Counterfactual Mean Embeddings

TL;DR

This paper introduces counterfactual mean embeddings (CME), a nonparametric Hilbert space representation for modeling entire counterfactual outcome distributions, enabling causal inference and distributional treatment effect estimation from observational data.

Contribution

The paper proposes a novel CME framework for modeling counterfactual distributions in a RKHS, allowing nonparametric causal inference and analysis of complex outcomes without distributional assumptions.

Findings

Effective in synthetic data experiments

Enables distributional treatment effect estimation

Applicable to complex outcomes like images and graphs

Abstract

Counterfactual inference has become a ubiquitous tool in online advertisement, recommendation systems, medical diagnosis, and econometrics. Accurate modeling of outcome distributions associated with different interventions -- known as counterfactual distributions -- is crucial for the success of these applications. In this work, we propose to model counterfactual distributions using a novel Hilbert space representation called counterfactual mean embedding (CME). The CME embeds the associated counterfactual distribution into a reproducing kernel Hilbert space (RKHS) endowed with a positive definite kernel, which allows us to perform causal inference over the entire landscape of the counterfactual distribution. Based on this representation, we propose a distributional treatment effect (DTE) that can quantify the causal effect over entire outcome distributions. Our approach is nonparametric as the CME can be estimated under the unconfoundedness assumption from observational data without requiring any parametric assumption about the underlying distributions. We also establish a rate of convergence of the proposed estimator which depends on the smoothness of the conditional mean and the Radon-Nikodym derivative of the underlying marginal distributions. Furthermore, our framework allows for more complex outcomes such as images, sequences, and graphs. Our experimental results on synthetic data and off-policy evaluation tasks demonstrate the advantages of the proposed estimator.