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The Ladder of Causation describes three qualitatively different types ofactivities an agent may be interested in engaging in, namely, seeing(observational), doing (interventional), and imagining (counterfactual) (Pearland Mackenzie, 2018). The inferential challenge imposed by the causal hierarchyis that data is collected by an agent observing or intervening in a system(layers 1 and 2), while its goal may be to understand what would have happenedhad it taken a different course of action, contrary to what factually ended uphappening (layer 3). While there exists a solid understanding of the conditionsunder which cross-layer inferences are allowed from observations tointerventions, the results are somewhat scarcer when targeting counterfactualquantities. In this paper, we study the identification of nestedcounterfactuals from an arbitrary combination of observations and experiments.Specifically, building on a more explicit definition of nested counterfactuals,we prove the counterfactual unnesting theorem (CUT), which allows one to maparbitrary nested counterfactuals to unnested ones. For instance, applicationsin mediation and fairness analysis usually evoke notions of direct, indirect,and spurious effects, which naturally require nesting. Second, we introduce asufficient and necessary graphical condition for counterfactual identificationfrom an arbitrary combination of observational and experimental distributions.Lastly, we develop an efficient and complete algorithm for identifying nestedcounterfactuals; failure of the algorithm returning an expression for a queryimplies it is not identifiable.

Nested Counterfactual Identification from Arbitrary Surrogate Experiments