Exponential families are statistical models which are the workhorses in
statistics, information theory, and machine learning. An exponential family can
either be normalized subtractively by its cumulant function or equivalently
normalized divisively by its partition function. Both subtractive and divisive
normalizers are strictly convex and smooth functions inducing pairs of Bregman
and Jensen divergences. It is well-known that skewed Bhattacharryya distances
between probability densities of an exponential family amounts to skewed Jensen
divergences induced by the cumulant function between their corresponding
natural parameters, and in limit cases that the sided Kullback-Leibler
divergences amount to reverse-sided Bregman divergences. In this note, we first
show that the $\alpha$-divergences between unnormalized densities of an
exponential family amounts scaled $\alpha$-skewed Jensen divergences induced by
the partition function. We then show how comparative convexity with respect to
a pair of quasi-arithmetic means allows to deform convex functions and define
dually flat spaces with corresponding divergences when ordinary convexity is
preserved.

指数族是统计学、信息论和机器学习中的主要模型，其可以通过累积函数或者分区函数进行标准化。减法式和除法式标准化都是严格凸函数，并且引发一对 Bregman 和 Jensen 散度。本研究首先证明了指数族非标准化密度之间的 α- 散度等于分区函数引发的 α- 偏倚 Jensen 散度。然后，研究展示了配对的准算术平均对比可用于定义凸函数变形和对应的对偶平面以及散度的比较凸性。