  Approximate Bayesian computation (ABC) is an approach for sampling from an
approximate posterior distribution in the presence of a computationally
intractable likelihood function. A common implementation is based on simulating
model, parameter and dataset triples, (m,\theta,y), from the prior, and then
accepting as samples from the approximate posterior, those pairs (m,\theta) for
which y, or a summary of y, is "close" to the observed data. Closeness is
typically determined though a distance measure and a kernel scale parameter,
\epsilon. Appropriate choice of \epsilon is important to producing a good
quality approximation. This paper proposes diagnostic tools for the choice of
\epsilon based on assessing the coverage property, which asserts that credible
intervals have the correct coverage levels. We provide theoretical results on
coverage for both model and parameter inference, and adapt these into
diagnostics for the ABC context. We re-analyse a study on human demographic
history to determine whether the adopted posterior approximation was
appropriate. R code implementing the proposed methodology is freely available
in the package "abc."
