  We consider Bayesian inference when only a limited number of noisy
log-likelihood evaluations can be obtained. This occurs for example when
complex simulator-based statistical models are fitted to data, and synthetic
likelihood (SL) method is used to form the noisy log-likelihood estimates using
computationally costly forward simulations. We frame the inference task as a
sequential Bayesian experimental design problem, where the log-likelihood
function is modelled with a hierarchical Gaussian process (GP) surrogate model,
which is used to efficiently select additional log-likelihood evaluation
locations. Motivated by recent progress in the related problem of batch
Bayesian optimisation, we develop various batch-sequential design strategies
which allow to run some of the potentially costly simulations in parallel. We
analyse the properties of the resulting method theoretically and empirically.
Experiments with several toy problems and simulation models suggest that our
method is robust, highly parallelisable, and sample-efficient.
