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

Abstract

One of the fundamental issues of estimation theory is the presence of deterministic nuisance parameters. While in the Bayesian paradigm the model parameters are random, introduction of deterministic nuisance parameters into the model exceeds the Bayesian framework to the hybrid framework. In this type of scenarios, the conventional Bayesian estimators are not valid, as they assume knowledge of the deterministic nuisance parameters. This paper is the second of a two-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. In part I, a new Cramer–Rao (CR)-type bound on the mean-square-error (MSE) for Bayesian estimation in the presence of deterministic nuisance parameters was established based on the concept of risk-unbiasedness. The proposed bound 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 maximum a posteriori probability (MAP) estimators with maximum likelihood (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 only in distinct cases. Simulations verify these results for the problem of blind separation of nonstationary sources. It is shown that unlike existing CR-type bounds, the RUB is asymptotically tight.