Consistency of Bayes estimators without the assumption that the model is correct

We examine a more general form of consistency which does not necessarily rely on the correct specification of the likelihood in the Bayesian setting, but we restrict the form of the likelihood to be in a minimal standard exponential family. First, we investigate the asymptotic behavior of the Bayes estimator of a parameter, and show that the Bayes estimator is consistent under the condition that the exponential family is full. However, we find that θ_i=θ_j and ∥θ_i−θ_j∥<ε, even for very small ε, behave differently, even in an asymptotic manner, when the model is not correct. We note that the distinction applies generally to Bayesian testing problems.