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Statistical Divergences between Densities of Truncated Exponential Families with Nested Supports: Duo Bregman and Duo Jensen Divergences

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By calculating the Kullback–Leibler divergence between two probability measures belonging to different exponential families dominated by the same measure, we obtain a formula that generalizes the ordinary Fenchel–Young divergence. Inspired… Click to show full abstract

By calculating the Kullback–Leibler divergence between two probability measures belonging to different exponential families dominated by the same measure, we obtain 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 strictly convex 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 how to use these duo divergences by calculating the Kullback–Leibler divergence between densities of truncated exponential families with nested supports, and report a formula for the Kullback–Leibler divergence between truncated normal distributions. Finally, we prove that the skewed Bhattacharyya distances between truncated exponential families amount to equivalent skewed duo Jensen divergences.

Keywords: duo; truncated exponential; duo bregman; divergence; exponential families; densities truncated

Journal Title: Entropy
Year Published: 2022

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