Approximate Bayesian computation with modified log-likelihood ratios

The aim of this contribution is to discuss approximate Bayesian computation based on the asymptotic theory of modified likelihood roots and log-likelihood ratios. Results on third-order approximations for univariate posterior distributions, also in the presence of nuisance parameters, are reviewed and the computation of asymptotic credible sets for a vector parameter of interest is illustrated. All these approximations are available at little additional computational cost over simple first-order approximations. Some illustrative examples are discussed, with particular attention to the use of matching priors.