Asymptotically Tight Misspecified Bayesian Cramér-Rao Bound

In many applications of estimation theory, the true data model is not perfectly known, leading to mismatch between the assumed model used for parameter estimation and the actual model. The non-Bayesian misspecified Cramér-Rao bound (MCRB) allows considering the effect of model misspecification on the estimator performance, and it has been extended to the Bayesian framework. Unlike the non-Bayesian MCRB, the corresponding Bayesian bound is asymptotically unattainable. In this paper, we derive an asymptotically tight misspecified Bayesian Cramér-Rao bound. We demonstrate that under some mild and common regularity conditions, this bound is asymptotically achieved by the maximum a-posteriori probability (MAP) estimator. The proposed bound is applied to the problems of variance estimation and direction-of-arrival estimation under model misspecification, illustrating its asymptotic attainability by the MAP estimator.