Bayesian Penalized Methods for High-Dimensional Data

Big data presents the overwhelming challenge of estimating a large number of parameters, which is much larger than the sample size. Even for a simple linear model, when the number of predictors is larger than or close to the sample size, such model may be unidentifiable and the least squares estimates of regression coefficients can be unstable. To deal with such issue, we systematically investigate three Bayesian regularization methods with applications in imaging genetics. First, we develop a Bayesian lasso estimator for the covariance matrix and propose a metropolis-based sampling scheme. This development is motivated by functional network exploration for the entire brain from magnetic resonance imaging (MRI) data. Second, we propose a Bayesian generalized low rank regression model (GLRR) for the mean parameter estimation and combine this with factor loading method of covariance estimation to capture the spatial correlation among the responses and jointly estimate the mean and covariance parameters. This development is motivated by performing genome-wide searches for associations between genetic variants and brain imaging phenotypes from data collected by Alzheimer's Disease Neuroimaging Initiative (ADNI). Third, we extend GLRR to longitudinal setting and propose a Bayesian longitudinal low rank regression (L2R2) to account for spatiotemporal correlation among the responses as well as estimation of full-rank coefficient matrix for standard prognostic factors. This development is motivated by genome-wide searches for associations between genetic variants and brain imaging phenotypes observed over time with a primary focus on role of aging and the interaction of age with genotype in affecting brain volume.Doctor of Philosoph