Conditions for Posterior Contraction in the Sparse Normal Means Problem

Abstract

The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as Laplace, but not too heavy, and a large amount of mass around zero relative to the tails, more so as the sparsity increases. These conditions give some general guidelines for choosing a shrinkage prior for estimation under a nearly black sparsity assumption. We verify these conditions for the class of priors considered by Ghosh and Chakrabarti (2015), which includes the horseshoe and the normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend the number of shrinkage priors which are known to lead to posterior contraction at the minimax estimation rate.

References in corpus (2)

• Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences
• The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors

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• Bayesian Effect Selection in Structured Additive Distributional Regression Models
• A Scalable Empirical Bayes Approach to Variable Selection in Generalized Linear Models
• Phase transition in compressed sensing with horseshoe prior
• Generalized Decomposition Priors on R2
• Global-Local Shrinkage Priors for Asymptotic Point and Interval Estimation of Normal Means under Sparsity
• Effect of global shrinkage parameter of horseshoe prior in compressed sensing
• Posterior Contraction rate and Asymptotic Bayes Optimality for one-group shrinkage priors in sparse normal means problem