Counterfactual Identifiability via Dynamic Optimal Transport
Fabio De Sousa Ribeiro, Ainkaran Santhirasekaram, Ben Glocker
We investigate the open question of counterfactual identifiability for high-dimensional variables given observational data. Pearl (2000) insists that counterfactuals must be identifiable—that is, uniquely recoverable from observed data—to support valid causal claims. A recent line of work prioritises empirical results but lacks identifiability analysis, raising concerns about the validity of its counterfactual claims due to confounding. To tackle this, we connect Optimal Transport (OT) with causality and prove that, under standard regularity, a mechanism within a Markovian structural causal model admits a unique, dynamic OT flow which: (i) pushes the observational distribution forward to the counterfactual; (ii) preserves a vector-monotone order that generalises scalar quantiles, ensuring counterfactual equivalence. Importantly, we provide a practical model recipe built on continuous-time OT flows, validate our theory in controlled scenarios, and demonstrate significant improvements in the axiomatic soundness of counterfactuals of real data.