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.

翻译：现有用于因果推断的机器学习方法通常估计以潜在结果均值表达的统计量（如平均处理效应）。然而，此类统计量无法捕捉潜在结果分布的完整信息。本文从观测数据出发，对干预后潜在结果的密度进行估计。为此，我们提出了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，该方法结合了两种归一化流：（i）用于估计干扰参数的干扰流，以及（ii）用于对潜在结果密度进行参数化估计的目标流。我们进一步基于单步偏差校正开发了一个可处理的优化目标，以实现目标流参数的高效且双重稳健估计。由此，我们的干预归一化流能够提供适当归一化的密度估计量。通过多项实验，我们证明了干预归一化流具有强大的表达能力和高效性，并能良好地适应样本量增长与高维混淆因素。据我们所知，干预归一化流是首个用于潜在结果密度估计的、恰当的全参数化深度学习方法。