Asymptotic properties of a sequence of posterior distributions with application to the dirichlet process mixture.

In this paper a sequence ot distributions on the set of all probability measures absolutely continuous with respect to σ-finite measure on a sample space is considered. The generic distribution can be viewed either as a pseudo-posterior distribution obtained by integrating the likelihood raisedto a positive power less than one with respect to a legitimate prior or as the posterior distribution corresponding to a certain data-dependent prior. The distribution is Hellinger consistent at each probability measure in the Kullback-Leibler support of the legitimate prior. Theoretical justification is provided for using the ad hoc data-dependent prior and the derived posterior, whose rate of convergence is assessed. It is shown how recourse to this distribution can be made to establish sufficient conditions for consistency of the posterior of location mixtures of normal densities, when the scale parameter has a sample- size dependent prior and the mixing measure has any distribution. If the mixing distribution is the trajectory of a Dirichlet process, then the posterior converges at the best known rate n (^−1/2) (In n) with respect to the Hellinger distance.