Abstract:
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We introduce the Spike-and-Slab LASSO, a variable selection method that bridges the gap between popular frequentist strategies (LASSO) and popular Bayesian strategies (spike-and-slab). The cornerstone of the approach is the new family of Spike-and-Slab LASSO (SS-LASSO) priors, which form a continuum between the Laplace prior and the point-mass spike-and-slab prior. We establish several appealing frequentist properties of the posterior distribution in the context of high-dimensional linear regression when p>n. In particular, the global posterior mode is shown to be rate-optimal under squared error loss. Going further, we show that the entire SS-LASSO posterior keeps pace with the global mode and concentrates at the optimal rate. Whereas the LASSO estimator is also known to be rate-optimal, the posterior distribution under a single Laplace prior is not. Up to now, the rate-optimal posterior concentration has been established only for spike-and-slab priors with a point mass at zero. Due to its continuity, the SS-LASSO prior is amenable to state-of-art non-convex optimization to rapidly elicit a series of posterior modes within a path-following scheme.
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