$f$ -发散的Sharp不等式
Sharp Inequalities for $f$-divergences
摘要：
$f$ -发散是概率测度之间的一类一般发散，它包括概率论、数理统计和信息论中的许多常用发散，如Kullback-Leibler发散、卡方发散、平方Hellinger距离、全变差距离等。摘要研究了在有限个$f$ -发散约束条件下，最大化或最小化两个概率测度之间的$f$ -发散的问题。我们表明，这些无限维的最优化问题都可以减少到最优化问题在小的有限维空间是易于处理的。我们的结果导致一个全面和统一的治疗问题，获得尖锐的不等式之间的$f$ -分歧。我们证明了许多现有的结果之间的不等式$f$ -分歧可以得到我们的结果的特殊情况下，我们也改进了一些现有的非尖锐的不等式。
英文摘要：
$f$-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 optimization problems over small finite dimensional spaces which are tractable. Our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between $f$-divergences. We demonstrate that many of the existing results on inequalities between $f$-divergences can be obtained as special cases of our results and we also improve on some existing non-sharp inequalities.
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