  High-dimensional feature selection arises in many areas of modern science.
For example, in genomic research we want to find the genes that can be used to
separate tissues of different classes (e.g. cancer and normal) from tens of
thousands of genes that are active (expressed) in certain tissue cells. To this
end, we wish to fit regression and classification models with a large number of
features (also called variables, predictors). In the past decade, penalized
likelihood methods for fitting regression models based on hyper-LASSO
penalization have received increasing attention in the literature. However,
fully Bayesian methods that use Markov chain Monte Carlo (MCMC) are still in
lack of development in the literature. In this paper we introduce an MCMC
(fully Bayesian) method for learning severely multi-modal posteriors of
logistic regression models based on hyper-LASSO priors (non-convex penalties).
Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling
framework; we call our method Bayesian logistic regression with hyper-LASSO
(BLRHL) priors. We have used simulation studies and real data analysis to
demonstrate the superior performance of hyper-LASSO priors, and to investigate
the issues of choosing heaviness and scale of hyper-LASSO priors.
