Fully Bayesian T-probit Regression with Heavy-tailed Priors for Selection in High-Dimensional Features with Grouping Structure

Feature selection is demanded in many modern scientific research problems that use high-dimensional data. A typical example is to find the genes that are most related to a certain disease (e.g., cancer) from high-dimensional gene expression profiles. There are tremendous difficulties in eliminating a large number of useless or redundant features. The expression levels of genes have structure; for example, a group of co-regulated genes that have similar biological functions tend to have similar mRNA expression levels. Many statistical methods have been proposed to take the grouping structure into consideration in feature selection and regression, including Group LASSO, Supervised Group LASSO, and regression on group representatives. In this thesis, we propose to use a sophisticated Markov chain Monte Carlo method (Hamiltonian Monte Carlo with restricted Gibbs sampling) to fit T-probit regression with heavy-tailed priors to make selection in the features with grouping structure. We will refer to this method as fully Bayesian T-probit. The main feature of fully Bayesian T-probit is that it can make feature selection within groups automatically without a pre-specification of the grouping structure and more efficiently discard noise features than LASSO (Least Absolute Shrinkage and Selection Operator). Therefore, the feature subsets selected by fully Bayesian T-probit are significantly more sparse than subsets selected by many other methods in the literature. Such succinct feature subsets are much easier to interpret or understand based on existing biological knowledge and further experimental investigations. In this thesis, we use simulated and real datasets to demonstrate that the predictive performances of the more sparse feature subsets selected by fully Bayesian T-probit are comparable with the much larger feature subsets selected by plain LASSO, Group LASSO, Supervised Group LASSO, random forest, penalized logistic regression and t-test. In addition, we demonstrate that the succinct feature subsets selected by fully Bayesian T-probit have significantly better predictive power than the feature subsets of the same size taken from the top features selected by the aforementioned methods