  It is known that fixed points of loopy belief propagation (BP) correspond to
stationary points of the Bethe variational problem, where we minimize the Bethe
free energy subject to normalization and marginalization constraints.
Unfortunately, this does not entirely explain BP because BP is a dual rather
than primal algorithm to solve the Bethe variational problem -- beliefs are
infeasible before convergence. Thus, we have no better understanding of BP than
as an algorithm to seek for a common zero of a system of non-linear functions,
not explicitly related to each other. In this theoretical paper, we show that
these functions are in fact explicitly related -- they are the partial
derivatives of a single function of reparameterizations. That means, BP seeks
for a stationary point of a single function, without any constraints. This
function has a very natural form: it is a linear combination of local
log-partition functions, exactly as the Bethe entropy is the same linear
combination of local entropies.
