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Exponential families are statistical models which are the workhorses instatistics, information theory, and machine learning among others. Anexponential family can either be normalized subtractively by its cumulant orfree energy function or equivalently normalized divisively by its partitionfunction. Both subtractive and divisive normalizers are strictly convex andsmooth functions inducing pairs of Bregman and Jensen divergences. It iswell-known that skewed Bhattacharryya distances between probability densitiesof an exponential family amounts to skewed Jensen divergences induced by thecumulant function between their corresponding natural parameters, and in limitcases that the sided Kullback-Leibler divergences amount to reverse-sidedBregman divergences. In this paper, we first show that the $\alpha$-divergencesbetween unnormalized densities of an exponential family amounts to scaled$\alpha$-skewed Jensen divergences induced by the partition function. We thenshow how comparative convexity with respect to a pair of quasi-arithmetic meansallows to deform both convex functions and their arguments, and thereby definedually flat spaces with corresponding divergences when ordinary convexity ispreserved.

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