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

Shahar Bar, Joseph Tabrikian

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 Cramér-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.

Original languageEnglish
Article number7202897
Pages (from-to)6647-6658
Number of pages12
JournalIEEE Transactions on Signal Processing
Issue number24
StatePublished - 15 Dec 2015


  • Bayesian Cramér-Rao bound
  • Bayesian estimators
  • MSE
  • combined minimum MSE-maximum likelihood (MS-ML)
  • hybrid Cramér-Rao bound
  • joint maximum a-posteriori probability-maximum likelihood (JMAP-ML)
  • maximum likelihood (ML)
  • nuisance parameters
  • performance bounds
  • risk-unbiasedness

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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