Performance bounds for parameter estimation play a crucial role in statistical signal processing theory and applications. Two widely recognized bounds are the Cram\'{e}r-Rao bound (CRB) in the non-Bayesian framework, and the Bayesian CRB (BCRB) in the Bayesian framework. However, unlike the CRB, the BCRB is asymptotically unattainable in general, and its equality condition is restrictive. This paper introduces an extension of the Bobrovsky--Mayer-Wolf--Zakai class of bounds, also known as the weighted BCRB (WBCRB). The WBCRB is optimized by tuning the weighting function in the scalar case. Based on this result, we propose an asymptotically tight version of the bound called AT-BCRB. We prove that the AT-BCRB is asymptotically attained by the maximum {\it a-posteriori} probability (MAP) estimator. Furthermore, we extend the WBCRB and the AT-BCRB to the case of vector parameters. The proposed bounds are evaluated in several fundamental signal processing examples, such as variance estimation of white Gaussian process, direction-of-arrival estimation, and mean estimation of Gaussian process with unknown variance and prior statistical information. It is shown that unlike the BCRB, the proposed bounds are asymptotically attainable and coincide with the expected CRB (ECRB). The ECRB, which imposes uniformly unbiasedness, cannot serve as a valid lower bound in the Bayesian framework, while the proposed bounds are valid for any estimator.

翻译：参数估计的性能界在统计信号处理理论与应用中扮演着关键角色。两个广泛认可的界分别是非贝叶斯框架下的克拉默-拉奥界（CRB）和贝叶斯框架下的贝叶斯CRB（BCRB）。然而，与CRB不同，BCRB通常无法渐近达到，且其等号成立条件具有限制性。本文引入了Bobrovsky–Mayer-Wolf–Zakai类界的扩展形式，即加权BCRB（WBCRB）。通过标量情形下对权函数的调节，WBCRB可达到优化。基于此结果，我们提出一种渐近紧致的界，称为AT-BCRB。我们证明AT-BCRB可通过最大后验概率（MAP）估计器渐近达到。此外，我们将WBCRB和AT-BCRB拓展至向量参数情形。所提出的界在多个基础信号处理实例中进行了评估，例如白高斯过程方差估计、波达方向估计以及含未知方差与先验统计信息的高斯过程均值估计。结果表明，与BCRB不同，所提出的界可渐近达到，并与期望CRB（ECRB）一致。ECRB要求无偏性一致成立，但在贝叶斯框架下无法作为有效下界，而所提出的界对任意估计器均有效。