  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. For
this, we propose a novel, fully-parametric deep learning method called
Interventional Normalizing Flows. Specifically, we combine two normalizing
flows, namely (i) a nuisance flow for estimating nuisance parameters and (ii) a
target flow for parametric estimation of the density of potential outcomes. We
further develop a tractable optimization objective based on a one-step bias
correction for efficient and doubly robust estimation of the target flow
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 density estimation of potential
outcomes.
