Counterfactual Mean Embeddings Krikamol Muandet∗
[email protected] Max Planck Institute for Intelligent Systems Tübingen, Germany Motonobu Kanagaway
[email protected] Data Science Department, EURECOM Sophia Antipolis, France Sorawit Saengkyongam
[email protected] University of Copenhagen Copenhagen, Denmark Sanparith Marukatat
[email protected] National Electronics and Computer Technology Center National Science and Technology Development Agency Pathumthani, Thailand Abstract Counterfactual inference has become a ubiquitous tool in online advertisement, recommen- dation systems, medical diagnosis, and econometrics. Accurate modelling of outcome distri- butions associated with different interventions—known as counterfactual distributions—is crucial for the success of these applications. In this work, we propose to model counter- factual 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 distri- bution. Based on this representation, we propose a distributional treatment effect (DTE) which 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 dis- tributions. 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 un- derlying marginal distributions. Furthermore, our framework allows for more complex arXiv:1805.08845v4 [stat.ML] 10 Jul 2021 outcomes such as images, sequences, and graphs.