Counterfactual Graphical Models: Constraints and Inference

Graphical models have been widely used as parsimonious encoders of the constraints underlying probability models. When organized in a structured way, these models can facilitate the derivation of non-trivial constraints, the inference of quantities of interest, and the optimization of their estimands. In particular, causal diagrams enable the efficient representation of the structural constraints of the underlying causal system. In this paper, we introduce an efficient graphical construction called Ancestral Multi-world Networks that is sound and complete for reading counterfactual independences from a causal diagram using d-separation. Moreover, we introduce the counterfactual (ctf-) calculus, which can be used to transform counterfactual quantities using three rules licensed by the constraints encoded in the diagram. This result generalizes Pearl's celebrated do-calculus from interventional to counterfactual reasoning.