Normalizing Flows for Interventional Density Estimation
This work addresses a gap in causal inference for researchers and practitioners needing full distributional information beyond average effects, though it is incremental as it builds on existing normalizing flow techniques.
The paper tackles the problem of estimating the full density of potential outcomes after interventions from observational data, rather than just mean effects like average treatment effect, by proposing Interventional Normalizing Flows, a novel deep learning method that combines two normalizing flows for nuisance parameter estimation and parametric density estimation, resulting in a properly normalized density estimator that scales well with sample size and high-dimensional confounding.
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.