How Many Needles in the Haystack? Adaptive Inference and Uncertainty Quantification for the Horseshoe

• Activity Number: 157
• Type: Topic Contributed
• Date/Time: Monday, August 1, 2016 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Stephanie Van Der Pas* and Botond Szabo and Aad van der Vaart
• Companies: and Leiden University and Leiden University
• Keywords: sparsity ; horseshoe ; credible sets ; nearly black vectors ; frequentist Bayes

Abstract

We investigate the frequentist properties of the hierarchical Bayes and empirical Bayes methods in the sparse multivariate mean model with unknown sparsity level. We consider the popular horseshoe prior introduced in Carvalho et al. (2010) and show that both adaptive Bayesian techniques lead to rate optimal posterior contraction without using any information on the sparsity level. Furthermore, we investigate the frequenstist coverage properties of Bayesian credible sets resulting from the horseshoe prior both in the non-adaptive and adaptive setting. We show that under a self-similarity type of assumption both the (slightly modified) hierarchical and empirical Bayes credible sets have (almost) rate adaptive size and good coverage. Our results require assumptions on the prior on the global parameter tau in the hierarchical Bayes setting, and on the estimator of tau in the empirical Bayes setting. We give examples of suitable priors for the former, and show that the maximum marginal likelihood estimator meets the conditions for the latter approach.