  The Ladder of Causation describes three qualitatively different types of
activities an agent may be interested in engaging in, namely, seeing
(observational), doing (interventional), and imagining (counterfactual) (Pearl
and Mackenzie, 2018). The inferential challenge imposed by the causal hierarchy
is 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 happened
had it taken a different course of action, contrary to what factually ended up
happening (layer 3). While there exists a solid understanding of the conditions
under which cross-layer inferences are allowed from observations to
interventions, the results are somewhat scarcer when targeting counterfactual
quantities. In this paper, we study the identification of nested
counterfactuals 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 map
arbitrary nested counterfactuals to unnested ones. For instance, applications
in mediation and fairness analysis usually evoke notions of direct, indirect,
and spurious effects, which naturally require nesting. Second, we introduce a
sufficient and necessary graphical condition for counterfactual identification
from an arbitrary combination of observational and experimental distributions.
Lastly, we develop an efficient and complete algorithm for identifying nested
counterfactuals; failure of the algorithm returning an expression for a query
implies it is not identifiable.
