Tight Lower Bounds for $\alpha$-Divergences Under Moment Constraints and Relations Between Different $\alpha$

Abstract

The $\alpha$-divergences include the well-known Kullback-Leibler divergence, Hellinger distance and ${\chi }^2$-divergence. In this paper, we derive differential and integral relations between the $\alpha$-divergences that are generalizations of the relation between the Kullback-Leibler divergence and the ${\chi }^2$-divergence. We also show tight lower bounds for the $\alpha$-divergences under given means and variances. In particular, we show a necessary and sufficient condition such that the binary divergences, which are divergences between probability measures on the same $2$-point set, always attain lower bounds. Kullback-Leibler divergence, Hellinger distance, and ${\chi }^2$-divergence satisfy this condition.

13 pages. arXiv admin note: text overlap with arXiv:2010.13548

References in corpus (3)

• Scalable Information Inequalities for Uncertainty Quantification
• On Relations Between the Relative entropy and ${\chi }^2$-Divergence, Generalizations and Applications
• A Tight Lower Bound for the Hellinger Distance with Given Means and Variances