  In this paper we study grouped variable selection problems by proposing a
specified prior, called the nested spike and slab prior, to model collective
behavior of regression coefficients. At the group level, the nested spike and
slab prior puts positive mass on the event that the l2-norm of the grouped
coefficients is equal to zero. At the individual level, each coefficient is
assumed to follow a spike and slab prior. We carry out maximum a posteriori
estimation for the model by applying blockwise coordinate descent algorithms to
solve an optimization problem involving an approximate objective modified by
majorization-minimization techniques. Simulation studies show that the proposed
estimator performs relatively well in the situations in which the true and
redundant covariates are both covered by the same group. Asymptotic analysis
under a frequentist's framework further shows that the l2 estimation error of
the proposed estimator can have a better upper bound if the group that covers
the true covariates does not cover too many redundant covariates. In addition,
given some regular conditions hold, the proposed estimator is asymptotically
invariant to group structures, and its model selection consistency can be
established without imposing irrepresentable-type conditions.
