Transporting the Past: An Optimal Transport View of Backtracking Counterfactuals

Zhangyi Liu, Pouria Fatemi, Suvrit Sra

We study backtracking counterfactual inference for Structural Causal Models (SCMs) through the lens of optimal transport. In SCMs, a factual observation generally induces a posterior distribution over exogenous variables, making counterfactual inference inherently distributional. We introduce two complementary frameworks that transport this posterior onto the set of exogenous configurations satisfying a desired counterfactual query. The first constructs a natural pointwise projection and is suitable when the transported exogenous distribution can be used directly. The second enforces mutual independence among counterfactual exogenous variables, so that the transported variables remain interpretable as genuine exogenous noises of the same SCM. We show that both frameworks are consistent with the original backtracking counterfactual principle, and we propose marginal, penalty-based, and hard-constrained solvers for finite-posterior settings. Synthetic experiments on non-bijective SCMs illustrate the geometry of the problem and reveal trade-offs among transport cost, counterfactual constraint satisfaction, and preservation of exogenous independence.