A Distribution Dependent Refinement of Pinsker's Inequality

Abstract

Given two probability distributions Q and P, let /spl par/Q-P/spl par//sub 1/ and D(Q/spl par/P), respectively, denote the L/sub 1/ distance and divergence between Q and P. We derive a refinement of Pinsker's inequality of the form D(Q/spl par/P)/spl ges/c(P)/spl par/Q-P/spl par//sub 1//sup 2/ and characterize the best P-dependent factor c(P). We apply the refined inequality to large deviations and measure concentration.