  This paper concerns message passing based approaches to sparse Bayesian
learning (SBL) with a linear model corrupted by additive white Gaussian noise
with unknown variance. With the conventional factor graph, mean field (MF)
message passing based algorithms have been proposed in the literature. In this
work, instead of using the conventional factor graph, we modify the factor
graph by adding some extra hard constraints (the graph looks like being
`stretched'), which enables the use of combined belief propagation (BP) and MF
message passing. We then propose a low complexity BP-MF SBL algorithm based on
which an approximate BP-MF SBL algorithm is also developed to further reduce
the complexity. Thanks to the use of BP, the BP-MF SBL algorithms show their
merits compared with state-of-the-art MF SBL algorithms: they deliver even
better performance with much lower complexity compared with the vector-form MF
SBL algorithm and they significantly outperform the scalar-form MF SBL
algorithm with similar complexity.
