  Many scientifically well-motivated statistical models in natural,
engineering, and environmental sciences are specified through a generative
process. However, in some cases, it may not be possible to write down the
likelihood for these models analytically. Approximate Bayesian computation
(ABC) methods allow Bayesian inference in such situations. The procedures are
nonetheless typically computationally intensive. Recently, computationally
attractive empirical likelihood-based ABC methods have been suggested in the
literature. All of these methods rely on the availability of several suitable
analytically tractable estimating equations, and this is sometimes problematic.
We propose an easy-to-use empirical likelihood ABC method in this article.
First, by using a variational approximation argument as a motivation, we show
that the target log-posterior can be approximated as a sum of an expected joint
log-likelihood and the differential entropy of the data generating density. The
expected log-likelihood is then estimated by an empirical likelihood where the
only inputs required are a choice of summary statistic, it's observed value,
and the ability to simulate the chosen summary statistics for any parameter
value under the model. The differential entropy is estimated from the simulated
summaries using traditional methods. Posterior consistency is established for
the method, and we discuss the bounds for the required number of simulated
summaries in detail. The performance of the proposed method is explored in
various examples.
