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December 2017 Uncertainty Quantification for the Horseshoe (with Discussion)
Stéphanie van der Pas, Botond Szabó, Aad van der Vaart
Bayesian Anal. 12(4): 1221-1274 (December 2017). DOI: 10.1214/17-BA1065

Abstract

We investigate the credible sets and marginal credible intervals resulting from the horseshoe prior in the sparse multivariate normal means model. We do so in an adaptive setting without assuming knowledge of the sparsity level (number of signals). We consider both the hierarchical Bayes method of putting a prior on the unknown sparsity level and the empirical Bayes method with the sparsity level estimated by maximum marginal likelihood. We show that credible balls and marginal credible intervals have good frequentist coverage and optimal size if the sparsity level of the prior is set correctly. By general theory honest confidence sets cannot adapt in size to an unknown sparsity level. Accordingly the hierarchical and empirical Bayes credible sets based on the horseshoe prior are not honest over the full parameter space. We show that this is due to over-shrinkage for certain parameters and characterise the set of parameters for which credible balls and marginal credible intervals do give correct uncertainty quantification. In particular we show that the fraction of false discoveries by the marginal Bayesian procedure is controlled by a correct choice of cut-off.

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Stéphanie van der Pas. Botond Szabó. Aad van der Vaart. "Uncertainty Quantification for the Horseshoe (with Discussion)." Bayesian Anal. 12 (4) 1221 - 1274, December 2017. https://doi.org/10.1214/17-BA1065

Information

Published: December 2017
First available in Project Euclid: 1 September 2017

zbMATH: 1384.62155
MathSciNet: MR3724985
Digital Object Identifier: 10.1214/17-BA1065

Subjects:
Primary: 62G15
Secondary: 62F15

Keywords: credible sets , frequentist Bayes , horseshoe , nearly black vectors , normal means problem , Sparsity

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Vol.12 • No. 4 • December 2017
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