Bayesian variable selection and hypothesis testing

In modern statistical and machine learning applications, there is an increasing need for developing methodologies and algorithms to analyze massive data sets. Coupled with the growing popularity of Bayesian methods in statistical analysis, range of new techniques have evolved that allow innovative model-building and inference. In this dissertation, we develop Bayesian methods for variable selection and hypothesis testing. One important theme of this work is to develop computationally efficient algorithms that also enjoy strong probabilistic guarantees of convergence in a frequentist sense. Another equally important theme is to bridge the gap of classical statistical inference and Bayesian inference, in particular, through a new approach of hypothesis testing which can justify the Bayesian interpretation of classical testing framework. These methods are validated and demonstrated through simulated examples and real data applications.