  Many statistical applications involve models for which it is difficult to
evaluate the likelihood, but from which it is relatively easy to sample.
Approximate Bayesian computation is a likelihood-free method for implementing
Bayesian inference in such cases. We present results on the asymptotic variance
of estimators obtained using approximate Bayesian computation in a large-data
limit. Our key assumption is that the data are summarized by a
fixed-dimensional summary statistic that obeys a central limit theorem. We
prove asymptotic normality of the mean of the approximate Bayesian computation
posterior. This result also shows that, in terms of asymptotic variance, we
should use a summary statistic that is the same dimension as the parameter
vector, p; and that any summary statistic of higher dimension can be reduced,
through a linear transformation, to dimension p in a way that can only reduce
the asymptotic variance of the posterior mean. We look at how the Monte Carlo
error of an importance sampling algorithm that samples from the approximate
Bayesian computation posterior affects the accuracy of estimators. We give
conditions on the importance sampling proposal distribution such that the
variance of the estimator will be the same order as that of the maximum
likelihood estimator based on the summary statistics used. This suggests an
iterative importance sampling algorithm, which we evaluate empirically on a
stochastic volatility model.
