Title: Frequentist coverage and sup-norm convergence rate in Gaussian process regression,
Abstract: Gaussian process (GP) regression is a powerful interpolation technique due to
its flexibility in capturing non-linearity. In this paper, we provide a general
framework for understanding the frequentist coverage of point-wise and
simultaneous Bayesian credible sets in GP regression. As an intermediate
result, we develop a Bernstein von-Mises type result under supremum norm in
random design GP regression. Identifying both the mean and covariance function
of the posterior distribution of the Gaussian process as regularized
$M$-estimators, we show that the sampling distribution of the posterior mean
function and the centered posterior distribution can be respectively
approximated by two population level GPs. By developing a comparison inequality
between two GPs, we provide exact characterization of frequentist coverage
probabilities of Bayesian point-wise credible intervals and simultaneous
credible bands of the regression function. Our results show that inference
based on GP regression tends to be conservative; when the prior is
under-smoothed, the resulting credible intervals and bands have minimax-optimal
sizes, with their frequentist coverage converging to a non-degenerate value
between their nominal level and one. As a byproduct of our theory, we show that
the GP regression also yields minimax-optimal posterior contraction rate
relative to the supremum norm, which provides a positive evidence to the long
standing problem on optimal supremum norm contraction rate in GP regression.