Sinkhorn Treatment Effects: A Causal Optimal Transport Measure

2026-05-08

AI Summary

This work addresses the limitation of traditional causal inference methods—such as average treatment effects—which capture only local differences in outcome distributions and thus fail to fully characterize the treatment’s impact on the entire distribution. The authors propose the Sinkhorn Treatment Effect, which for the first time integrates entropy-regularized optimal transport into causal inference. By constructing a smooth transformation of counterfactual mean embeddings, they derive a differentiable functional representation of distributional treatment effects. Building on this framework, they develop a debiased estimator with asymptotic efficiency and a multi-regularization-parameter aggregation test. Both theoretical analysis and empirical experiments demonstrate that the proposed approach substantially enhances the identification and detection of distributional causal effects on synthetic and image data.

Abstract

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual distributions. Unlike classical quantities such as the average treatment effect, this measure captures differences across entire distributions. We analyze this divergence as a statistical functional and show it can be written as a smooth transformation of counterfactual mean embeddings with an appropriate kernel. This characterization allows us to establish first-order pathwise differentiability in general, and second-order pathwise differentiability under the null hypothesis of equal counterfactual distributions. Leveraging this smoothness, we construct debiased estimators and use them to obtain asymptotically valid tests for distributional treatment effects with a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we further propose an aggregated test that combines evidence across a grid of regularization choices. Experiments on simulated and image data demonstrate the practical advantages of our estimator and testing procedure.

Problem

• treatment effect
• distributional divergence
• counterfactual distributions
• causal inference
• optimal transport

Innovation

• Sinkhorn treatment effect
• entropic optimal transport
• counterfactual distributions
• pathwise differentiability
• debiased estimation