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