By calculating the Kullback-Leibler divergence between two probability measures belonging to different exponential families, we end up with a formula that generalizes the ordinary Fenchel-Young divergence. Inspired by this formula, we define the duo Fenchel-Young divergence and report a majorization condition on its pair of generators which guarantees that this divergence is always non-negative. The duo Fenchel-Young divergence is also equivalent to a duo Bregman divergence. We show the use of these duo divergences by calculating the Kullback-Leibler divergence between densities of nested exponential families, and report a formula for the Kullback-Leibler divergence between truncated normal distributions. Finally, we prove that the skewed Bhattacharyya distance between nested exponential families amounts to an equivalent skewed duo Jensen divergence.

翻译：通过计算属于不同指数族的两个概率测度之间的Kullback-Leibler散度，我们得到了一个推广普通Fenchel-Young散度的公式。受此公式启发，我们定义了双重Fenchel-Young散度，并给出了其生成函数对的一个优化条件，该条件保证该散度始终非负。双重Fenchel-Young散度也等价于双重Bregman散度。我们通过计算嵌套指数族密度之间的Kullback-Leibler散度来展示这些双重散度的应用，并给出了截断正态分布之间Kullback-Leibler散度的计算公式。最后，我们证明了嵌套指数族之间的偏斜Bhattacharyya距离等价于一个相应的偏斜双重Jensen散度。