Mutual information between two random variables is a well-studied notion, whose understanding is fairly complete. Mutual information between one random variable and a pair of other random variables, however, is a far more involved notion. Specifically, Shannon's mutual information does not capture fine-grained interactions between those three variables, resulting in limited insights in complex systems. To capture these fine-grained interactions, in 2010 Williams and Beer proposed to decompose this mutual information to information atoms, called unique, redundant, and synergistic, and proposed several operational axioms that these atoms must satisfy. In spite of numerous efforts, a general formula which satisfies these axioms has yet to be found. Inspired by Judea Pearl's do-calculus, we resolve this open problem by introducing the do-operation, an operation over the variable system which sets a certain marginal to a desired value, which is distinct from any existing approaches. Using this operation, we provide the first explicit formula for calculating the information atoms so that Williams and Beer's axioms are satisfied, as well as additional properties from subsequent studies in the field.

翻译：两个随机变量之间的互信息是一个研究充分的概念，其理解已相当完整。然而，一个随机变量与另一对随机变量之间的互信息则是一个更为复杂的概念。具体而言，香农互信息无法捕捉这三个变量之间的细粒度交互作用，从而限制了在复杂系统中的洞察力。为捕捉这些细粒度交互，威廉姆斯和比尔于2010年提出将该互信息分解为信息原子，即唯一信息、冗余信息和协同信息，并提出了这些原子必须满足的几个操作公理。尽管付出了诸多努力，但满足这些公理的一般公式至今仍未找到。受朱迪亚·珀尔的do-演算启发，我们通过引入do操作（一种对变量系统的操作，可将特定边缘分布设置为期望值，这不同于任何现有方法）解决了这一开放问题。利用该操作，我们首次提出了计算信息原子的显式公式，该公式不仅满足威廉姆斯和比尔的公理，还满足该领域后续研究中的其他性质。