Optimal Linear MMSE Estimation Under Correlation Uncertainty in Restricted Bayesian Framework

A restricted Bayes approach is proposed for linear estimation of a scalar random parameter based on a scalar observation under uncertainty regarding the correlation between the parameter and the observation. In particular, the optimal linear estimator that minimizes the average mean-squared error (MSE) is derived under a constraint on the worst-case MSE by considering possible values of the correlation coefficient and its probability distribution. A closed-form expression is derived for the optimal linear estimator in the proposed restricted Bayesian framework by considering a generic statistical characterization of the correlation coefficient. Performance of the proposed estimator is evaluated via numerical examples and its benefits are illustrated in various scenarios. The proposed framework is also extended to the case of vector-valued observation and the properties of the optimal linear estimator are characterized.