Application of Gaussian Process Priors on Bayesian Regression

This dissertation aims at introducing Gaussian process priors on the regression to capture features of dataset more adequately. Three different types of problems occur often in the regression. 1) For the dataset with missing covariates in the semiparametric regression, we utilize Gaussian process priors on the nonparametric component of the regression function to perform imputations of missing covariates. For the Bayesian inference of parameters, we specify objective priors on the Gaussian process parameters.Posteriorpropriety of the model under the objective priors is also demonstrated. 2) For modeling binary and ordinal data, we proposed a flexible nonparametric regression model that combines flexible power link function with a Gaussian process prior on the latent regression function. We develop an efficient sampling algorithm for posterior inference and prove the posterior consistency of the proposed model. 3) In the high dimensional dataset, the estimation of regression coefficients especially when the covariates are highly multicollinear is very challenging. Therefore, we develop a model by using structured spike an slab prior on regression coefficients. Prior information of similarity between covariates can be encoded into the covariance structure of Gaussian process which can be used to induce sparsity. Hyperparameters of the Gaussian process can be used to control different sparsity pattern. The superiority of the proposed model is demonstrated using various simulation studies and real data examples