CAREER: Bayesian Generalized Shrinkage: An Encompassing Model Approach

随着在不同应用领域中收集的数据集的复杂性不断增加，统计学家不得不考虑具有不同复杂程度的大量不同统计模型。本研究的主要动机是通过提供计算效率和统计准确的解决方案，扩大经典收缩方法在新的统计问题中的模型选择和推理目标的范围。该项目开发的方法具有广泛的应用，从遗传学和流行病学研究到通信网络。首席研究员致力于增加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开发的统计方法的影响。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在与项目目标相关的应用程序中分析结构化数据集。首席研究员提出了一类广义收缩方法，用于结构化高维模型的模型选择和推理。在贝叶斯框架中操作，所提出的方法将包含模型的参数收缩到子模型的家族。给定一类统计模型，主要研究者将包含模型定义为将正先验概率分配给指定类别中任何模型的任意小邻域的模型，该邻域根据某些一般统计散度度量（如Kullback-Leibler散度）定义。这个一般的配方扩大了传统的收缩机制的范围，采用收缩对潜在的非嵌套模型。模型选择是通过对包含模型的后验总结进行后处理来进行的。主要研究者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结构识别，作为一般方法的各种应用。主要研究者制定了信息理论技术，用于研究非渐近频率论行为的后验分布，特别强调为该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.