Convergence of Gaussian process emulators with estimated hyper-parameters and applications in Bayesian inverse problems
We consider hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. Using results from scattered data approximation, we provide a convergence analysis of the method applied to a fixed, unknown function of interest. Finally, we consider the use of Gaussian process emulators to approximate the mathematical model in an inverse problem, and discuss related stability properties.