A Causal Calculus for Statistical Research

A calculus is proposed that admits two conditioning operators: ordinary Bayes conditioning, P (yjX = x), and causal conditioning, P (yjset(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 ..