  We present an efficient approach for doing approximate Bayesian inference
when only a limited number of noisy likelihood evaluations can be obtained due
to computational constraints, which is becoming increasingly common for
applications of complex models. Our main methodological innovation is to model
the log-likelihood function using a Gaussian process (GP) in a local fashion
and apply this model to emulate the progression that an exact
Metropolis-Hastings (MH) algorithm would take if it was applicable. New
log-likelihood evaluation locations are selected using sequential experimental
design strategies such that each MH accept/reject decision is done within a
pre-specified error tolerance. The resulting approach is conceptually simple
and sample-efficient as it takes full advantage of the GP model. It is also
more robust to violations of GP modelling assumptions and better suited for the
typical situation where the posterior is substantially more concentrated than
the prior, compared with various existing inference methods based on global GP
surrogate modelling. We discuss the probabilistic interpretations and central
theoretical aspects of our approach, and we then demonstrate the benefits of
the resulting algorithm in the context of likelihood-free inference for
simulator-based statistical models.
