  Conditional inference provides a rigorous approach to counter bias when data
from automated model selections is reused for inference. We develop in this
paper a statistically consistent Bayesian framework to assess uncertainties
within linear models that are informed by grouped sparsities in covariates.
Finding wide applications when genes, proteins, genetic variants, neuroimaging
measurements are grouped respectively by their biological pathways, molecular
functions, regulatory regions, cognitive roles, these models are selected
through a useful class of group-sparse learning algorithms. An adjustment
factor to account precisely for the selection of promising groups, deployed
with a generalized version of Laplace-type approximations is the centerpiece of
our new methods. Accommodating well known group-sparse models such as those
selected by the Group LASSO, the overlapping Group LASSO, the sparse Group
LASSO etc., we illustrate the efficacy of our methodology in extensive
experiments and on data from a human neuroimaging application.
