A Causal Calculus for Statistical Research

Many statisticians are reluctant to deal with problems involving causal considerations because we lack the mathematical notation for distinguishing causal influence from statistical association. To address this problem, a notation is proposed that admits two conditioning operators: ordinary Bayes conditioning, $P(y \mid X=x)$, and causal conditioning, $P(y \mid \operatorname{set}(X=x))$, that is, conditioning $P(y)$ on holding $X$ constant (at $x)$ by external intervention. This distinction, which will be supported by three rules of inference, will permit us to derive probability expressions for the combined effect of observations and interventions. The resulting calculus yields simple solutions to a number of interesting problems in causal inference and should allow rank-and-file researchers to tackle practical problems that are generally considered too hard, or impossible. Examples are: 1. Deciding whether the information available in a given observational study is sufficient for obtaining consistent estimates of causal effects. 2. Deriving algebraic expressions for causal effect estimands. 3. Selecting measurements that would render randomized experiments unnecessary. 4. Selecting a set of indirect (randomized) experiments to replace direct experiments that are either infeasible or too expensive. 5. Predicting (or bounding) the efficacy of treatments from randomized trials with imperfect compliance. Starting with nonparametric specification of structural equations, the paper establishes the semantics necessary for a theory of interventions, presents the three rules of inference, demonstrates the use of the resulting calculus on a number of examples, and establishes an operational definition of structural equations.