Robustness of Sparse Bayesian Learning in Correlated Environments

In this paper we analyse the performance of Sparse Bayesian Learning (SBL) in an environment with correlated sources. We provide two new perspectives to understand SBL's strategy for handling correlated sources. Using a Minimum Power Distortionless Response (MPDR) beamformer-based perspective of SBL, it is shown that the measured signal covariance strucure based on uncorrelated sources assumption used by SBL, infact provides immunity to source correlation. The MPDR based perspective is extended to formulate SBL's misspecified model. This misspecified model is compared to the underlying true data distribution using Kullback-Leibler (KL) divergence metric and it is shown that SBL attempts to find an uncorrelated source covariance matrix that best fits the data. Theory on performance of misspecified models for the two source case is studied to gain further insights into the problem. Numerical results are provided to demonstrate SBL's robustness as well as to quantify the model misfit when sources are correlated.