Asymptotic Behaviour Of Bayes Estimates Under Possibly Incorrect Models

Introduction The frequentist asymptotic properties of Bayes estimators and of posterior distributions are well-known and have been investigated in different directions, see e.g. Bickel and Yahav (1969), Ibragimov and Has'minskii (1981), Strasser (1981) or Lehmann (1983). The interesting generalization to a possibly incorrect model has been treated by Berk (1966), who proved under regularity conditions, that a.s. the posterior distribution converges weakly toward the Dirac measure at the pseudotrue parameter, assuming its uniqueness. This is the parameter value corresponding to the distribution in the model, which is nearest to the true distribution in the sense of the information distance. The result is proven in the general case of possibly non-unique pseudo-true parameters for a corresponding generalization of the above mentioned weak convergence. Unfortunately, many standard models with unbounded parameter space are not covered by his theorems (see Remark 1 in our Section