Bernoulli

  • Bernoulli
  • Volume 26, Number 1 (2020), 616-641.

On frequentist coverage errors of Bayesian credible sets in moderately high dimensions

Keisuke Yano and Kengo Kato

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Abstract

In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry–Esseen type bound on quasi-posterior distributions and recent results on high-dimensional CLT on hyperrectangles. We use this general result to quantify coverage errors of Castillo–Nickl and $L^{\infty}$-credible bands for Gaussian white noise models, linear inverse problems, and (possibly non-Gaussian) nonparametric regression models. In particular, we show that Bayesian credible bands for those nonparametric models have coverage errors decaying polynomially fast in the sample size, implying advantages of Bayesian credible bands over confidence bands based on extreme value theory.

Article information

Source
Bernoulli, Volume 26, Number 1 (2020), 616-641.

Dates
Received: August 2018
Revised: June 2019
First available in Project Euclid: 26 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1574758840

Digital Object Identifier
doi:10.3150/19-BEJ1142

Mathematical Reviews number (MathSciNet)
MR4036046

Zentralblatt MATH identifier
07140511

Keywords
Castillo–Nickl band credible rectangle sieve prior

Citation

Yano, Keisuke; Kato, Kengo. On frequentist coverage errors of Bayesian credible sets in moderately high dimensions. Bernoulli 26 (2020), no. 1, 616--641. doi:10.3150/19-BEJ1142. https://projecteuclid.org/euclid.bj/1574758840


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Supplemental materials

  • Supplement to “On frequentist coverage errors of Bayesian credible sets in high dimensions”. The supplementary material contains the proofs omitted in the main text.