Low Complexity Static and Dynamic Sparse Bayesian Learning Combining BP, VB and EP Message Passing

Sparse Bayesian Learning (SBL) provides sophisticated (state) model order selection with unknown support distribution. This allows to handle problems with big state dimensions and relatively limited data by exploiting variations in parameter importance. The techniques proposed in this paper allow to handle the extension of SBL to time-varying states, modeled as diagonal first-order auto-regressive (DAR(1)) processes with unknown parameters to be estimated also. Adding the parameters to the state leads to an augmented state and a non-linear (at least bilinear) state-space model. The proposed approach, which applies also to more general non-linear models, uses a combination of belief propagation (BP), Variational Bayes (VB) or mean field (MF) techniques, and Expectation Propagation (EP) to approximate the posterior marginal distributions of the scalar factors. We propose Fisher Information Matrix analysis to determine the variable split between the use of BP and VB allowing to stay optimal in terms of Laplace approximation.