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，必须在参数估计之前或同时进行模型选择，以减少需要考虑的参数数量。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。正则化和稀疏性等工具是一些常用的概念，用于获得解释观测数据的简约模型。近年来，统计领域见证了高维问题的频率论和贝叶斯方法的爆炸式增长。尽管有这些和其他的进步，在高维问题中，贝叶斯模型选择在“客观”意义上仍然是一个重要的问题，尚未得到令人满意的解决。对客观性的需求转化为对指定非信息性的不适当先验的需求，这反过来又使传统的贝叶斯因子无法使用。该项目提出在大量高维图形模型中推导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旨在为高维图形模型的客观贝叶斯模型选择领域提供急需的理论。在此过程中，本方法学研究了客观贝叶斯方法在此背景下的优缺点。所开发的理论为高维设置中模型选择/估计的开发算法和计算技术提供了反馈。吞吐量或高维数据的可用性几乎触及了科学的每个领域。需要制定正确的模型来解释观测到的高维数据渗透到许多科学领域。事实上，这样的数据，其中的变量的数量往往远远高于样本的数量，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，提出可以解释这些数据的正确模型，以及在这些高维设置中进行参数估计，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，以及各种生物标志物在癌症研究中的相关性。该项目建议开发统计方法，专门针对识别以客观方式解释高维数据的模型。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，具有广泛的应用前景。PI和联合PI与应用领域的科学家合作，特别是与医学院、工程学院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项合作研究的一个组成部分。该项目的科学成果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.