  Approximate Bayesian Computation (ABC) methods are applicable to statistical
models specified by generative processes with analytically intractable
likelihoods. These methods try to approximate the posterior density of a model
parameter by comparing the observed data with additional process-generated
simulated datasets. For computational benefit, only the values of certain
well-chosen summary statistics are usually compared, instead of the whole
dataset. Most ABC procedures are computationally expensive, justified only
heuristically, and have poor asymptotic properties. In this article, we
introduce a new empirical likelihood-based approach to the ABC paradigm called
ABCel. The proposed procedure is computationally tractable and approximates the
target log posterior of the parameter as a sum of two functions of the data --
namely, the mean of the optimal log-empirical likelihood weights and the
estimated differential entropy of the summary functions. We rigorously justify
the procedure via direct and reverse information projections onto appropriate
classes of probability densities. Past applications of empirical likelihood in
ABC demanded constraints based on analytically tractable estimating functions
that involve both the data and the parameter; although by the nature of the ABC
problem such functions may not be available in general. In contrast, we use
constraints that are functions of the summary statistics only. Equally
importantly, we show that our construction directly connects to the reverse
information projection. We show that ABCel is posterior consistent and has
highly favourable asymptotic properties. Its construction justifies the use of
simple summary statistics like moments, quantiles, etc, which in practice
produce an accurate approximation of the posterior density. We illustrate the
performance of the proposed procedure in a range of applications.
