  Many modern statistical applications involve inference for complex stochastic
models, where it is easy to simulate from the models, but impossible to
calculate likelihoods. Approximate Bayesian computation (ABC) is a method of
inference for such models. It replaces calculation of the likelihood by a step
which involves simulating artificial data for different parameter values, and
comparing summary statistics of the simulated data to summary statistics of the
observed data. Here we show how to construct appropriate summary statistics for
ABC in a semi-automatic manner. We aim for summary statistics which will enable
inference about certain parameters of interest to be as accurate as possible.
Theoretical results show that optimal summary statistics are the posterior
means of the parameters. While these cannot be calculated analytically, we use
an extra stage of simulation to estimate how the posterior means vary as a
function of the data; and then use these estimates of our summary statistics
within ABC. Empirical results show that our approach is a robust method for
choosing summary statistics, that can result in substantially more accurate ABC
analyses than the ad-hoc choices of summary statistics proposed in the
literature. We also demonstrate advantages over two alternative methods of
simulation-based inference.
