A Causal Calculus for Statistical Research

Abstract

A calculus is proposed that admits two conditioning operators: ordinary Bayes conditioning, P (y|X = x), and causal conditioning, P (y|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, and proposes an operational definition of structural equations.