  We propose a theoretical framework for non redundant reconstruction of a
global loss from a collection of local ones under constraints given by a
functor; we call this loss the regionalized loss in honor to Yedidia, Freeman,
Weiss' celebrated article `Constructing free-energy approximations and
generalized belief propagation algorithms' where a first example of
regionalized loss, for entropy and the marginal functor, is built. We show how
one can associate to these regionalized losses message passing algorithms for
finding their critical points. It is a natural mathematical framework for
optimization problems where there are multiple points of views on a dataset and
replaces message passing algorithms as canonical ways of finding the optima of
these problems. We explain how Generalized Belief propagation algorithms fall
into the framework we propose and propose novel message passing algorithms for
noisy channel networks.
