Asymptotic normality of the posterior given a statistic

The authors establish the asymptotic normality and determine the limiting variance of the posterior density for a multivariate parameter, given the value of a consistent and asymptotically Gaussian statistic satisfying a uniform local central limit theorem. Their proof is given in the continuous case but generalizes to lattice-valued random variables. It hinges on a uniform Edgeworth expansion used to control the behaviour of the conditioning statistic. They provide examples and show how their result can help in identifying reference priors.