Sinkhorn Treatment Effects: A Causal Optimal Transport Measure
Authors: Medha Agarwal, Alex Luedtke
Summary
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
Relevance
Not directly related to my persona/midtraining work—this is about causal inference and treatment effect estimation in general statistical settings, not about language model behavior or conditioning.
Abstract
arXiv:2605.08485v1 Announce Type: new 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.