  Belief propagation (BP) is an iterative method to perform approximate
inference on arbitrary graphical models. Whether BP converges and if the
solution is a unique fixed point depends on both the structure and the
parametrization of the model. To understand this dependence it is interesting
to find \emph{all} fixed points. In this work, we formulate a set of polynomial
equations, the solutions of which correspond to BP fixed points. To solve such
a nonlinear system we present the numerical polynomial-homotopy-continuation
(NPHC) method. Experiments on binary Ising models and on error-correcting codes
show how our method is capable of obtaining all BP fixed points. On Ising
models with fixed parameters we show how the structure influences both the
number of fixed points and the convergence properties. We further asses the
accuracy of the marginals and weighted combinations thereof. Weighting
marginals with their respective partition function increases the accuracy in
all experiments. Contrary to the conjecture that uniqueness of BP fixed points
implies convergence, we find graphs for which BP fails to converge, even though
a unique fixed point exists. Moreover, we show that this fixed point gives a
good approximation, and the NPHC method is able to obtain this fixed point.

Machine Learning;Algebraic Geometry