  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.
