On the use of approximate Bayesian computation Markov chain Monte Carlo with inflated tolerance and post-correction

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.