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.

翻译：指数族是统计学、信息论和机器学习中重要的统计模型。指数族可通过累积函数实现双减归一化，或等价地通过配分函数实现双除归一化。这两种归一化函数均为严格凸且光滑的函数，可生成Bregman散度和Jensen散度对。已知指数族概率密度之间的偏斜Bhattacharryya距离等价于由累积函数在其自然参数之间诱导的偏斜Jensen散度，且在极限情况下，单侧Kullback-Leibler散度等价于反向单侧Bregman散度。本文首先证明：指数族非归一化密度之间的α-散度等价于由配分函数诱导的缩放α-偏斜Jensen散度；随后展示基于拟算术平均对的比较凸性如何能在保持普通凸性的条件下变形凸函数，并构造具有相应散度的对偶平坦空间。