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，为了减少所考虑的参数数量，必须在参数估计之前或同时进行模型选择。事实上，模型选择是处理高维数据的统计学家面临的主要挑战之一。正则化和稀疏化等工具是用来获得简约模型来解释观测数据的一些常见概念。近年来，统计领域见证了高维问题的频数和贝叶斯方法的爆炸性增长。尽管取得了这些进展，但高维问题中“客观”意义上的贝叶斯模型选择仍然是一个重要的问题，尚未得到令人满意的解决。对客观性的需要转化为需要指定非信息性的不适当的先验，这反过来又使得传统的贝叶斯因子无法使用。该项目建议在一大类高维图形模型中推导出客观贝叶斯估计和模型选择过程。因此，本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中亟需的理论做出贡献。在这一过程中，该方法论研究了客观贝叶斯方法在这方面的优点和缺点。发展起来的理论为高维环境下模型选择/估计的开发算法和计算技术提供了支持。吞吐量或高维数据的可用性几乎触及了科学的每一个领域。需要制定正确的模型来解释观察到的高维数据，这一需求渗透到了许多科学领域。事实上，这种变量数量往往比样本数量高得多的数据，被称为“大p小n”问题，现在比以往任何时候都更加普遍。发现高维数据中的统计信号，提出能够解释这些数据的正确模型，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，这样的挑战也出现在高风险的辩论中，如气候变化，临床试验中某些药物的有效性，以及各种生物标记物在癌症研究中的相关性。该项目建议开发专门针对确定客观地解释高维数据的模型的统计方法。特别是，该项目旨在开发高维问题中更好的客观贝叶斯模型选择和参数估计方法，并具有广泛的应用前景。PI和共同PI与应用领域的科学家合作，特别是与其医学院、工程学院和环境科学学院的教职员工/研究人员合作。对研究生的培训和指导是这项合作研究不可或缺的一部分。该项目的科学成果将在影响较大的同行评议期刊上发表。

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.