Asymptotic behavior of Bayes estimates under possibly incorrect models

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Abstract
We prove that the posterior distribution in a possibly incorrect parametric model a.s. concentrates in a strong sense on the set of pseudotrue parameters determined by the true distribution. As a consequence, we obtain in the case of a unique pseudotrue parameter the strong consistency of pseudo-Bayes estimators w.r.t. general loss functions. Further, we present a simple example based on normal distributions and having two different pseudotrue parameters, where pseudo-Bayes estimators have an essentially different asymptotic behavior than the pseudomaximum likelihood estimator. While the MLE is strongly consistent, the sequence of posterior means is strongly inconsistent and a.s. almost all its accumulation points are not pseudotrue. Finally, we give conditions under which a pseudo-Bayes estimator for a unique pseudotrue parameter has an asymptotic normal distribution.

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