Objective Bayesian higher-order asymptotics in models with nuisance parameters

A higher-order approximation to the marginal posterior distribution for a scalar parameter of interest in the presence of nuisance parameters is proposed. The approximation is obtained using a matching prior. The procedure improves the normal first-order approximation and has several advantages. It does not require the elicitation on the nuisance parameters, neither numerical integration nor Monte Carlo simulation, and it enables us to perform accurate Bayesian inference even for small sample sizes. Numerical illustrations are given for models of practical interest, such as linear non-normal models and logistic regression. Finally, it is shown how the proposed approximation can routinely be applied in practice using results from likelihood asymptotics and the R package bundle hoa.