Adaptive posterior contraction rates for the horseshoe

We investigate the frequentist properties of Bayesian procedures for estimation based on the horseshoe prior in the sparse multivariate normal means model. Previous theoretical results assumed that the sparsity level, that is, the number of signals, was known. We drop this assumption and characterize the behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter of the horseshoe prior. We prove that the MMLE is an effective estimator of the sparsity level, in the sense that it leads to (near) minimax optimal estimation of the underlying mean vector generating the data. Besides this empirical Bayes procedure, we consider the hierarchical Bayes method of putting a prior on the unknown sparsity level as well. We show that both Bayesian techniques lead to rate-adaptive optimal posterior contraction, which implies that the horseshoe posterior is a good candidate for generating rate-adaptive credible sets.Comment: arXiv admin note: substantial text overlap with arXiv:1607.0189

Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection

This paper describes an approach for Bayesian modeling in spherical datasets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.Business Administratio