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

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.

53 pages; the supplement is included in pp. 32-53

References in corpus (19)

• An MCMC Approach to Classical Estimation
• Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
• Bayesian linear regression with sparse priors
• Bayesian inverse problems with Gaussian priors
• Frequentist coverage of adaptive nonparametric Bayesian credible sets
• Confidence bands in density estimation
• On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures
• Anti-concentration and honest, adaptive confidence bands
• Nonparametric Bernstein-von Mises theorems in Gaussian white noise
• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• On Bayesian supremum norm contraction rates
• Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors
• A Bernstein-von Mises theorem for smooth functionals in semiparametric models
• Adaptive Bernstein-von Mises theorems in Gaussian white noise
• Quasi-Bayesian analysis of nonparametric instrumental variables models
• Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
• On adaptive posterior concentration rates
• A Bernstein-Von Mises Theorem for discrete probability distributions
• Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems