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February 2020 On frequentist coverage errors of Bayesian credible sets in moderately high dimensions
Keisuke Yano, Kengo Kato
Bernoulli 26(1): 616-641 (February 2020). DOI: 10.3150/19-BEJ1142

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.

Citation

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Keisuke Yano. Kengo Kato. "On frequentist coverage errors of Bayesian credible sets in moderately high dimensions." Bernoulli 26 (1) 616 - 641, February 2020. https://doi.org/10.3150/19-BEJ1142

Information

Received: 1 August 2018; Revised: 1 June 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140511
MathSciNet: MR4036046
Digital Object Identifier: 10.3150/19-BEJ1142

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 1 • February 2020
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