Performance bounds for parameter estimation play a crucial role in statistical signal processing theory and applications. Two widely recognized bounds are the Cramér-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 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.

中文翻译：

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渐进紧贝叶斯克拉默-饶界


参数估计的性能界限在统计信号处理理论和应用中起着至关重要的作用。两个广泛认可的界限是非贝叶斯框架中的 Cramér-Rao 界限 (CRB) 和贝叶斯框架中的贝叶斯 CRB (BCRB)。然而，与CRB不同的是，BCRB一般来说是渐近不可达到的，并且其等式条件是有限制的。本文介绍了 Bobrovsky-Mayer-Wolf-Zakai 类界限的扩展，也称为加权 BCRB (WBCRB)。 WBCRB 通过调整标量情况下的加权函数来优化。基于这个结果，我们提出了一个称为 AT-BCRB 的渐近紧边界版本。我们证明 AT-BCRB 是通过最大后验概率 (MAP) 估计器渐近获得的。此外，我们将 WBCRB 和 AT-BCRB 扩展到矢量参数的情况。所提出的界限在几个基本信号处理示例中进行了评估，例如白高斯过程的方差估计、到达方向估计以及具有未知方差和先验统计信息的高斯过程的均值估计。结果表明，与 BCRB 不同，所提出的边界是渐近可达到的，并且与预期的 CRB (ECRB) 一致。 ECRB 强制要求一致无偏，不能作为贝叶斯框架中的有效下限，而提议的界限对于任何估计量都有效。