Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 3 / b / Solution

Under the usual regular, identifiable, fixed-dimensional setting, the maximum-likelihood estimator is an interior point with vanishing log-likelihood gradient, and the observed information is positive definite. DefineThe second-order Taylor expansion of the Bayesian deviance isWith a prior density approximately constant over the likelihood's concentration region, the posterior density is therefore proportional toRecognizing the multivariate normal density givesTypically $J$ grows at rate $n$ and the posterior concentrates in a region of radius $n^{-1/2}$, making higher-order terms negligible under the regularity assumptions. The local flatness assumption concerns this region, not necessarily the entire parameter space. Boundary maxima, unidentified directions, or persistent multiple modes can invalidate this approximation.