  Approximate Bayesian computation (ABC) can be used for model fitting when the
likelihood function is intractable but simulating from the model is feasible.
However, even a single evaluation of a complex model may take several hours,
limiting the number of model evaluations available. Modelling the discrepancy
between the simulated and observed data using a Gaussian process (GP) can be
used to reduce the number of model evaluations required by ABC, but the
sensitivity of this approach to a specific GP formulation has not yet been
thoroughly investigated. We begin with a comprehensive empirical evaluation of
using GPs in ABC, including various transformations of the discrepancies and
two novel GP formulations. Our results indicate the choice of GP may
significantly affect the accuracy of the estimated posterior distribution.
Selection of an appropriate GP model is thus important. We formulate expected
utility to measure the accuracy of classifying discrepancies below or above the
ABC threshold, and show that it can be used to automate the GP model selection
step. Finally, based on the understanding gained with toy examples, we fit a
population genetic model for bacteria, providing insight into horizontal gene
transfer events within the population and from external origins.
