  The paper revisits the Bayesian group lasso and uses spike and slab priors
for group variable selection. In the process, the connection of our model with
penalized regression is demonstrated, and the role of posterior median for
thresholding is pointed out. We show that the posterior median estimator has
the oracle property for group variable selection and estimation under
orthogonal designs, while the group lasso has suboptimal asymptotic estimation
rate when variable selection consistency is achieved. Next we consider bi-level
selection problem and propose the Bayesian sparse group selection again with
spike and slab priors to select variables both at the group level and also
within a group. We demonstrate via simulation that the posterior median
estimator of our spike and slab models has excellent performance for both
variable selection and estimation.
