Bayesian Modeling and Variable Selection for Complex Data

As we routinely encounter high-throughput datasets in complex biological and environment research, developing novel models and methods for variable selection has received widespread attention. In this dissertation, we addressed a few key challenges in Bayesian modeling and variable selection for high-dimensional data with complex spatial structures. a) Most Bayesian variable selection methods are restricted to mixture priors having separate components for characterizing the signal and the noise. However, such priors encounter computational issues in high dimensions. This has motivated continuous shrinkage priors, resembling the two-component priors facilitating computation and interpretability. While such priors are widely used for estimating high-dimensional sparse vectors, selecting a subset of variables remains a daunting task. b) Spatial/spatial-temporal data sets with complex structures are nowadays commonly encountered in various scientific research fields ranging from atmospheric sciences, forestry, environmental science, biological science, and social science. Selecting important spatial variables that have significant influences on occurrences of events is undoubtedly necessary and essential for providing insights to researchers. Self-excitation, which is a feature that occurrence of an event increases the likelihood of more occurrences of the same type of events nearby in time and space, can be found in many natural/social events. Research on modeling data with self-excitation feature has increasingly drawn interests recently. However, existing literature on self-exciting models with inclusion of high-dimensional spatial covariates is still underdeveloped. c) Gaussian Process is among the most powerful model frames for spatial data. Its major bottleneck is the computational complexity which stems from inversion of dense matrices associated with a Gaussian process covariance. Hierarchical divide-conquer Gaussian Process models have been investigated for ultra large data sets. However, computation associated with scaling the distributing computing algorithm to handle a large number of sub-groups poses a serious bottleneck. In chapter 2 of this dissertation, we propose a general approach for variable selection with shrinkage priors. The presence of very few tuning parameters makes our method attractive in comparison to ad hoc thresholding approaches. The applicability of the approach is not limited to continuous shrinkage priors, but can be used along with any shrinkage prior. Theoretical properties for near-collinear design matrices are investigated and the method is shown to have good performance in a wide range of synthetic data examples and in a real data example on selecting genes affecting survival due to lymphoma. In Chapter 3 of this dissertation, we propose a new self-exciting model that allows the inclusion of spatial covariates. We develop algorithms which are effective in obtaining accurate estimation and variable selection results in a variety of synthetic data examples. Our proposed model is applied on Chicago crime data where the influence of various spatial features is investigated. In Chapter 4, we focus on a hierarchical Gaussian Process regression model for ultra-high dimensional spatial datasets. By evaluating the latent Gaussian process on a regular grid, we propose an efficient computational algorithm through circulant embedding. The latent Gaussian process borrows information across multiple sub-groups, thereby obtaining a more accurate prediction. The hierarchical model and our proposed algorithm are studied through simulation examples.