Inference post Selection of Group-sparse Regression Models

We develop a post-selective Bayesian framework to jointly and consistently estimate parameters within automatic group-sparse regression models. Selected through an indispensable class of learning algorithms, e.g. the Group LASSO, the overlapping Group LASSO, the sparse Group LASSO etc., uncertainty estimates for the matched parameters are unreliable in the absence of adjustments for selection bias. Limiting however the application of state of the art tools for the group-sparse problem include estimation strictly tailored to (i) real-valued projections onto very specific selected subspaces, (ii) selection events admitting representations as linear inequalities in the data variables. Our Bayesian methods address these gaps by deriving an adjustment factor in an easily feasible analytic form that eliminates bias from the selection of promising groups. Paying a very nominal price for this adjustment, experiments on simulated data and the Human Connectome Project demonstrate the efficacy of our methods at a joint estimation of group-sparse parameters learned from data.