Normalizing Flows for Interventional Density Estimation

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of potential outcomes. In this work, we estimate the density of potential outcomes after interventions from observational data. Specifically, we propose a novel, fully-parametric deep learning method for this purpose, called Interventional Normalizing Flows. Our Interventional Normalizing Flows offer a properly normalized density estimator. For this, we introduce an iterative training of two normalizing flows, namely (i) a teacher flow for estimation of nuisance parameters and (ii) a student flow for parametric estimation of the density of potential outcomes. For efficient and doubly-robust estimation of the student flow parameters, we develop a custom tractable optimization objective based on a one-step bias correction. Across various experiments, we demonstrate that our Interventional Normalizing Flows are expressive and highly effective, and scale well with both sample size and high-dimensional confounding. To the best of our knowledge, our Interventional Normalizing Flows are the first fully-parametric, deep learning method for density estimation of potential outcomes.