Some inequalities relating different measures of divergence between two probability distributions

This note presents new inequalities relating different divergence measures in the family of “convex likelihood-ratio expectation” measures of Csiszar (1967), Ali and Silvey (1966), and especially in the single-parameter family of “AM-GM” divergence measures. The most prominent result is that θ² ⩽¼J, where θ is the Bhattacharyya angle of divergence (a true distance metric), and J is the symmetric cross-entropy. A pair of “log Γ” divergences is also introduced and related to the cross-entropies I and J