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It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler,$\alpha$-divergences, and Jeffreys' divergences between densities belonging toa same exponential family have generic closed-form formulas relying on thestrictly convex and real-analytic cumulant function characterizing theexponential family. In this work, we report (dis)similarity formulas whichbypass the explicit use of the cumulant function and highlight the role ofquasi-arithmetic means and their multivariate mean operator extensions. Inpractice, these cumulant-free formulas are handy when implementing these(dis)similarities using legacy Application Programming Interfaces (APIs) sinceour method requires only to partially factorize the densities canonically ofthe considered exponential family.

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