Bayesian estimation and uncertainty quantification for high dimensional data

现代背景下的统计数据呈现出越来越大的规模、形式和复杂性，如图像、视频、功能、来自不同来源的树，包括条形码、互联网搜索、社交网络、移动设备、卫星、基因组学、医学扫描等。这些数据通常在大小和维度上都很巨大。然而，一些低维结构通常隐藏在这样的数据中。在这种情况下，决策的贝叶斯方法特别有用，因为数据中的结构属性可以很容易地纳入其框架，并可以自动量化决策过程中的不确定性。然而，计算在大数据领域仍然是一个挑战，因为常见的计算方法不能很好地扩展，特别是在分析涉及大量模型的情况下。在拟议的研究中，将使用一些最新的尖端计算和评估方法。这对于研究人脑发育过程中变量间的关系、基因通路分析等应用具有重要意义。将开发计算程序包，并将允许用户免费访问。结果将通过在不同地方举行的文章、研讨会和讲座来传播。这项拟议的研究将把各种概念联系在一起，并综合成一种强大的方法，用于分析适合于STEM学科中特定学科和跨学科研究的高维数据。该提案的教育部分将以研究生咨询和提供专题课程的形式影响人力资源开发。国际学生协会致力于让女性学生和代表不足群体的学生参与，以促进多样性。拟议的研究将全面涉及各种类型的高维数据的贝叶斯分析的理论、计算和应用。将同时考虑参数模型和非参数模型，并将讨论各种数据类型(包括图形、网络、路径和树)的估计、预测、聚类和评估模型不确定性的重要问题。将开发预先构造、可伸缩计算和不确定性量化的技术，并将开始研究由此产生的过程的频率收敛特性。一些最新的关于连续收缩先验的想法在高维环境下具有计算优势，将被用于拟议的研究中。在非参数和高维模型中，后验收敛性质的研究是非常精细的。利用PI和其他研究人员开发的理论工具，将被用来研究后验分布的收敛性质，从而有助于确定最有效的方法。从拟议的研究中开发的方法将被应用于研究大脑图像、癌症研究和各种其他背景。

Statistical data in modern context appear in increasing size, form and complexity such as images, videos, functions, trees from diverse sources including barcodes, internet searches, social networks, mobile devices, satellites, genomics, medical scans etc. Such data are typically huge in size and dimension. Nevertheless, some lower dimensional structures is commonly hidden within such data. The Bayesian approach to decision making is particularly useful in the context since structural property in the data can be easily incorporated in its framework and can automatically quantify the uncertainty in the decision making process. Computation however remains a challenge in the big data regime since common computing methods do not scale well, especially when a large number of models are involved in the analysis. Some of the newest cutting-edge techniques for computing and evaluating methods will be used in the proposed research. It will have significant impact on studying relations between variables in human brain development, gene-pathway analysis and other applications. Computational packages will be developed and users will be given free access. Results will be disseminated through articles, seminars and talks given at various places. The proposed research will connect various concepts together and synthesize into a powerful approach for analyzing high dimensional data appropriate for subject specific and interdisciplinary research in STEM disciplines. The educational component of the proposal will impact human resource development in the form of graduate student advising and offering of special topics courses. The PI is committed to involving female students and students from under-represented groups to promote diversity.The proposed research will have all round involvement in theory, computation and application concerning Bayesian analysis of high dimensional data of various types. Both parametric and nonparametric models will be considered and important issues of estimation, prediction, clustering and assessing model uncertainty will be addressed for a variety of data types including graphs, networks, pathways and trees. Techniques of prior construction, scalable computation and uncertainty quantification will be developed and study of frequentist convergence properties of the resulting procedures will be initiated. Some of the most recent ideas on continuous shrinkage priors which have computational advantage in the high dimensional setting will be employed in the proposed research. Study of posterior convergence properties is extremely delicate in nonparametric and high dimensional models. Using the theoretical tools developed by the PI and other researchers will be employed to study convergence properties of the posterior distributions, and thus will help identify the most efficient methods. The methods developed from the proposed research will be applied in studying brain images, cancer studies and various other contexts.