On frequentist coverage errors of Bayesian credible sets in high dimensions
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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 Bayesian inference on the slope vector by fitting a Gaussian distribution on the error term and putting priors on the slope vector together with the error variance. The Gaussian specification on the error distribution may be incorrect, so that we work with quasi-likelihoods. 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 hyper-rectangles. We use this general result to quantify coverage errors of Castillo-Nickl and L^∞-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.
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