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被充分研究的概念，其理解已相当完备。然而，一个随机变量与另外两个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，从而在复杂系统分析中导致局限性。为了捕捉这些细粒度交互，Williams和Beer于2010年提出将这种互信息分解为信息原子，即独有信息、冗余信息和协同信息，并提出了这些信息原子必须满足的若干操作公理。尽管已有众多努力，但满足这些公理的通用公式仍未找到。受Judea Pearl的do-演算启发，我们通过引入do-操作解决了这一开放性问题。该操作作用于变量系统，可将某一边际分布设定为期望值，这与现有方法截然不同。利用这一操作，我们首次给出了计算信息原子的显式公式，使其满足Williams和Beer的公理，以及该领域后续研究中提出的附加性质。