Estimation of Optimal Causal Bounds via Covariate-Assisted Optimal Transport

2025-05-30

AI Summary

This paper addresses the problem of tightening bounds on causal quantities under partial identification (PI) using pre-treatment covariates. Specifically, it targets the partially identified set of the joint distribution of potential outcomes for a single unit. To this end, we propose a covariate-assisted bounding framework grounded in Conditional Optimal Transport (COT). Our key theoretical contribution is the first proof of continuity of the COT functional under the Wasserstein topology, enabling rigorous asymptotic analysis. Building on this, we construct the first direct, consistent, and nuisance-parameter-free nonparametric estimator for PI bounds. We establish theoretical convergence rates for the estimator and demonstrate—through extensive simulations—that the proposed method substantially improves both accuracy and stability of causal bound estimation compared to existing PI approaches.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

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.

Problem

• Estimating causal bounds with unobservable joint outcomes
• Addressing discontinuity in conditional optimal transport estimation
• Proposing consistent non-parametric estimator without nuisance parameters

Innovation

• Uses adapted Wasserstein distance for continuity
• Proposes direct non-parametric COT estimator
• Avoids nuisance parameter estimation effectively