Normalizing Flows for Interventional Density Estimation

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of <PRE_TAG>potential outcomes</POST_TAG> (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of <PRE_TAG>potential outcomes</POST_TAG>. In this work, we estimate the density of <PRE_TAG>potential outcomes</POST_TAG> after interventions from observational data. For this, we propose a novel, fully-parametric deep learning method called Interventional Normalizing Flows. Specifically, we combine two normalizing flows, namely (i) a <PRE_TAG>nuisance flow</POST_TAG> for estimating nuisance parameters and (ii) a <PRE_TAG>target flow</POST_TAG> for parametric estimation of the density of <PRE_TAG>potential outcomes</POST_TAG>. We further develop a tractable optimization objective based on a one-step bias correction for efficient and <PRE_TAG>doubly robust estimation</POST_TAG> of the <PRE_TAG>target flow</POST_TAG> parameters. As a result, our Interventional Normalizing Flows offer a properly normalized density estimator. 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 proper fully-parametric, deep learning method for <PRE_TAG>density estimation</POST_TAG> of potential outcomes.