On the Asymptotic Behaviour of Posterior Distributions

SUMMARY Let a random sample of size n be taken from a distribution having a density depending on a real parameter 0, and let 0 have an absolutely continuous prior distribution with density ir(G). We give a rigorous proof that, under suitable regularity conditions, the posterior distribution of 0 will, when n tends to infinity, be asymptotically normal with mean equal to the maximumlikelihood estimator and variance equal to the reciprocal of the second derivative of the logarithm of the likelihood function evaluated at the maximum-likelihood estimator, independently of the form of 7r(G).