On Parametric Misspecified Bayesian Cramér–Rao bound: An application to linear/Gaussian systems

A lower bound is an important tool for predicting the performance that an estimator can achieve under a particular statistical model. Bayesian bounds are a kind of such bounds which not only utilizes the observation statistics but also includes the prior model information. In reality, however, the true model generating the data is either unknown or simplified when deriving estimators, which motivates the works to derive estimation bounds under modeling mismatch situations. This paper provides a derivation of a Bayesian Cram\'{e}r-Rao bound under model misspecification, by introducing important concepts such as the pseudotrue parameter in a Bayesian context which was not identified in previous works. The general result is particularized in linear and Gaussian problems, where closed-forms are available and results are used to validate the results.