  In this article, we propose a novel spatial global-local spike-and-slab
selection prior for image-on-scalar regression. We consider a Bayesian
hierarchical Gaussian process model for image smoothing, that uses a flexible
Inverse-Wishart process prior to handle within-image dependency, and propose a
general global-local spatial selection prior that extends a rich class of
well-studied selection priors. Unlike existing constructions, we achieve
simultaneous global (i.e, at covariate-level) and local (i.e., at
pixel/voxel-level) selection by introducing `participation rate' parameters
that measure the probability for the individual covariates to affect the
observed images. This along with a hard-thresholding strategy leads to
dependency between selections at the two levels, introduces extra sparsity at
the local level, and allows the global selection to be informed by the local
selection, all in a model-based manner. We design an efficient Gibbs sampler
that allows inference for large image data. We show on simulated data that
parameters are interpretable and lead to efficient selection. Finally, we
demonstrate performance of the proposed model by using data from the Autism
Brain Imaging Data Exchange (ABIDE) study. To the best of our knowledge, the
proposed model construction is the first in the Bayesian literature to
simultaneously achieve image smoothing, parameter estimation and a two-level
variable selection for image-on-scalar regression.
