CAREER: Bayesian Generalized Shrinkage: An Encompassing Model Approach

随着在不同应用领域收集的数据集的日益复杂，统计学家不得不考虑太多复杂程度不同的不同统计模型。这项研究的一个主要动机是扩大经典收缩方法的范围，通过提供计算高效和统计上准确的解决方案来选择新的统计问题的模型和推断目标。该项目开发的方法具有广泛的应用范围，从遗传和流行病学研究到通信网络。首席调查员致力于加强与更广泛的科学界的互动和合作，以最大限度地发挥通过该项目制定的统计方法的影响。该项目的成果包括用户友好的软件包，使科学界的用户能够分析与项目目标相关的应用程序中的结构化数据集。主要研究人员提出了一类用于结构化高维模型的模型选择和推理的广义收缩方法。在贝叶斯框架下运行，所提出的方法将包含模型的参数缩小为一族子模型。在给定一类统计模型的情况下，主要研究者将包含模型定义为这样一种模型：它为指定类中任意模型的任意小邻域分配一个正的先验概率，这些邻域是根据一些一般的统计发散度量来定义的，例如Kullback-Leibler发散。这一通用公式通过对潜在的非嵌套模型使用收缩来扩大传统收缩机制的范围。通过后处理来自包含模型的后验摘要来执行模型选择。主要研究人员开发了新的后处理方案，用于变量选择和聚类、张量分解、树结构模型和网络拓扑识别，作为一般方法的各种应用。主要研究人员阐述了用于研究包含后验分布的非渐近频率行为的信息理论技术，特别强调为本项目中开发的模型选择程序提供理论支持。

With ever-increasing complexities of datasets collected in diverse application areas, statisticians are compelled to entertain a plethora of different statistical models with varying degrees of complexity. A primary motivation for this research is to broaden the scope of classical shrinkage approaches to model selection and inferential goals in new statistical problems by providing computationally efficient and statistically accurate solutions. The methodology developed in the project has broad applications ranging from genetic and epidemiological studies to communication networks. The principal investigator is committed to increased interactions and collaborations with the broader scientific community to maximize the impact of the statistical methodology developed through this project. The deliverables of the project include user-friendly software packages that enable users in the scientific community to analyze structured data sets in applications relevant to the project goals.The principal investigator proposes a class of generalized shrinkage methods for model selection and inference in structured high-dimensional models. Operating in a Bayesian framework, the proposed approach shrinks the parameters of an encompassing model towards a family of sub-models. Given a class of statistical models, the principal investigator defines an encompassing model as one which assigns a positive prior probability to arbitrarily small neighborhoods of any model in the specified class, with the neighborhoods defined in terms of some general statistical divergence measure such as the Kullback-Leibler divergence. This general formulation enlarges the scope of traditional shrinkage mechanisms by employing shrinkage towards potentially non-nested models. Model selection is performed by post-processing the posterior summaries from the encompassing model. The principal investigator develops novel post-processing schemes for variable selection and clustering, tensor decompositions, tree-structured models and network topology identification as various applications of the general methodology. The principal investigator formulates information theoretical techniques for studying non-asymptotic frequentist behavior of the encompassing posterior distributions, with specific emphasis on providing theoretical support for the model selection procedures developed in this project.