An Improvement on Vajda’s Inequality

Abstract

Let D and V be respectively information divergence and variational distance. It is shown that $D \geqslant \log \tfrac{2} {{2 - V}} - \tfrac{{2 - V}} {2}\log \tfrac{{2 + V}} {2}$, hence improving Vajda’s inequality $D \geqslant \log \tfrac{{2 + V}} {{2 - V}} - \tfrac{{2V}} {{2 + V}}$. The proof is based on a lemma which states that for any f-divergence symmetric in the sense that D _f (P, Q)=D _f (Q, P), one has that inf$\left\{ {D_f \left( {P,Q} \right):V\left( {P,Q} \right) = v} \right\} = \tfrac{{2 - v}} {2}f\tfrac{{2 + v}} {{2 - v}} - f'\left( 1 \right)v$. This lemma has interest on its own and implies precise lower bounds for several well-known divergences.

Research partially supported by CAPES, CNP_q and FINATEC grants.