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We study the use of Gaussian process emulators to approximate theparameter-to-observation map or the negative log-likelihood in Bayesian inverseproblems. We prove error bounds on the Hellinger distance between the trueposterior distribution and various approximations based on the Gaussian processemulator. Our analysis includes approximations based on the mean of thepredictive process, as well as approximations based on the full Gaussianprocess emulator. Our results show that the Hellinger distance between the trueposterior and its approximations can be bounded by moments of the error in theemulator. Numerical results confirm our theoretical findings.

Posterior Consistency for Gaussian Process Approximations of Bayesian Posterior Distributions