Conditional Counterfactual Mean Embeddings: Doubly Robust Estimation and Learning Rates

A complete understanding of heterogeneous treatment effects involves characterizing the full conditional distribution of potential outcomes. To this end, we propose the Conditional Counterfactual Mean Embeddings (CCME), a framework that embeds conditional distributions of counterfactual outcomes into a reproducing kernel Hilbert space (RKHS). Under this framework, we develop a two-stage meta-estimator for CCME that accommodates any RKHS-valued regression in each stage. Based on this meta-estimator, we develop three practical CCME estimators: (1) Ridge Regression estimator, (2) Deep Feature estimator that parameterizes the feature map by a neural network, and (3) Neural-Kernel estimator that performs RKHS-valued regression, with the coefficients parameterized by a neural network. We provide finite-sample convergence rates for all estimators, establishing that they possess the double robustness property. Our experiments demonstrate that our estimators accurately recover distributional features including multimodal structure of conditional counterfactual distributions.

💡 Research Summary

This paper introduces Conditional Counterfactual Mean Embeddings (CCME), a novel framework for representing the full conditional distribution of a potential outcome in a reproducing kernel Hilbert space (RKHS). While prior work on Counterfactual Mean Embeddings (CME) focused on marginal (unconditional) counterfactual distributions, CCME extends the idea to distributions conditioned on a set of covariates V = η(X), thereby enabling a richer description of heterogeneous treatment effects that goes beyond scalar summaries such as average treatment effect (ATE) or conditional average treatment effect (CATE).

Key methodological contributions

1. Double‑robust identification in RKHS – The authors define two nuisance functions: the propensity score π(x)=P(A=1|X=x) and the treated‑group conditional mean embedding μ₀(x)=E