Estimation of Optimal Causal Bounds via Covariate-Assisted Optimal Transport

Abstract: We study the estimation of causal estimand involving the joint distribution of treatment and control outcomes for a single unit. In typical causal inference settings, it is impossible to observe both outcomes simultaneously, which places our estimation within the domain of partial identification (PI). Pre-treatment covariates can substantially reduce estimation uncertainty by shrinking the partially identified set. Recently, it was shown that covariate-assisted PI sets can be characterized through conditional optimal transport (COT) problems. However, finite-sample estimation of COT poses significant challenges, primarily because, as we explain, the COT functional is discontinuous under the weak topology, rendering the direct plug-in estimator inconsistent. To address this issue, existing literature relies on relaxations or indirect methods involving the estimation of non-parametric nuisance statistics. In this work, we demonstrate the continuity of the COT functional under a stronger topology induced by the adapted Wasserstein distance. Leveraging this result, we propose a direct, consistent, non-parametric estimator for COT value that avoids nuisance parameter estimation. We derive the convergence rate for our estimator and validate its effectiveness through comprehensive simulations, demonstrating its improved performance compared to existing approaches.