Bayesian Variable Selection Using Continuous Shrinkage Priors for Nonparametric Models and Non-Gaussian Data.

WEI, RAN. Bayesian Variable Selection Using Continuous Shrinkage Priors for Nonparametric Models and Non-Gaussian Data. (Under the direction of Subhashis Ghoshal and Brian Reich.) In this thesis, we study the properties and applications of Bayesian variable selection in regression models. We focus on Bayesian shrinkage priors that are global-local mixture priors to select an appropriate subset of covariates. First, a Bayesian non-parametric regression model is proposed with multivariate continuous shrinkage prior. This approach is to decompose the commonly-used linear regression model into the summation of nonlinear main effects and two-way interaction terms and apply the proposed computationallyadvantageous continuous shrinkage prior to identify important effects. We construct a multivariate Dirichlet-Laplace prior that aggressively shrinks many of the terms towards zero, thus mitigating the noise of including unimportant exposures and allowing us to isolate the effects of important exposures. Theoretical studies demonstrate the asymptotic prediction and variable selection consistency properties, while numerical simulations present model performance of prediction and variable selection under practical scenarios. The method is applied on neurobehavioral data from Agricultural Health Study that investigates the associations between pesticide use and neurobehavioral outcomes in farmers. The proposed method shows improved accuracy in predicting the joint effects on neurobehavioral responses, while restricting the number of covariates included in the model through variable selection technique. Next, we investigate the contraction properties of shrinkage priors in logistic regression model when the number of covariates is high. For the prior distribution that is heavy-tailed and concentrated around around zero with large probability, the logistic coefficient estimates are asymptotically concentrated around the true sparse vector in L2-error contraction rate. It is shown that the proposed contraction rate is comparable with point mass prior that is studied in Atchadé (2017). The simulation study under logistic regression model verifies the theoretical results by showing that shrinkage priors such as horseshoe prior and Dirichlet-Laplace prior perform quite similar as the point mass prior in estimation, variable selection and prediction, but yield much better results than Bayesian lasso and non-informative normal prior. Last, we propose spatial extension of Bayesian shrinkage prior in the application of forecasting calibration for fine particulate matter from wildland fire smoke. To improve the forecast, we consider expanding the space-time forecasts by computing spatial summaries of the forecast in surrounding areas and using the constructed covariates in a multivariate regression model. We apply an additive nonparametric regression model to delineate the associations between the fine particulate measurements at monitoring stations and multivariate spatial summaries. Our proposed model incorporates both spatial variation on the additive nonparametric effects and Bayesian variable selection across locations. The simulation study evaluates the additive regression model under different model assumptions (nonlinearity, spatial dependence in the mean functions and residuals) and demonstrates its advantages in both prediction and variable selection. The regression model is implemented to downscale the fine particulate matter forecasts of wildland fire smoke in Washington state in 2015 and find an improvement of 18% in mean squared error compared to the standard linear regression calibration. © Copyright 2017 by Ran Wei