On Pinsker's Type Inequalities and Csiszar's f-divergences. Part I: Second and Fourth-Order Inequalities

Abstract

We study conditions on $f$ under which an $f$-divergence $D_f$ will satisfy $D_f \geq c_f V^2$ or $D_f \geq c_{2,f} V^2 + c_{4,f} V^4$, where $V$ denotes variational distance and the coefficients $c_f$, $c_{2,f}$ and $c_{4,f}$ are {\em best possible}. As a consequence, we obtain lower bounds in terms of $V$ for many well known distance and divergence measures. For instance, let $D_{(\alpha)} (P,Q) = [\alpha (\alpha-1)]^{-1} [\int q^{\alpha} p^{1-\alpha} d \mu -1]$ and ${\cal I}_\alpha (P,Q) = (\alpha -1)^{-1} \log [\int p^\alpha q^{1-\alpha} d \mu]$ be respectively the {\em relative information of type} ($1-\alpha$) and {\em R\'{e}nyi's information gain of order} $\alpha$. We show that $D_{(\alpha)} \geq {1/2} V^2 + {1/72} (\alpha+1)(2-\alpha) V^4$ whenever $-1 \leq \alpha \leq 2$, $\alpha \not= 0,1$ and that ${\cal I}_{\alpha} = \frac{\alpha}{2} V^2 + {1/36} \alpha (1 + 5 \alpha - 5 \alpha^2) V^4$ for $0 < \alpha < 1$. Pinsker's inequality $D \geq {1/2} V^2$ and its extension $D \geq {1/2} V^2 + {1/36} V^4$ are special cases of each one of these.