do-calculus

Suppose you want a probability that you could only measure by running an experiment, written P(Y given do(X=x)), but all you actually have is observational data and a causal graph. The do-calculus is a small set of rewrite rules that lets you push, pull, and erase the do-operator until the expression contains only ordinary conditional probabilities you can estimate from the data you have. If you succeed, the causal quantity is identified; if no sequence of moves works, it is provably not identifiable from that graph alone.

It consists of three rules, each licensing a transformation when a specific d-separation holds in a graph that has been surgically modified by deleting arrows into or out of the intervened nodes. Rule 1 inserts or deletes an observation; Rule 2 exchanges an intervention for an observation (the heart of adjustment); Rule 3 inserts or deletes an intervention. Iterating these rules, together with the laws of probability, can reduce a do-expression to a do-free one whenever identification is possible.

The deep result is that do-calculus is complete: for any graph and any interventional query, if the query is identifiable then the rules can derive it, and if they cannot, no method can. Familiar tools such as the backdoor and frontdoor adjustments are special cases that fall out of the calculus rather than separate tricks.