Approximate Bayesian computing (ABC) is a likelihood-free method that has
grown increasingly popular since early applications in population genetics.
However, the theoretical justification for Bayesian inference (e.g.
construction of credible intervals) based on this method has not yet been fully
addressed when using non-sufficient summary statistics. We introduce a more
general computational technique, approximate confidence distribution computing
(ACC), to overcome a few issues associated with the ABC method, for instance,
the lack of theory supporting for constructing credible (or confidence)
intervals when the ACC method uses non-sufficient summary statistics, the long
computing time, and the necessity of a prior assumption. Specifically, we
establish frequentist coverage properties for the outcome of the ACC method by
using the theory of confidence distributions, and thus inference based on ACC
is justified, even if reliant upon a non-sufficient summary statistic.
Furthermore, the ACC method is very broadly applicable; in fact, the ABC
algorithm can be viewed as a special case of an ACC method without damaging the
integrity of ACC based inference. We supplement the theory with simulation
studies and an epidemiological application to illustrate the benefits of the
ACC method. It is demonstrated that a well-tended ACC algorithm can greatly
increase its computing efficiency over a typical ABC algorithm.