Belief propagation: accurate marginals or accurate partition function—where is the difference?

We analyze belief propagation on patch potential models—attractive models with varying local potentials—obtain all of the potentially many fixed points, and gather novel insights into belief propagation properties. In particular, we observe and theoretically explain several regions in the parameter space that behave fundamentally differently. We specify and elaborate on one specific region that, despite the existence of multiple fixed points, is relatively well behaved and provides insights into the relationship between the accuracy of the marginals and the partition function. We demonstrate the inexistence of a principal relationship between both quantities and provide sufficient conditions for a fixed point to be optimal with respect to approximating both the marginals and the partition function.