Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family
This provides a more practical method for statisticians and machine learning practitioners working with exponential families, though it is incremental as it builds on existing divergence formulas.
The paper tackled the problem of computing common divergences between exponential family densities without needing the cumulant function, resulting in formulas that rely on quasi-arithmetic means and are easier to implement with legacy APIs.
It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $α$-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.