Inference for Functionals in High-Dimensional Regression

现代科学和工程应用涉及具有大量变量或特征的大型数据集。这方面的一个关键挑战是区分科学相关变量和不相关变量--换句话说，区分信号和噪声。这些变量之间微妙的非线性关系加剧了这一挑战。广义线性模型是经典统计学中发现这种非线性关系最常用的工具，即使在当代大数据环境中也经常使用。不幸的是，传统上用来证明这些方法有效性的经典统计理论在这一领域失败了。该项目将开发新的方法，用于在广义线性模型的框架内推断科学相关的参数，适用于高维或大数据的设置。所开发的理论将有助于在基因组学，计算神经科学，信号和图像处理等应用中观察到的变量之间的关系的原则性推理。主要研究者还将通过指导研究生参与该项目，并开发将纳入该项目成果的课程。该研究项目将围绕两大主题开发用于推断高维广义线性模型中科学相关的低维泛函的统计理论和方法：（1）信噪比型泛函的频率论推断;（2）连续收缩先验下泛函的贝叶斯推断。第一个主题将开发新的估计的信噪比和遗传相关性，一个泛化的信噪比，测量统计遗传学中的多个性状之间的共享遗传基础。第二个推力将构建数据驱动的可信区间的基础信号的组件下计算易处理的连续收缩先验。这两个推力将开发推理程序不可知的稀疏水平的基础信号。为了实现这一目标，该研究将集中在比例渐近高维制度，并利用近似的消息传递理论，最初在概率，信息论和统计物理学的新见解。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Modern science and engineering applications involve large datasets with a multitude of variables or features. A key challenge in this context is to distinguish the scientifically relevant variables from the irrelevant ones - in other words, the signal from the noise. The challenge is compounded by subtle nonlinear relationships among these variables. Generalized linear models are the most often used tools in classical statistics for discovering such nonlinear relationships and they are routinely employed, even in contemporary big data settings. Unfortunately, classical statistical theory, traditionally used to justify the validity of these methods, fails in this regime. This project will develop novel approaches for inferring scientifically relevant parameters in the framework of generalized linear models, adapted to the setting of high-dimensional or big data. The theory developed will facilitate principled inference regarding the relations among observed variables in applications such as genomics, computational neuroscience, signal and image processing. The principal investigator will also engage graduate students in the project by mentoring them and develop courses that will incorporate results from this project.This research project will develop statistical theory and methods for inferring scientifically relevant low-dimensional functionals in high-dimensional generalized linear models, organized around two broad themes: (1) frequentist inference for signal-to-noise ratio type functionals; (2) Bayesian inference for functionals under continuous shrinkage priors. The first theme will develop novel estimators for the signal-to-noise ratio and the genetic relatedness, a generalization of the signal-to-noise ratio that measures the shared genetic basis between multiple traits in statistical genetics. The second thrust will construct data-driven credible intervals for components of the underlying signal under computationally tractable continuous shrinkage priors. Both thrusts will develop inference procedures agnostic to sparsity level of the underlying signal. To achieve this, the research will focus on the proportional asymptotics high-dimensional regime and utilize novel insights from approximate message passing theory, developed originally in probability, information theory, and statistical physics.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.