Separate but equal: Equality in belief propagation for single-cycle graphs

Belief propagation is a widely used, incomplete optimization algorithm whose main theoretical properties hold only under the assumption that beliefs are not equal. Nevertheless, there is substantial evidence to suggest that equality between beliefs does occur. A published method to overcome belief equality, which is based on the use of unary function-nodes, is commonly assumed to resolve the problem.

In this study, we focus on min-sum, the version of belief propagation that is used to solve constraint optimization problems. We prove that for the case of a single-cycle graph, belief equality can only be avoided when the algorithm converges to the optimal solution. Under any other circumstances, the unary function method will not prevent equality, indicating that some of the existing results presented in the literature are in need of reassessment. We differentiate between belief equality, which refers to equal beliefs in a single message, and assignment equality, which prevents the coherent assignment of values to the variables, and we provide conditions for both.