  Approximate Bayesian computation allows for inference of complicated
probabilistic models with intractable likelihoods using model simulations. The
Markov chain Monte Carlo implementation of approximate Bayesian computation is
often sensitive to the tolerance parameter: low tolerance leads to poor mixing
and large tolerance entails excess bias. We consider an approach using a
relatively large tolerance for the Markov chain Monte Carlo sampler to ensure
its sufficient mixing, and post-processing the output leading to estimators for
a range of finer tolerances. We introduce an approximate confidence interval
for the related post-corrected estimators, and propose an adaptive approximate
Bayesian computation Markov chain Monte Carlo, which finds a `balanced'
tolerance level automatically, based on acceptance rate optimisation. Our
experiments show that post-processing based estimators can perform better than
direct Markov chain targetting a fine tolerance, that our confidence intervals
are reliable, and that our adaptive algorithm leads to reliable inference with
little user specification.
