  The computational efficiency of approximate Bayesian computation (ABC) has
been improved by using surrogate models such as Gaussian processes (GP). In one
such promising framework the discrepancy between the simulated and observed
data is modelled with a GP which is further used to form a model-based
estimator for the intractable posterior. In this article we improve this
approach in several ways. We develop batch-sequential Bayesian experimental
design strategies to parallellise the expensive simulations. In earlier work
only sequential strategies have been used. Current surrogate-based ABC methods
also do not fully account the uncertainty due to the limited budget of
simulations as they output only a point estimate of the ABC posterior. We
propose a numerical method to fully quantify the uncertainty in, for example,
ABC posterior moments. We also provide some new analysis on the GP modelling
assumptions in the resulting improved framework called Bayesian ABC and discuss
its connection to Bayesian quadrature (BQ) and Bayesian optimisation (BO).
Experiments with toy and real-world simulation models demonstrate advantages of
the proposed techniques.
