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In random parameter estimation, Bayesian lower bounds (BLBs) for themean-square error have been noticed to not be tight in a number of cases, evenwhen the sample size, or the signal-to-noise ratio, grow to infinity. In thispaper, we study alternative forms of BLBs obtained from a covarianceinequality, where the inner product is based on the \textit{a posteriori}instead of the joint probability density function. We hence obtain a family ofBLBs, which is shown to form a counterpart at least as tight as the well-knownWeiss-Weinstein family of BLBs, and we extend it to the general case of vectorparameter estimation. Conditions for equality between these two families areprovided. Focusing on the Bayesian Cram\'er-Rao bound (BCRB), a definition ofefficiency is proposed relatively to its tighter form, and efficient estimatorsare described for various types of common estimation problems, e.g., scalar,exponential family model parameter estimation. Finally, an example is provided,for which the classical BCRB is known to not be tight, while we show itstighter form is, based on formal proofs of asymptotic efficiency of Bayesianestimators. This analysis is finally corroborated by numerical results.

Some Results on Tighter Bayesian Lower Bounds on the Mean-Square Error