A Reverse Pinsker Inequality

Research output: Working paper/PreprintPreprint

Abstract

Pinsker’s widely used inequality upper-bounds the total variation distance kP − Qk1in terms of the Kullback-Leibler divergence D(P||Q). Although in general a bound in the reverse direction is impossible, in many applications the quantity of interest is actually D∗ (v, Q) — defined, for an arbitrary fixed Q, as the
infimum of D(P||Q) over all distributions P that are v-far away from Q in total variation. We show that D∗ (v, Q) ≤ Cv2 + O(v3), where C = C(Q) = 1/2 for “balanced” distributions, thereby providing a kind of reverse Pinsker inequality. Some of the structural results obtained in the course of the proof may be of
independent interest. An application to large deviations is given
Original languageEnglish
Number of pages8
StatePublished - 2012

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