We discuss higher-order approximations to the marginal posterior distribution for a scalar parameter of interest in the presence of nuisance parameters. These higher-order approximations are obtained using a suitable matching prior. The proposed procedure has several advantages since it does not require the elicitation on the nuisance parameter, neither numerical integration or MCMC simulation, and it enables us to perform accurate Bayesian inference even for very small sample sizes. Numerical illustrations are given for models of practical interest, such as linear non-normal models and logistic regression. We also illustrate how the proposed accurate approximation can routinely be applied in practice using results from likelihood asymptotics and the R package bundle ho
