  Variable selection techniques have become increasingly popular amongst
statisticians due to an increased number of regression and classification
applications involving high-dimensional data where we expect some predictors to
be unimportant. In this context, Bayesian variable selection techniques
involving Markov chain Monte Carlo exploration of the posterior distribution
over models can be prohibitively computationally expensive and so there has
been attention paid to quasi-Bayesian approaches such as maximum a posteriori
(MAP) estimation using priors that induce sparsity in such estimates. We focus
on this latter approach, expanding on the hierarchies proposed to date to
provide a Bayesian interpretation and generalization of state-of-the-art
penalized optimization approaches and providing simultaneously a natural way to
include prior information about parameters within this framework. We give
examples of how to use this hierarchy to compute MAP estimates for linear and
logistic regression as well as sparse precision-matrix estimates in Gaussian
graphical models. In addition, an adaptive group lasso method is derived using
the framework.
