# On Reverse Pinsker Inequalities

## TL;DR

This paper introduces new upper bounds on relative entropy and Rényi divergence based on total variation distance, refining existing inequalities and extending results to broader settings.

## Contribution

It provides novel upper bounds on relative entropy and Rényi divergence as functions of total variation distance, improving and extending previous inequalities.

## Key findings

- Refined inequality by Verdú for general probability measures.
- Improved tightness of Csiszár and Talata's inequality.
- Extended bounds to Rényi divergence of arbitrary order.

## Abstract

New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verd\'{u} for general probability measures. A second bound improves the tightness of an inequality by Csisz\'{a}r and Talata for arbitrary probability measures that are defined on a common finite set. The latter result is further extended, for probability measures on a finite set, leading to an upper bound on the R\'{e}nyi divergence of an arbitrary non-negative order (including $\infty$) as a function of the total variation distance. Another lower bound by Verd\'{u} on the total variation distance, expressed in terms of the distribution of the relative information, is tightened and it is attained under some conditions. The effect of these improvements is exemplified.