The Annals of Statistics

Frequentist coverage of adaptive nonparametric Bayesian credible sets

Botond Szabó, A. W. van der Vaart, and J. H. van Zanten

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Abstract

We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of “polished tail” parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale.

Article information

Source
Ann. Statist., Volume 43, Number 4 (2015), 1391-1428.

Dates
Received: November 2013
Revised: April 2014
First available in Project Euclid: 17 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1434546205

Digital Object Identifier
doi:10.1214/14-AOS1270

Mathematical Reviews number (MathSciNet)
MR3357861

Zentralblatt MATH identifier
1317.62040

Subjects
Primary: 62G15: Tolerance and confidence regions 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Keywords
Credible set coverage uncertainty quantification

Citation

Szabó, Botond; van der Vaart, A. W.; van Zanten, J. H. Frequentist coverage of adaptive nonparametric Bayesian credible sets. Ann. Statist. 43 (2015), no. 4, 1391--1428. doi:10.1214/14-AOS1270. https://projecteuclid.org/euclid.aos/1434546205


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See also

  • Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".
  • Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".
  • Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".
  • Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".
  • Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".
  • Rejoinder to discussions of "Frequentist coverage of adaptive nonparametric Bayesian credible sets".