Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means

Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the (skew) Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when considering quasiarithmetic means. Finally, we introduce a generalization of the Bhattacharyya statistical distances based on comparative means using relative convexity.