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Gaussian process (GP) regression is a powerful interpolation technique due toits flexibility in capturing non-linearity. In this paper, we provide a generalframework for understanding the frequentist coverage of point-wise andsimultaneous Bayesian credible sets in GP regression. As an intermediateresult, we develop a Bernstein von-Mises type result under supremum norm inrandom design GP regression. Identifying both the mean and covariance functionof the posterior distribution of the Gaussian process as regularized$M$-estimators, we show that the sampling distribution of the posterior meanfunction and the centered posterior distribution can be respectivelyapproximated by two population level GPs. By developing a comparison inequalitybetween two GPs, we provide exact characterization of frequentist coverageprobabilities of Bayesian point-wise credible intervals and simultaneouscredible bands of the regression function. Our results show that inferencebased on GP regression tends to be conservative; when the prior isunder-smoothed, the resulting credible intervals and bands have minimax-optimalsizes, with their frequentist coverage converging to a non-degenerate valuebetween their nominal level and one. As a byproduct of our theory, we show thatthe GP regression also yields minimax-optimal posterior contraction raterelative to the supremum norm, which provides a positive evidence to the longstanding problem on optimal supremum norm contraction rate in GP regression.

Frequentist coverage and sup-norm convergence rate in Gaussian process regression