The duo Bregman and Fenchel-Young divergences

TL;DR

This paper introduces the duo Fenchel-Young and Bregman divergences, generalizing classical divergences for exponential family distributions, with applications to calculating Kullback-Leibler divergence and Bhattacharyya distance.

Contribution

It defines the duo Fenchel-Young divergence, establishes conditions for non-negativity, and links it to other divergences, expanding the theoretical framework for exponential family distributions.

Findings

Derived a formula for KL divergence between nested exponential families.

Established the duo Fenchel-Young divergence as a non-negative measure.

Connected skewed Bhattacharyya distance to duo Jensen divergence.

Abstract

By calculating the Kullback-Leibler divergence between two probability measures belonging to different exponential families, we end up with a formula that generalizes the ordinary Fenchel-Young divergence. Inspired by this formula, we define the duo Fenchel-Young divergence and report a majorization condition on its pair of generators which guarantees that this divergence is always non-negative. The duo Fenchel-Young divergence is also equivalent to a duo Bregman divergence. We show the use of these duo divergences by calculating the Kullback-Leibler divergence between densities of nested exponential families, and report a formula for the Kullback-Leibler divergence between truncated normal distributions. Finally, we prove that the skewed Bhattacharyya distance between nested exponential families amounts to an equivalent skewed duo Jensen divergence.