Mean Field Asymptotics in Statistical Inference: Variational Approach, Multiple Testing, and Predictive Inference

大数据时代对高维统计推断提出了前所未有的统计计算挑战。一个挑战是各种统计推断任务的“双重目标”性质：统计学家希望设计的程序能够实现近乎最优的统计效率，并在模型错误规范的情况下满足期望的有效性保证。此外，许多统计推断过程涉及贝叶斯成分，在大规模数据集上执行精确的贝叶斯推断在计算上具有挑战性。本项目将在一些高维统计推断任务中解决这些挑战。该项目开发的技术和方法将进一步推动包括高维统计、统计物理、优化、信息论和统计机器学习在内的广泛领域之间的相互作用。这个项目的结果预计在计算生物学、计算机视觉、神经科学、自然语言处理和多重测试中具有适用性。研究生和本科生将通过参与项目接触到这些结果，这些结果将被纳入课程。本项目旨在利用统计推理的平均场渐近理论，解决多重测试和预测推理中的统计和计算挑战。该项目主要研究三个方面的问题：1)分析thoulless - anderson - palmer （TAP）变分推理目标函数的非凸景观，并设计优化这些函数的高效算法；2)在错误发现率（FDR）控制任务中，在模型正确指定的情况下，设计程序使发现数量最大化，同时在模型错误指定的情况下控制频率的FDR；3)在预测推理任务中，设计给出合理小的预测集的程序，同时在存在模型错误规范的情况下保持覆盖的频率有效性。本研究将开发研究高维统计模型的平均场渐近性的新技术，这将可能适用于特定的统计模型之外，并将在其他科学和工程领域相关。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 era of big data poses unprecedented statistical and computational challenges in high-dimensional statistical inference. One challenge is the “dual objective” nature of various statistical inference tasks: statisticians hope to design procedures that achieve near-optimal statistical efficiency and satisfy desired validity guarantees even under model misspecification. Furthermore, many statistical inference procedures involve a Bayesian component, and performing exact Bayesian inference on large-scale datasets is computationally challenging. This project will address these challenges in some high dimensional statistical inference tasks. The techniques and methods developed in the project will further advance the interplay between a broad range of areas including high-dimensional statistics, statistical physics, optimization, information theory, and statistical machine learning. Results from this project are anticipated to have applicability in computational biology, computer vision, neuroscience, natural language processing, and multiple testing. Graduate and undergraduate students will be exposed to these results through involvement in the project, and the results will be incorporated in courses.This project aims to resolve statistical and computational challenges in multiple testing and predictive inference, using the mean field asymptotic theory of statistical inference. Focusing on a few stylized problems, the program consists of three major research thrusts: 1) analyze the non-convex landscape of Thouless-Anderson-Palmer (TAP) variational inference objective functions and design efficient algorithms for optimizing these functions; 2) in the task of false discovery rate (FDR) control, design procedures that maximize the number of discoveries when models are correctly specified while controlling the frequentist FDR even under model misspecification; and 3) in the task of predictive inference, design procedures that give reasonably small prediction sets while maintaining the frequentist validity of coverage in the presence of model misspecification. This research will develop new techniques for studying the mean field asymptotics of high-dimensional statistical models, which will likely be applicable beyond the specific statistical models and will be relevant in other areas of science and engineering.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.