On the Inference of Applying Gaussian Process Modeling to a Deterministic Function
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The Gaussian process modeling is a standard tool for building emulators for computer experiments, which is usually a deterministic function, for example, solution to a partial differential equations system. In this work, we investigate applying Gaussian process models to a deterministic function from prediction and uncertainty quantification perspectives. While the upper bounds and optimal convergence rates of prediction in Gaussian process modeling have been extensively studied in the literature, a thorough exploration of the convergence rate and theoretical study of uncertainty quantification is lacking. We prove that, if one uses maximum likelihood estimation to estimate the variance, under different choices of nugget parameters, the predictor is not optimal and/or the confidence interval is not reliable. In particular, lower bounds of the predictor under different choices of nugget parameters are obtained. The results suggest that, if one applies Gaussian process models to a deterministic function, the reliability of the confidence interval and the optimality of predictors cannot be achieved at the same time.
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