Information decompositions quantify how the Shannon information about a given random variable is distributed among several other random variables. Various requirements have been proposed that such a decomposition should satisfy, leading to different candidate solutions. Curiously, however, only two of the original requirements that determined the Shannon information have been considered, namely monotonicity and normalization. Two other important properties, continuity and additivity, have not been considered. In this contribution, we focus on the mutual information of two finite variables $Y,Z$ about a third finite variable $S$ and check which of the decompositions satisfy these two properties. While most of them satisfy continuity, only one of them is both continuous and additive.

翻译：信息分解量化了关于给定随机变量的香农信息如何在多个其他随机变量之间分布。已有研究提出了此类分解应满足的各种要求，从而衍生出不同的候选方案。然而，值得注意的是，目前仅考虑了决定香农信息的两个原始要求，即单调性和归一化。另外两个重要性质——连续性和可加性——尚未被探讨。本文聚焦于两个有限变量$Y,Z$关于第三个有限变量$S$的互信息，并检验其中哪些分解满足这两个性质。结果表明，虽然大多数分解满足连续性，但仅有一种分解兼具连续性与可加性。