Bayesian Variable Selection and Inference for Nonparametric Kernel Machine and Functional Models

Abstract

In this dissertation, we have focused on developing three methods to address the challenges in highly correlated high-dimensional and functional data. In the first study, the Bayesian variable selection method is developed under a generalized fused multi-kernel machine regression. This method can apply to continuous/binary/ordered categorical response variables. We demonstrate the advantage of our method using bio-photonics Raman spectroscopy to identify which molecular fingerprinting wavenumber is associated with drug dosages of brain tumors. In the second study, we propose a Bayesian inference based on the Bayes factor. Our approach employs a generalized fused multi-kernel machine regression to adjust for multiple tests and identify significant pathways. The advantage of this method is illustrated by using genetic pathway data to test significantly correlated multiple pathways associated with Type II diabetes, estimating nonlinear relationships. Finally, we introduce a testing procedure for the departure of nonlinearity using a functional single index model. This procedure employs a randomly projected empirical process to reduce dimensionality while preserving essential statistical properties. The method is applied to autism brain imaging data to test whether fMRI signals are related to the autism diagnostic observation schedule. Therefore, the proposed three methods advance the field of variable selection and inference by offering innovative solutions to problems associated with correlated high-dimensional and functional data with practical applications across various domains.