Asymptotic expansions of posterior distributions are derived for a two-dimensional exponential family, which includes normal, gamma, inverse gamma and inverse Gaussian distributions. Reparameterization allows us to use a data-dependent transformation, convert the likelihood function to the two-dimensional standard normal density and apply a version of Stein's identity to assess the posterior distributions. Applications are given to characterize optimal noninformative priors in the sense of Stein, to suggest the form of a high-order correction to the distribution function of a sequential likelihood ratio statistic and to provide confidence intervals for one parameter in the presence of other nuisance parameters.
"Integrable Expansions for Posterior Distributions for a Two-Parameter Exponential Family." Ann. Statist. 22 (4) 1808 - 1830, December, 1994. https://doi.org/10.1214/aos/1176325758