On Improved Bounds for Probability Metrics and $f$-Divergences

Derivation of tight bounds for probability metrics and f -divergences is of interest in information theory and statistics. This paper provides elementary proofs that lead, in some cases, to significant improvements over existing bounds; they also lead to the derivation of some existing bounds in a simplified way. The inequalities derived in this paper relate between the Bhattacharyya parameter, capacitory discrimination, chi-squared divergence, Chernoff information, Hellinger distance, relative entropy, and the total variation distance. The presentation is aimed to be self-contained. Index Terms – Bhattacharyya parameter, capacitory discrimination, Chernoff information, chi-squared divergence, f -divergence, Hellinger distance, relative entropy, total variation distance.