  Variational message passing (VMP), belief propagation (BP) and expectation
propagation (EP) have found their wide applications in complex statistical
signal processing problems. In addition to viewing them as a class of
algorithms operating on graphical models, this paper unifies them under an
optimization framework, namely, Bethe free energy minimization with differently
and appropriately imposed constraints. This new perspective in terms of
constraint manipulation can offer additional insights on the connection between
different message passing algorithms and is valid for a generic statistical
model. It also founds a theoretical framework to systematically derive message
passing variants. Taking the sparse signal recovery (SSR) problem as an
example, a low-complexity EP variant can be obtained by simple constraint
reformulation, delivering better estimation performance with lower complexity
than the standard EP algorithm. Furthermore, we can resort to the framework for
the systematic derivation of hybrid message passing for complex inference
tasks. Notably, a hybrid message passing algorithm is exemplarily derived for
joint SSR and statistical model learning with near-optimal inference
performance and scalable complexity.
