Optimal Nonparametric Estimation of High-Dimensional Functionals in Causal Inference

因果关系是科学和政策中许多最重要问题的核心：哪种癌症治疗对哪种患者最有效？更严格的枪支法律会导致更少的凶杀案吗？因果推理关注的是用数学方法来表达这些问题，探索是否可以从数据中收集答案，如果可以，确定如何使用统计方法。因果推理的经典方法倾向于着眼于简单的总结效应，例如，如果一种治疗方法适用于整个人群，而不是完全适用于整个人群，平均结果将如何变化。然而，利用大数据，研究人员可以提出更复杂的问题，例如治疗效果如何随复杂的协变量信息而变化，或者结果密度如何随多个时间点的顺序治疗而变化。在这个项目中，PI将开发灵活的统计方法来回答这些问题，而不强加强大的假设，并将研究最优性，即一个人能多好地回答这些问题。上述问题可以看作是高维函数估计问题。这里的经典方法是使用强参数假设将这些问题简化为有限维问题。这允许使用标准方法和对最优性的深入理解，但是当真正的参数结构未知时，不正确的假设可能导致相当大的偏差和无关的效率界限。事实上，在非参数情况下所知甚少。因此，PI将开发新的高维因果函数的非参数估计，研究它们的风险，提供置信带和推理工具，并探索极小极大下界。所有方法都将在R软件中提供。本建议侧重于(a)估计反事实密度和(b)异质性处理效应的基本问题，在因果推理的三个突出领域：(i)无混淆点处理，（ii）工具变量，（iii）时变处理。此外，PI将为高维泛函的偏差校正估计和推理建立一个一般框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Causality is central to many of the most important questions in science and policy: Which cancer treatments are most effective for which patients? Would more strict gun laws result in fewer homicides? Causal inference is concerned with formulating such questions mathematically, exploring whether answers can be gleaned from data, and if so, determining how well and with what statistical methods. Classical methods in causal inference tend to aim at simple summary effects, such as how outcomes would change on average if a treatment were applied to an entire population versus not at all. However, with big data, investigators can ask more complicated questions, such as how treatment effects vary with complex covariate information, or how outcome densities would change with sequential treatments applied over many timepoints. In this project the PI will develop flexible statistical methods for answering such questions without imposing strong assumptions, and will study optimality, i.e., how well one can possibly answer such questions. The above questions can be framed as high-dimensional functional estimation problems. The classical approach here is to use strong parametric assumptions to reduce these problems to finite-dimensional ones. This allows for standard methods and a deep understanding of optimality, but when true parametric structure is unknown, incorrect assumptions can result in sizable bias and irrelevant efficiency bounds. In fact, little is known in the nonparametric case. Thus, the PI will develop novel nonparametric estimators of high-dimensional causal functionals, study their risk, provide confidence bands and inferential tools, and explore minimax lower bounds. All methods will be made available in R software. This proposal focuses on the foundational problems of (a) estimating counterfactual densities and (b) heterogeneous treatment effects, in three prominent domains of causal inference: (i) unconfounded point treatments, (ii) instrumental variables, and (iii) time-varying treatments. In addition, the PI will establish a general framework for bias-corrected estimation and inference for high-dimensional functionals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.