A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Asymptotic arguments are widely used in Bayesian inference, and in recent years there has been considerable developments of the so-called higher-order asymptotics. This theory provides very accurate approximations to posterior distributions, and to related quantities, in a variety of parametric statistical problems, even for small sample sizes. The aim of this contribution is to discuss recent advances in approximate Bayesian computations based on the asymptotic theory of modified loglikelihood ratios, both from theoretical and practical point of views. Results on third-order approximations for univariate posterior distributions, also in the presence of nuisance parameters, are reviewed and a new formula for a vector parameter of interest is presented. All these approximations may routinely be applied in practice for Bayesian inference, since they require little more than standard likelihood quantities for their implementation, and hence they may be available at little additional computational cost over simple first-order approximations. Moreover, these approximations give rise to a simple simulation scheme, alternative to MCMC, for Bayesian computation of marginal posterior distributions for a scalar parameter of interest. In addition, they can be used for testing precise null hypothesis and to define accurate Bayesian credible sets. Some illustrative examples are discussed, with particular attention to the use of matching priors.