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Approximate Bayesian computation allows for inference of complicatedprobabilistic models with intractable likelihoods using model simulations. TheMarkov chain Monte Carlo implementation of approximate Bayesian computation isoften sensitive to the tolerance parameter: low tolerance leads to poor mixingand large tolerance entails excess bias. We consider an approach using arelatively large tolerance for the Markov chain Monte Carlo sampler to ensureits sufficient mixing, and post-processing the output leading to estimators fora range of finer tolerances. We introduce an approximate confidence intervalfor the related post-corrected estimators, and propose an adaptive approximateBayesian computation Markov chain Monte Carlo, which finds a `balanced'tolerance level automatically, based on acceptance rate optimisation. Ourexperiments show that post-processing based estimators can perform better thandirect Markov chain targetting a fine tolerance, that our confidence intervalsare reliable, and that our adaptive algorithm leads to reliable inference withlittle user specification.

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