Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs

In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.