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Some of the most prominent results in causal inference have been developed in the context of atomic interventions, following the semantics of the do-operator and the inferential power of the do-calculus. In practice, many real-world settings require more complex types of interventions that cannot be represented by a simple atomic intervention. In this paper, we investigate a general class of interventions that covers some non-trivial types of policies (conditional and stochastic), which goes beyond the atomic class. Our goal is to develop general understanding and formal machinery to be able to reason about the effects of those policies, similar to the robust treatment developed to handle the atomic case. Specifically, in this paper, we introduce a new set of inference rules (akin to do-calculus) that can be used to derive claims about general interventions, which we call σ-calculus. We develop a systematic and efficient procedure for finding estimands of the effect of general policies as a function of the available observational and experimental distributions. We then prove that our algorithm and σ-calculus are both sound for the tasks of identification (Pearl, 1995) and z-identification (Bareinboim and Pearl, 2012) under this class of interventions.

A Calculus for Stochastic Interventions:Causal Effect Identification and Surrogate Experiments