Sinkhorn Treatment Effects: A Causal Optimal Transport Measure

TL;DR

This paper introduces the Sinkhorn treatment effect, a new divergence measure between counterfactual distributions based on entropic optimal transport, enabling distributional treatment effect testing.

Contribution

It develops a smooth, differentiable divergence measure for counterfactual distributions, along with debiased estimators and tests, including an aggregated test over regularization parameters.

Findings

The Sinkhorn treatment effect captures distributional differences effectively.

Proposed estimators are asymptotically valid and demonstrate practical advantages.

Experiments show improved performance on simulated 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.