Bayesian Hypothesis Testing and Variable Selection in High Dimensional Regression

This dissertation consists of three distinct but related research projects. First of all, we study the Bayesian approach to model selection in the class of normal regression models. We propose an explicit closed-form expression of the Bayes factor with the use of Zellner\u27s g-prior and the beta-prime prior for g. Noting that linear models with a growing number of unknown parameters have recently gained increasing popularity in practice, such as the spline problem, we shall thus be particularly interested in studying the model selection consistency of the Bayes factor under the scenario in which the dimension of the parameter space increases with the sample size. Our results show that the proposed Bayes factor is always consistent under the null model and is consistent under the alternative model except for a small set of alternative models which can be characterized. It is noteworthy that the results mentioned above can be applied to the analysis of variance (ANOVA) model, which has been widely used in many areas of science, such as ecology, psychology, and behavioral research. For the one-way unbalanced ANOVA model, we propose an explicit closed-form expression of the Bayes factor which is thus easy to compute. In addition, its corresponding model selection consistency has been investigated under different asymptotic situations. For the one-way random effects models, we also propose a closed-form Bayes factor without integral representation which has reasonable model selection consistency under different asymptotic scenarios. Moreover, the performance of the proposed Bayes factor is examined by numerical studies. The second project deals with the intrinsic Bayesian inference for the correlation coefficient between the disturbances in the system of two seemingly unrelated regression equations. This work was inspired by the observation that considerable attention has been paid to the improved estimation of the regression coefficients of each model, whereas little attention has just been paid for making inference of the correlation coefficient, even though most of the improved estimation of the regression coefficients depend on the correlation coefficient. We propose an objective Bayesian solution to the problems of hypothesis testing and point estimation for the correlation coefficient based on combined use of the invariant loss function and the objective prior distribution for the unknown model parameters. This new solution possesses an invariance property under monotonic reparameterization of the quantity of interest. Some simulation studies and one real-data example are given for illustrative purpose. In the third project, we propose a new Bayesian strength of evidence built on divergence measures for testing point null hypotheses. Our proposed approach can be viewed as an objective and automatic approach to the problem of testing a point null hypothesis. It is shown that the new evidence successfully reconciles the disagreement between frequentists and Bayesians in many classical examples in which Lindley\u27s paradox often occurs. In particular, note that the proposed Bayesian approach under the noninformative prior often recovers the frequentist P-values. From a Bayesian decision-theoretical viewpoint, it is justified that the new evidence is a formal Bayes test for some specific loss functions. The performance of the proposed approach is illustrated through several numerical examples. Possible applications of the new evidence for a variety of point null hypothesis testing problems are also briefly discussed