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We investigate the frequentist coverage properties of credible sets resultingin from Gaussian process priors with squared exponential covariance kernel.First we show that by selecting the scaling hyper-parameter using the maximummarginal likelihood estimator in the (slightly modified) squared exponentialcovariance kernel the corresponding credible sets will provide overconfident,misleading uncertainty statements for a large, representative subclass of thefunctional parameters in context of the Gaussian white noise model. Then weshow that by either blowing up the credible sets with a logarithmic factor ormodifying the maximum marginal likelihood estimator with a logarithmic term onecan get reliable uncertainty statement and adaptive size of the credible setsunder some additional restriction. Finally we demonstrate on a numerical studythat the derived negative and positive results extend beyond the Gaussian whitenoise model to the nonparametric regression and classification models for smallsample sizes as well.

Can we trust Bayesian uncertainty quantification from Gaussian process priors with squared exponential covariance kernel?