Honest Bayesian confidence sets for the L2-norm

Botond Szabó*, Aad van der Vaart, Harry van Zanten

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We investigate the problem of constructing Bayesian credible sets that are honest and adaptive for the L2-loss over a scale of Sobolev classes with regularity ranging between [D, 2D], for some given D in the context of the signal-in-white-noise model. We consider a scale of prior distributions indexed by a regularity hyper-parameter and choose the hyper-parameter both by marginal likelihood empirical Bayes and by hierarchical Bayes method, respectively. Next we consider a ball centred around the corresponding posterior mean with prescribed posterior probability. We show by theory and examples that both the empirical Bayes and the hierarchical Bayes credible sets give misleading, overconfident uncertainty quantification for certain oddly behaving truth. Then we construct a new empirical Bayes method based on risk estimation, which provides the correct uncertainty quantification and optimal size.

Original languageEnglish
Pages (from-to)36-51
Number of pages16
JournalJournal of Statistical Planning and Inference
Volume166
DOIs
Publication statusPublished - 1 Nov 2015
Externally publishedYes

Keywords

  • Coverage
  • Credible sets
  • Uncertainty quantification

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