  We tackle modelling and inference for variable selection in regression
problems with many predictors and many responses. We focus on detecting
hotspots, i.e., predictors associated with several responses. Such a task is
critical in statistical genetics, as hotspot genetic variants shape the
architecture of the genome by controlling the expression of many genes and may
initiate decisive functional mechanisms underlying disease endpoints. Existing
hierarchical regression approaches designed to model hotspots suffer from two
limitations: their discrimination of hotspots is sensitive to the choice of
top-level scale parameters for the propensity of predictors to be hotspots, and
they do not scale to large predictor and response vectors, e.g., of dimensions
$10^3-10^5$ in genetic applications. We address these shortcomings by
introducing a flexible hierarchical regression framework that is tailored to
the detection of hotspots and scalable to the above dimensions. Our proposal
implements a fully Bayesian model for hotspots based on the horseshoe shrinkage
prior. Its global-local formulation shrinks noise globally and hence
accommodates the highly sparse nature of genetic analyses, while being robust
to individual signals, thus leaving the effects of hotspots unshrunk. Inference
is carried out using a fast variational algorithm coupled with a novel
simulated annealing procedure that allows efficient exploration of multimodal
distributions.
