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Conditions for posterior contraction in the sparse normal means problem

机译：稀疏正常均值问题中的后收缩情况

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摘要

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 in [12], 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.
机译：稀疏法线均值问题的第一个贝叶斯结果已针对尖峰和台阶先验证明。但是，从计算的角度来看，这些先验并不方便。同时，已经提出了大量的连续收缩先验。这些收缩先验中的许多可以写成法线的比例混合，这使得它们特别容易实现。我们根据法线比例混合的局部方差提出先验的一般条件，以确保以最小最大速率进行后收缩。这些条件要求尾巴至少与拉普拉斯一样重，但不要太重，并且相对于尾巴而言，零附近存在大量质量，随着稀疏度的增加，质量会更大。这些条件为在近似黑色稀疏性假设下选择收缩率进行估算提供了一些一般性指导。我们针对[12]中考虑的先验类别验证了这些条件，其中包括马蹄形和正指数伽玛先验，以及对于马蹄+，反高斯先验，正伽玛先验和尖峰和-平板套索，因此扩展了先验收缩的数量，已知先验收缩的数量以minimax估计速率导致后收缩。

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