Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family

• URL: http://arxiv.org/abs/2003.02469v3
• Date: Tue, 7 Apr 2020 04:11:00 GMT
• Title: Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family
• Authors: Frank Nielsen and Richard Nock
• Abstract summary: In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function. Our method requires only to partially factorize the densities canonically of the considered exponential family.
• Score: 38.13659821903422
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $\alpha$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similarities using legacy Application Programming Interfaces (APIs) since our method requires only to partially factorize the densities canonically of the considered exponential family.

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