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 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 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.

翻译：现有的机器学习因果推断方法通常估计以潜在结果均值表达的因果量（例如平均处理效应）。然而，此类量无法完全捕捉潜在结果分布的全部信息。本研究通过观测数据估计干预后潜在结果的密度分布。为此，我们提出一种新颖的全参数化深度学习方法——干预归一化流。具体而言，我们结合两种归一化流：(i) 用于估计干扰参数的干扰流；(ii) 用于参数化估计潜在结果密度的目标流。我们进一步基于一步偏差校正推导出可计算的优化目标，以实现目标流参数的高效双稳健估计。最终，我们的干预归一化流提供了一种具有适当归一化特性的密度估计器。通过多项实验，我们证明干预归一化流兼具表达力与高效性，且能良好适应样本规模与高维混杂因素。据我们所知，这是首个用于潜在结果密度估计的全参数化深度学习方法。