The authors introduce the Sinkhorn treatment effect, a measure that uses optimal transport to capture how entire counterfactual distributions differ (not just means). They show it can be written as a smooth transformation of counterfactual mean embeddings in a reproducing kernel Hilbert space, prove it's pathwise differentiable, and use this to build debiased estimators and valid statistical tests. They also propose an aggregated test that combines evidence across multiple regularization parameters.
Main takeaways:
- The Sinkhorn treatment effect measures distributional differences between counterfactual outcomes using entropic optimal transport, capturing more than average treatment effects
- It's a smooth functional of counterfactual mean embeddings with an appropriate kernel, which enables statistical analysis
- First-order pathwise differentiable in general; second-order under the null hypothesis (equal distributions)
- This smoothness allows construction of debiased estimators and asymptotically valid hypothesis tests
- An aggregated test combines evidence across a grid of regularization choices since test power depends on the unknown regularization parameter