Bayesian Estimation of Covariance Matrices in Non-Homogeneous Environments

In many applications, it is required to detect, from a primary vector, the presence of a signal of interest embedded in noise with unknown statistics. We consider a situation where the training samples used to infer the noise statistics do not share the same covariance matrix as the vector under test. A Bayesian model is proposed where the covariance matrices of the primary and the secondary data are assumed to be random, with some appropriate joint distribution. The prior distributions of these matrices reflect a rough knowledge about the environment. Within this framework, the minimum mean-square error (MMSE) estimator and the maximum a posteriori (MAP) estimator of the primary data covariance matrix are derived. A Gibbs sampling strategy is presented for the implementation of the MMSE estimator. Numerical simulations illustrate the performances of these estimators and compare them with those of the sample covariance matrix estimator.