CAREER: Beyond Conditional Independence: New Model-Free Targets for High-Dimensional Inference

几十年来，统计学领域取得了巨大的成功，从简单、易于理解的模型中汲取科学见解。但是，计算机技术的进步使研究人员能够同时测量和存储大量数据，从而为理解和操纵比以往任何时候都复杂得多的系统打开了大门。事实上，机器学习领域在将预测模型拟合到这些数据集方面非常成功，但这些方法的黑箱性质使得使用通常的统计方法很难从中获得科学见解。事实上，不仅传统的统计方法在现代大数据环境中失败了，而且它们所回答的问题甚至不再有意义，因为它们所基于的模型甚至不具有近似性。在这项研究中，PI将首先提出新的统计方法来提出科学问题，这些问题在复杂的数据中有意义，但仍然有可解释的答案。其次，PI将寻找新的统计方法，以严格的方式回答这些问题，并研究这些方法的数学和计算特性，以便尽可能有效地使用它们。最后，PI将与基因组学、微生物组和政治学领域的专家合作，利用这些方法在这些领域获得新的科学见解。在整个项目中，PI还将提供免费的统计咨询服务，以帮助更广泛的研究界，与高中教师一起开发新课程，并为本科生和研究生提供丰富的教育和研究经验。为了理解协变量在高维回归中的重要性，对响应的条件独立性进行假设检验已经变得越来越流行。这种测试的吸引力在于，无论响应如何依赖于协变量，包括它们的关系是高度非线性的，包括可能是高阶的相互作用的情况，它都提供了统计上严格的洞察力，这种洞察力是定义良好的，并且可以科学地解释。然而，作为无模型推理目标的条件独立性只提供了一种科学见解，在某些应用程序中可能用处不大。PI将条件独立性扩展到新的无模型目标，用于两种类型的数据，即具有高度局部依赖协变量的数据和具有组合协变量的数据。然后，PI将完全超越条件独立性，提出新的无模型目标，而不仅仅是识别（如在变量选择中）数据中的实际数字量化关系，例如协变量与响应之间的关系或响应条件分布中两个协变量之间的相互作用。随着每一个新的目标，PI将开发全新的方法来进行强大且可证明有效的推理。所提出的目标和相关方法的新颖性将提供与其他统计领域的新联系，包括贝叶斯计算，测量理论，统计物理，实验设计，因果推理和图形模型估计。最终，本研究旨在为研究人员提供一套新工具，以超越参数目标的限制，而是利用最先进的机器学习工具，以统计原则的方式回答有关其数据的新颖和重要问题。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。

The field of statistics has seen great success over many decades drawing scientific insights from simple, easy-to-understand models. But progress in computing is now allowing researchers to measure and store huge amounts of data at once, opening the door to understanding and manipulating much more complex systems than ever before. Indeed, the field of machine learning has been very successful at fitting predictive models to such data sets, but the black-box nature of these methods makes it hard to draw scientific insights from them using the usual statistical approach. In fact, not only do classical statistical methods fail in this modern big data setting, but the questions they answer no longer even make sense because the models they are based on do not hold even approximately. In this research, the PI will first come up with new statistical ways of posing scientific questions that make sense in complex data but still have interpretable answers. Second, the PI will find new statistical methods to answer those questions in a rigorous way, and study the mathematical and computational properties of these methods so that they can be used as effectively as possible. And finally, the PI will work with experts in the areas of genomics, the microbiome, and political science to use these methods to gain new scientific insights in these fields. Throughout the project, the PI will also run a free statistical consulting service to help the broader research community, develop new curricula with high school teachers, and provide enriching educational and research experiences for undergraduate and graduate students.To understand the importance of a covariate in a high-dimensional regression, it has become increasingly popular to perform a hypothesis test for conditional independence with the response. The appeal of such a test is that it provides statistically rigorous insight that is well-defined and scientifically interpretable no matter how the response depends on the covariates, including the case when their relationship is highly nonlinear and includes interactions, possibly of high order. However, conditional independence as a model-free inferential target only provides a type of scientific insight that can be of little use in some applications. The PI will extend conditional independence to new model-free targets for two types of data on which it does not provide a useful inferential target, namely, data with highly-locally-dependent covariates and data with compositional covariates. Then, the PI will move past conditional independence entirely to propose novel model-free targets that instead of just identifying (as in variable selection) actually numerically quantify relationships in the data, such as the relationship between a covariate and a response or the interaction between two covariates in the response's conditional distribution. Along with each new target, the PI will develop entirely new methods for powerful and provably valid inference. The novelty of the proposed targets and associated methods will provide for new connections with other fields of statistics including Bayesian computation, measure theory, statistical physics, experimental design, causal inference, and graphical model estimation. Ultimately, this research aims to provide a suite of new tools for researchers to move beyond the constraints of parametric targets and instead leverage state-of-the-art machine learning tools to answer novel and important questions about their data in a statistically principled way.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.