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.
Highlights
We show that the horseshoe credible sets and intervals are effective tools for uncertainty quantification, unless the underlying signals are too close to the universal threshold in a sense that is made precise in this work
We show in this work that the uncertainty quantification given by the horseshoe posterior distribution is “honest” only under certain prior assumptions on the parameters
The question is whether these Bayesian credible sets are appropriate for uncertainty quantification from a frequentist point of view
Summary
Despite the ubiquity of problems with sparse structures, and the large amount of research effort into finding consistent and minimax optimal estimators for the underlying sparse structures Tibshirani (1996); Johnstone and Silverman (2004); Castillo and Van der Vaart (2012); Castillo et al (2015); Jiang and Zhang (2009); Griffin and Brown (2010); Johnson and Rossell (2010); Ghosh and Chakrabarti (2015); Caron and Doucet (2008); Bhattacharya et al (2014); Bhadra et al (2017); Rockova (2015), the number of options for uncertainty quantification in the sparse normal means problem is very limited. The results in van der Pas et al (2014) show that τ can be interpreted as the proportion of nonzero parameters, up to a logarithmic factor If it is set at a value of the order (pn/n) log(n/pn), the horseshoe posterior contracts around the true θ0 at the (near) minimax estimation rate for quadratic loss. We characterise the parameters for which the credible sets of the horseshoe posterior distribution give good coverage, and the ones for which they do not We investigate this both for the empirical and hierarchical Bayes approaches, both when τ is set deterministically, and in adaptive settings where the number of nonzero means is unknown. Uncertainty quantification in the case of the sparse normal means model was addressed in the recent paper Belitser and Nurushev (2015) These authors consider a mixed Bayesian-frequentist procedure, which leads to a mixture over sets I ⊂ {1, 2, . A supplement (van der Pas et al, 2017b) contains the proofs of the other results, as a sequence of appendices
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
