We propose a new concept of codivergence, which quantifies the similarity between two probability measures $P_1, P_2$ relative to a reference probability measure $P_0$. In the neighborhood of the reference measure $P_0$, a codivergence behaves like an inner product between the measures $P_1 - P_0$ and $P_2 - P_0$. Codivergences of covariance-type and correlation-type are introduced and studied with a focus on two specific correlation-type codivergences, the $\chi^2$-codivergence and the Hellinger codivergence. We derive explicit expressions for several common parametric families of probability distributions. For a codivergence, we introduce moreover the divergence matrix as an analogue of the Gram matrix. It is shown that the $\chi^2$-divergence matrix satisfies a data-processing inequality.

翻译：我们提出共散度（codivergence）这一新概念，用于量化两个概率测度 $P_1, P_2$ 相对于参考概率测度 $P_0$ 的相似性。在参考测度 $P_0$ 的邻域内，共散度的行为类似于测度 $P_1 - P_0$ 与 $P_2 - P_0$ 之间的内积。本文引入并研究了协方差型与相关型共散度，重点关注两类特定相关型共散度：$\chi^2$ 共散度与 Hellinger 共散度。针对若干常见参数概率分布族，我们推导出显式表达式。此外，对于共散度，我们引入散度矩阵作为 Gram 矩阵的类比，并证明 $\chi^2$ 散度矩阵满足数据处理不等式。