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Gaussian process modeling is a standard tool for building emulators forcomputer 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 deterministicfunction from prediction and uncertainty quantification perspectives, where theGaussian process model is misspecified. Specifically, we consider the casewhere the underlying function is fixed and from a reproducing kernel Hilbertspace generated by some kernel function, and the same kernel function is usedin the Gaussian process modeling as the correlation function for prediction anduncertainty quantification. While upper bounds and the optimal convergence rateof prediction in the Gaussian process modeling have been extensively studied inthe literature, a comprehensive exploration of convergence rates andtheoretical study of uncertainty quantification is lacking. We prove that, ifone uses maximum likelihood estimation to estimate the variance in Gaussianprocess modeling, under different choices of the regularization parametervalue, the predictor is not optimal and/or the confidence interval is notreliable. In particular, lower bounds of the prediction error under differentchoices of the regularization parameter value are obtained. The resultsindicate that, if one directly applies Gaussian process modeling to a fixedfunction, the reliability of the confidence interval and the optimality of thepredictor cannot be achieved at the same time.

On the Inference of Applying Gaussian Process Modeling to a Deterministic Function