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Abstract:
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In the Bayesian setting, the first general theoretical results for high dimensional sparse models have been obtained for the Spike and Slab priors in a Gaussian setting. However other approaches to model sparsity are now used, mainly to tackle computational issues. Among them the so called "one group" model have grown more and more popular. In the Gaussian set up, most of the more popular one-group prior can be written as a scale mixture of Gaussian, shrinkage properties being induced by choosing a mixing distribution with a lot of mass near 0. In this work we will study asymptotic properties of these type of priors for estimation and testing and give conditions under which the posterior contracts at the minimax rate and lead to efficient testing procedures. The conditions requires that the prior puts sufficient mass near 0 relative to the tails, and more so as the sparsity increases. In addition the prior tails should be at least as heavy as Laplace but not too heavy. These conditions give general guidelines for constructing a shrinkage prior. We verify them on different classes of priors, and thus extend the family of priors which are known to contract at the minimax rate.
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