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, defining important concepts such as pseudotrue parameter that were not clearly 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.

翻译：下界是预测估计器在特定统计模型下所能达到性能的重要工具。贝叶斯界限是这类界限的一种，它不仅利用观测统计量，还包含先验模型信息。然而在实际中，生成数据的真实模型要么未知，要么在推导估计器时被简化，这促使了在模型失配情形下推导估计界限的研究。本文推导了模型误设定条件下的贝叶斯克拉美-罗界，定义了先前工作中未明确识别的伪真实参数等重要概念。将一般结果特化到线性高斯问题中，此时可获得闭式解，并利用这些结果验证了结论的有效性。