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Bayesian posterior distributions arising in modern applications, includinginverse problems in partial differential equation models in tomography andsubsurface flow, are often computationally intractable due to the largecomputational cost of evaluating the data likelihood. To alleviate thisproblem, we consider using Gaussian process regression to build a surrogatemodel for the likelihood, resulting in an approximate posterior distributionthat is amenable to computations in practice. This work serves as anintroduction to Gaussian process regression, in particular in the context ofbuilding surrogate models for inverse problems, and presents new insights intoa suitable choice of training points. We show that the error between the trueand approximate posterior distribution can be bounded by the error between thetrue and approximate likelihood, measured in the $L^2$-norm weighted by thetrue posterior, and that efficiently bounding the error between the true andapproximate likelihood in this norm suggests choosing the training points inthe Gaussian process surrogate model based on the true posterior.

Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measures