CAREER: Bayesian Generalized Shrinkage: An Encompassing Model Approach
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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.