernce}和Taneja \cite{tan3} \textit{Arithemtic-Geometric divergence}。这三个指标之间存在着有趣的关系。像\textit{Hellinger discrimination}、\textit{symmetric}、$χ^2 - $、\textit{divergence}和\textit{triangular discrimination}这样的发散度量在文献中也是已知的。所有这些措施可以写为Csiszár的\textit{f-divergence}的特殊情况。最近，作者证明了一个与所有六个测度有关的不等式。在本文中，我们的目的是给一个参数推广的上述措施，并建立它们之间的关系。并导出了一个类似于\textit{Hellinger's}和\textit{triangular discriminations}的新测度.

Sharp Inequalities for $f$-divergences

$f$-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler divergence, chi-squared divergence, squared Hellinger distance, total variation distance etc. In this paper, we study the problem of maximizing or minimizing an $f$-divergence between two probability measures subject to a finite number of constraints on other $f$-divergences. We show that these infinite-dimensional optimization problems can all be reduced to optimizat…

Lower bounds for the minimax risk using $f$-divergences and applications

Lower bounds involving $f$-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Our proofs just use simple convexity facts. Special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as Fano's inequality, Pinsker's inequality and inequalities based on global entropy conditions. Two applications are provided: a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower …

$f$-Divergence Inequalities via Functional Domination

This paper considers derivation of $f$-divergence inequalities via the approach of functional domination. Bounds on an $f$-divergence based on one or several other $f$-divergences are introduced, dealing with pairs of probability measures defined on arbitrary alphabets. In addition, a variety of bounds are shown to hold under boundedness assumptions on the relative information. The journal paper, which includes more approaches for the derivation of f-divergence inequalities and proofs, is available on the arXiv at https://arxiv.org/abs/1508.00335, and it has been published in the IEEE Trans. o…

相对Jensen-Shannon和算术-几何散度测度的统一推广及其性质

On Unified Generalizations of Relative Jensen--Shannon and Arithmetic--Geometric Divergence Measures, and Their Properties

在本文中，我们将考虑一些非对称散度测度的参数推广。\textit{non-symmetric divergence measures}如：Kullback-Leibler \textit{relative information}、$χ^2-$ \textit{divergence}、\textit{relative J -- divergence}、\textit{relative Jensen -- Shannon divergence}和\textit{relative Arithmetic -- Geometric divergence}。所有考虑的推广都可以写成Csiszár的\textit{f-divergence}的特殊情况。通过对概率分布施加一些条件，本文的目的是发展这些测度的界及其参数推广。