A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family

Abstract

Abstract-Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky-MayerWolf-Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakaï bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakaï. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakaï bound, and the Bayesian Cramér-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.Index Terms-Bayesian bounds on the MSE, Weiss-Weinstein family.

Performance bounds under misspecification model for MIMO radar application

“…The UCRB is the lowest bound in both figures which means that, obviously, if the model is correctly specified, estimators can provide better performance in term of the MSE than in the misspecified case. Note that, the CRB which takes into account the bias of the MLE and the MCRB exhibit a threshold effect[14][15][16]. The origin of this phenomenom is only due to the bias of the estimators as we can see the difference between the UCRB and the CRB in both figures.Fig.…”

Parameter bounds under misspecified models

“…This theorem applies to any viable inner product space. It is the basis for [7], and proof is given in [8].…”

“…Several bounds derive from the covariance inequality and the minimum norm theorem via varied choices of score function and constraints [6], [8]. Let ζ(x) = θ f (x) − μ g and define the score function as…”

“…The goal herein, however, is to consider the class of non-Bayesian parameter bounds emerging from the covariance inequality [6] under the assumption of model misspecification. Casting the bound problem as one of constrained minimization [7], [8] is likewise considered. Primary attention is devoted to the Cramér-Rao bound (CRB).…”

Hybrid lower bound on the MSE based on the Barankin and Weiss-Weinstein bounds

“…Our motivation comes from the fact that, among the Bayesian bounds, the WeissWeinstein bound is known to be one of the tightest [25]. So, one can expect that the combination of these bounds will lead to a bound tighter than the HBB.…”