Bayesian Decision Theoretic Methods for Some High-Dimensional Multiple Inference Problems

该项目涵盖了统计学家在分析来自脑成像、下一代测序、大气科学、天文研究和许多其他科学研究的高维数据时面临的一些突出和重要的推理问题。在这些实验中，探测不到强信号，特别是当信号稀疏时，往往比探测不到弱信号更严重，而且随着信号变得更强，这种错误会变得更严重。这是一个重要的问题，但在为同时检验多个假设而设计的现有程序中尚未得到充分利用。此外，使用多个置信区间的选择性推断是统计研究的一个新兴领域，其重要性最近才认识到。然而，在分析具有稀疏信号的高维数据时，如果没有适当考虑到数据的稀疏性，则设计用于提供所选重要信号估计的现有区间可能会在遗漏真实信号或过于频繁地覆盖零的意义上变得无信息。该研究项目旨在开发新的创新方法，采用贝叶斯决策理论的观点，特别适合解决这些问题。它侧重于以下两个广泛的研究领域：(i)开发新的多重测试方法来控制包含II型错误严重程度的错误发现，以及（II）在零膨胀混合先验下为选定参数开发新的多重置信区间。预计该项目将对统计学的理论和实践产生广泛的影响。它可以产生新的方法来发现现代和高维科学调查中的真实信号，并为更好地利用统计来满足现代社会和科学需求铺平道路。例如，了解季节性变化下的植被变化对于在应对气候变化和粮食安全时更有效地进行土地利用管理至关重要。这个项目可能为解决可持续性问题提供新的方法。此外，人们越来越需要复杂的统计工具，以便更好地理解新技术带来的大量数据所带来的天文行为。同样，这个项目有可能满足这种需求。研究结果将通过在国家和国际会议上的演讲和讨论以及访问其他机构来传播。在这个项目下开发的软件将免费提供给科学界。

The project covers some outstanding and important inference problems that statisticians face when analyzing high-dimensional data from brain imaging, next generation sequencing, atmospheric science, astronomical studies, and many other scientific investigations. Miss-detecting a strong signal in these experiments, particularly when the signals are sparse, is often a more severe error than miss-detecting a weak signal, and this error gets more severe as the signal gets stronger. This is an important issue which has not been fully utilized in the existing procedures designed for simultaneous testing of multiple hypotheses. Also, selective inference using multiple confidence intervals is an emerging area of statistical research whose importance is being realized very recently. However, while analyzing high-dimensional data with sparse signals, the existing intervals designed to provide estimates of the selected significant signals can become non-informative in the sense of miss-covering the true signal or covering zero too often if the sparse nature of the data is not properly taken into account. This research project seeks to develop new and innovative methods taking a Bayesian decision theoretic viewpoint which is particularly well suited to tackle these issues. It focuses on the following two broad areas of research: (i) Developing new multiple testing methods controlling false discoveries incorporating the severity of type II errors, and (ii) developing new multiple confidence intervals for selected parameters under zero-inflated mixture prior. This project will be expected to have a broad impact on the theory and practice of statistics. It can produce novel methodologies to detect true signals in modern and high-dimensional scientific investigations, and pave the way for better use of statistics towards meeting modern societal and scientific needs. For instance, understanding vegetation changes under seasonal variability is crucial for more effective land use management when coping with climate changes and food security. This project can potentially offer new methodologies towards addressing that sustainability issue. Also, there is an increasing demand for sophisticated statistical tools to have better understanding of astronomical behaviors based on the influx of data created by the advent of new technologies. Again, this project can potentially meet that demand. The results will be disseminated through presentations and discussions at national and international conferences, and visits to other institutions. The software to be developed under this project will be made available, free of charge, to the scientific community.