Collaborative Research: Objective Bayesian Model Selection and Estimation in High Dimensional Statistical Models

人们普遍认为，在许多高维情况下，模型选择必须在参数估计之前或同时进行，以减少所考虑的参数的数量。事实上，模型选择是统计学家在处理高维数据时面临的主要挑战之一。工具，如正则化和稀疏性是一些常见的概念，用于获得简约模型来解释观测数据。近年来，统计领域已经见证了爆炸的频率和贝叶斯方法的高维问题。尽管有这些和其他的进步，贝叶斯模型选择在“客观”的意义上在高维问题仍然是一个重要的问题，尚未得到满意的解决。对客观性的需求转化为对指定无信息的不适当先验的需求，这反过来又使传统的贝叶斯因子不可用。该项目提出了在一个大类的高维图形模型推导出客观贝叶斯估计和模型选择程序。因此，在这个项目中提出的方法的目的是有助于急需的理论在该地区的客观贝叶斯模型选择高维图形模型。在这个过程中，方法学研究的优点和缺点的客观贝叶斯方法在这种情况下。所开发的理论为高维环境中的模型选择/估计提供了算法和计算技术。吞吐量或高维数据的可用性几乎涉及科学的每个领域。需要制定正确的模型，解释观察到的高维数据渗透到许多科学领域。事实上，这种数据的变量数量往往比样本数量高得多，被称为“大p小n”问题，现在比以往任何时候都更加普遍。在高维数据中发现统计信号，提出正确的模型来解释这些数据，以及在这些高维环境中进行参数估计是现代统计学家必须应对的一些主要挑战。此外，这些挑战也出现在高风险的辩论中，如气候变化，某些药物在临床试验中的有效性，以及癌症研究中各种生物标志物的相关性。该项目建议开发专门针对识别模型的统计方法，这些模型以客观的方式解释高维数据。特别是该项目旨在发展高维问题中更好的客观贝叶斯模型选择和参数估计方法，具有广泛的应用。PI和co-PI与应用领域的科学家合作，特别是与医学院，工程和环境科学学院的教师/研究人员合作。研究生的培训和指导是这种合作研究的一个组成部分。该项目的科学产出将在高影响力的同行评审期刊上发表。

It is widely accepted that in many high dimensional situations, model selection has to be performed either before parameter estimation or simultaneously, in order to reduce the number of parameters under consideration. Indeed, model selection is one of the major challenges facing statisticians working with high dimensional data. Tools such as regularization and sparsity are some of the common notions employed to obtain parsimonious models to explain observed data. In recent years, the field of statistics has witnessed an explosion of frequentist and Bayesian methods for high dimensional problems. Despite these and other advances, Bayesian model selection in an "objective" sense in high dimensional problems remains an important problem that has yet to be solved satisfactorily. The need for objectivity translates into a need for specifying noninformative improper priors, which in turn renders the traditional Bayes factors unusable. The project proposes to derive objective Bayesian estimation and model selection procedures in a large class of high dimensional graphical models. The methodology that is proposed in this project therefore aims to contribute to much needed theory in the area of objective Bayesian model selection for high dimensional graphical models. In the process the methodology studies the benefits and shortcomings of objective Bayesian methods in this context. The theory that is developed feeds into developing algorithms and computational techniques for model selection/estimation in high dimensional settings.The availability of throughput or high dimensional data has touched almost every field of science. The need to formulate correct models that explain observed high dimensional data permeates through many scientific fields. Indeed, such data where the number of variables is often much higher than the number of samples, referred to as the "large p small n" problem, is now more pervasive than it has ever been. Discovering statistical signals in high dimensional data, proposing correct models that can explain such data, and parameter estimation in these high dimensional settings are some of the major challenges that modern day statisticians have to contend with. Moreover, such challenges also feature in high stakes debates such as climate change, effectiveness of certain drugs in clinical trials, and relevance of various biomarkers in cancer studies. This project proposes to develop statistical methodology which is specifically targeted towards identifying models which explain high dimensional data in an objective manner. In particular the project is designed to develop better objective Bayesian model selection and parameter estimation methods in high dimensional problems, and has widespread applications. The PI and co-PI collaborate with scientists in applied fields, especially with faculty/researchers in their Medical Schools, Schools of Engineering and Environmental Sciences. Training of graduate students and mentoring is an integral part of this collaborative research. Scientific output from the project is intended for publication in high impact peer-reviewed journals.