  Gaussian process modeling is a standard tool for building emulators for
computer experiments, which are usually used to study deterministic functions,
for example, a solution to a given system of partial differential equations.
This work investigates applying Gaussian process modeling to a deterministic
function from prediction and uncertainty quantification perspectives, where the
Gaussian process model is misspecified. Specifically, we consider the case
where the underlying function is fixed and from a reproducing kernel Hilbert
space generated by some kernel function, and the same kernel function is used
in the Gaussian process modeling as the correlation function for prediction and
uncertainty quantification. While upper bounds and the optimal convergence rate
of prediction in the Gaussian process modeling have been extensively studied in
the literature, a comprehensive exploration of convergence rates and
theoretical study of uncertainty quantification is lacking. We prove that, if
one uses maximum likelihood estimation to estimate the variance in Gaussian
process modeling, under different choices of the regularization parameter
value, the predictor is not optimal and/or the confidence interval is not
reliable. In particular, lower bounds of the prediction error under different
choices of the regularization parameter value are obtained. The results
indicate that, if one directly applies Gaussian process modeling to a fixed
function, the reliability of the confidence interval and the optimality of the
predictor cannot be achieved at the same time.
