Bayesian Inference with Unknown Noise Covariance
Doctoral thesis, 2004
This thesis studies Bayesian methods in statistical signal processing. A central theme is that the treatment of the unknown noise covariance matrix is of chief concern for all scenarios.
As a first application, the detection of land mines using infrared techniques is investigated. In this setting, the dimensions of the involved noise covariance matrices are often enormous. By exploiting an assumption of rotational invariance, the number of free parameters is decreased substantially. This enables accurate estimation of the remaining parameters. Furthermore, the signal component is handled using Bayesian techniques. This facilitates the incorporation of partial knowledge about external parameters such as conditions in the soil, or the burial depth of the mine. Finally, additional visible images are conveniently included in the framework through a joint treatment with the infrared images.
The remaining part of the thesis focuses more specifically on the Bayesian treatment of the unknown noise color. Several different non-informative prior distributions for covariance matrices are discussed and evaluated. For the linear signal model, a relation between the standard Adaptive Maximum Likelihood (AML) estimator and the usage of the Jeffreys prior distribution is established. Moreover, a connection between the regularized version of the AML estimator and an adjustment to the Jeffreys prior is also recognized. Apart from providing insight regarding the classical estimators, these results allow for a formal treatment of the regularization parameter. To improve the treatment of the eigenvalues of the covariance matrix, the reference prior is also derived. In difference to the Jeffreys and the regularized priors, the reference prior does not allow for analytical solutions. Considerable efforts are therefore dedicated to the design of efficient implementations for this prior. Simulations verify that the reference prior generally outperforms both the Jeffreys and the regularized solutions. Ultimately, the Jeffreys and the regularized solutions are applied for detection of linear signals. To circumvent difficulties with the usage of an improper prior for the signal parameter, different versions of Intrinsic Bayes Factors are exploited. Simulations indicate that the resulting Bayes Factors outperform the Generalized Likelihood Ration Test (GLRT) in difficult scenarios involving small training data sets.