Parallel Gaussian process surrogate method to accelerate likelihood-free inference

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) is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a Bayesian sequential 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 batch Bayesian optimisation, we develop various batch-sequential strategies where multiple simulations are adaptively selected to minimise either the expected or median loss function measuring the uncertainty in the resulting posterior. We analyse the properties of the resulting method theoretically and empirically. Experiments with toy problems and three simulation models suggest that our method is robust, highly parallelisable, and sample-efficient.