Bayesian estimation and uncertainty quantification for high dimensional data

现代背景下的统计数据以越来越大的尺寸、形式和复杂性出现，例如来自不同来源的图像、视频、函数、树，包括条形码、互联网搜索、社交网络、移动的设备、卫星、基因组学、医学扫描等。然而，一些低维结构通常隐藏在这些数据中。贝叶斯决策方法在上下文中特别有用，因为数据中的结构属性可以很容易地纳入其框架中，并且可以自动量化决策过程中的不确定性。 然而，在大数据领域，计算仍然是一个挑战，因为普通的计算方法不能很好地扩展，特别是当分析中涉及大量模型时。一些最新的尖端技术的计算和评估方法将被用于拟议的研究。这将对人脑发育变量间关系的研究、基因通路分析等应用产生重要影响。将开发计算软件包，用户可免费使用。将通过在各地发表文章、举办研讨会和讲座来传播成果。拟议的研究将把各种概念连接在一起，并综合成一种强大的方法，用于分析适合STEM学科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。PI致力于让女性学生和来自代表性不足群体的学生参与进来，以促进多样性。拟议的研究将全面涉及各种类型高维数据的贝叶斯分析的理论，计算和应用。参数和非参数模型将被考虑和估计，预测，聚类和评估模型的不确定性的重要问题将被解决的各种数据类型，包括图形，网络，路径和树木。将开发事先建设，可扩展的计算和不确定性量化的技术，并将开始研究所产生的程序的频率收敛特性。一些最新的想法，连续收缩先验的计算优势，在高维设置将在拟议的研究。在非参数和高维模型中，后验收敛性质的研究是非常微妙的。使用PI和其他研究人员开发的理论工具将用于研究后验分布的收敛特性，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应用于研究大脑图像，癌症研究和其他各种情况。

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.