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Belief propagation is shown to be an instance of a hybrid between two projection algorithms in the convex programming literature: Dykstra's algorithm with cyclic Bregman projections and an alternating Bregman projections algorithm. Via this connection, new results concerning the convergence and performance of belief propagation can be proven by exploiting the corresponding literature about the two projections algorithms it hybridizes. In this regard, it is identified that the lack of guaranteed convergence for belief propagation results from the asymmetry of its Bregman divergence by proving that when the associated hybrid projection algorithm generalization is used with a symmetric Bregman divergence, it always converges. Additionally, by characterizing factorizations that are close to acyclic in a manner independent of their girth, a new collection of distributions for which belief propagation is guaranteed to perform well is identified using the new projection algorithm framework.