The duo Bregman and Fenchel-Young divergences

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.

arXiv comment22 pages, 7 figures

References in corpus (6)

• An elementary introduction to information geometry
• The Burbea-Rao and Bhattacharyya centroids
• On a Variational Definition for the Jensen-Shannon Symmetrization of Distances based on the Information Radius
• Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions
• An Extension of Young's Inequality
• A Computationally Efficient Method for Learning Exponential Family Distributions

Cited by in corpus (1)

• Divergences induced by dual subtractive and divisive normalizations of exponential families and their convex deformations