  Mathematical models implemented on a computer have become the driving force
behind the acceleration of the cycle of scientific processes. This is because
computer models are typically much faster and economical to run than physical
experiments. In this work, we develop an empirical Bayes approach to
predictions of physical quantities using a computer model, where we assume that
the computer model under consideration needs to be calibrated and is
computationally expensive. We propose a Gaussian process emulator and a
Gaussian process model for the systematic discrepancy between the computer
model and the underlying physical process. This allows for closed-form and
easy-to-compute predictions given by a conditional distribution induced by the
Gaussian processes. We provide a rigorous theoretical justification of the
proposed approach by establishing posterior consistency of the estimated
physical process. The computational efficiency of the methods is demonstrated
in an extensive simulation study and a real data example. The newly established
approach makes enhanced use of computer models both from practical and
theoretical standpoints.
