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 teacher flow for estimating nuisance parameters and (ii) a student flow for a parametric estimation of the density of potential outcomes. We further develop a tractable optimization objective based on a one-step bias correction for an efficient and doubly robust estimation of the student 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 fully-parametric, deep learning method for density estimation of potential outcomes.

翻译：现有的因果推断机器学习方法通常估计以潜在结果均值表示的量（例如平均处理效应）。然而，此类量无法捕捉潜在结果分布的完整信息。本研究旨在从观测数据中估计干预后潜在结果的密度函数。为此，我们提出一种新颖的全参数深度学习方法——干预标准化流。具体而言，该方法融合了两个标准化流：（i）用于估计冗余参数的教师流，以及（ii）用于对潜在结果密度进行参数估计的学生流。我们进一步开发了基于单步偏差校正的可优化目标函数，以实现对学生流参数的高效且鲁棒的双稳健估计。由此，我们的干预标准化流提供了具备正规化特性的密度估计器。通过多项实验证明，该模型具有高表达力与高效性，且能良好适应样本规模增长及高维混杂因素。据我们所知，干预标准化流是首个针对潜在结果密度估计的全参数化深度学习方法。