Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part II: Estimation Methods


One in all the elemental issues of estimation theory is the presence of deterministic nuisance parameters. Whereas within the Bayesian paradigm the model parameters are random, introduction of deterministic nuisance parameters into the model exceeds the Bayesian framework to the hybrid framework. During this type of eventualities, the conventional Bayesian estimators don't seem to be valid, as they assume data of the deterministic nuisance parameters. This paper is the second of a 2-half study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. In part I, a replacement Cramér–Rao (CR)-type bound on the mean-square-error (MSE) for Bayesian estimation in the presence of deterministic nuisance parameters was established based mostly on the concept of risk-unbiasedness. The proposed certain was named risk-unbiased bound (RUB). This paper presents properties of asymptotic uniform mean- and risk-unbiasedness of some Bayesian estimators: 1) the minimum MSE (MMSE) or most a posteriori probability (MAP) estimators with maximum chance (ML) estimates substituting the deterministic parameters, named MS-ML and MAP-ML, respectively, and 2) joint MAP and ML estimator, named JMAP-ML. Furthermore, an asymptotic performance analysis of the MS-ML and MAP-ML estimators is presented. These estimators are shown to asymptotically achieve the RUB, while the existing CR-type bounds can be achieved solely in distinct cases. Simulations verify these results for the problem of blind separation of nonstationary sources. It is shown that not like existing CR-kind bounds, the RUB is asymptotically tight.

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