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We present a framework for approximate Bayesian inference when only a limitednumber of noisy log-likelihood evaluations can be obtained due to computationalconstraints, which is becoming increasingly common for applications of complexmodels. We model the log-likelihood function using a Gaussian process (GP) andthe main methodological innovation is to apply this model to emulate theprogression that an exact Metropolis-Hastings (MH) sampler would take if it wasapplicable. Informative log-likelihood evaluation locations are selected usinga sequential experimental design strategy until the MH accept/reject decisionis done accurately enough according to the GP model. The resulting approximatesampler is conceptually simple and sample-efficient. It is also more robust toviolations of GP modelling assumptions compared with earlier, related "Bayesianoptimisation-like" methods tailored for Bayesian inference. We discuss sometheoretical aspects and various interpretations of the resulting approximate MHsampler, and demonstrate its benefits in the context of Bayesian andgeneralised Bayesian likelihood-free inference for simulator-based statisticalmodels.

Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC