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Empirical and Variational Bayesian Methods in High Dimensional Data Analysis
Bayesian methods have been widely used nowadays. This dissertation presents new research within the fields of empirical Bayes (EB) and variational Bayes (VB) methods. EB is a data driven method to determine an optimal prior. Often nonparametric method is used to implement EB. In the first topic in this dissertation, EB is applied on multiple testing where test size is in thousands scale. The framework is applied on allelic imbalance (ALI) detection at single nucleotide polymorphisms (SNPs). It is an immediate extension of [Zhang and Keles, 2018], which focuses on an experiment with one replicate. Work in this dissertation focuses on an experiment with two replicates, yielding a 2 dimensional (2D) joint model. The proposed 2D model is also extended to gene level ALI detection. VB is an optimization method to determine the posterior. Conventional methods, such as Markov chain Monte Carlo (MCMC), can suffer in data with sufficiently large number of dimensions. In the second topic in this dissertation, VB is applied on variable selection in high dimensional linear regression. VB approach to variable selection (VBVS) might have to concern large matrix computation, such as inversion [Ormerod et al., 2017]. The calculation is inaccurate given sufficiently large number of dimensions. The proposed method applies clustering on VBVS, and thus such large matrix computation can be avoided. In addition, a novel initialization by EB in the first project is developed to help VBVS avoid saddle point convergence. Data examples show the proposed method outperforms or ties with the existing methods.
Sun, Liangrui, "Empirical and Variational Bayesian Methods in High Dimensional Data Analysis" (2021). ETD collection for University of Nebraska - Lincoln. AAI28713269.