  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.
