Sinkhorn Treatment Effects: A Causal Optimal Transport Measure

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual outcome distributions. Unlike classical quantities such as the average treatment effect, it captures differences across entire distributions. We show that this estimand 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 asymptotically valid tests for distributional treatment effects at a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we 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.