  Feature selection is demanded in many modern scientific research problems
that use high-dimensional data. A typical example is to find the most useful
genes that are related to a certain disease (eg, cancer) from high-dimensional
gene expressions. The expressions of genes have grouping structures, for
example, a group of co-regulated genes that have similar biological functions
tend to have similar expressions. Many statistical methods have been proposed
to take the grouping structure into consideration in feature selection,
including group LASSO, supervised group LASSO, and regression on group
representatives. In this paper, we propose a fully Bayesian Robit regression
method with heavy-tailed (sparsity) priors (shortened by FBRHT) for selecting
features with grouping structure. The main features of FBRHT include that it
discards more aggressively unrelated features than LASSO, and it can make
feature selection within groups automatically without a pre-specified grouping
structure. In this paper, we use simulated and real datasets to demonstrate
that the predictive power of the sparse feature subsets selected by FBRHT are
comparable with other much larger feature subsets selected by LASSO, group
LASSO, supervised group LASSO, penalized logistic regression and random forest,
and that the succinct feature subsets selected by FBRHT have significantly
better predictive power than the feature subsets of the same size taken from
the top features selected by the aforementioned methods.
