On the Asymptotic Efficiency of ABC Estimators
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Many statistical applications involve models for which it is difficult to evaluate the likelihood, but relatively easy to sample from. Approximate Bayesian computation (ABC) is a likelihood-free method for implementing Bayesian inference in such cases. It avoids evaluating the likelihood function and generates samples from an approximate posterior distribution by jointly simulating the parameter and the data, and accepting parameter values that give simulated data close to the observed data. We present results on the asymptotic variance of ABC estimators in a large-data limit. Our key assumption is that we summarise the data by a fixed dimensional summary statistic and that this summary statistic obeys a central limit theorem. We prove asymptotic normality of the ABC posterior mean. This improves on recent results on consistency for the ABC posterior mean, and in particular specifies its rate of convergence. 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 ABC posterior mean. We then look at how the Monte Carlo error of an importance sampling algorithm that samples from the ABC posterior effects the accuracy of the ABC estimator. We give conditions on the importance sampling proposal distribution such that the variance of the ABC estimator will be the same order as that of the MLE based on the summary statistics used by ABC. This result suggests an iterative importance sampling algorithm, which we then evaluate empirically on a stochastic volatility model.