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We introduce a differential complex of local observables given adecomposition of a global set of random variables into subsets. Its boundaryoperator allows us to define a transport equation equivalent to BeliefPropagation. This definition reveals a set of conserved quantities under BeliefPropagation and gives new insight on the relationship of its equilibria withthe critical points of Bethe free energy.

A Homological Approach to Belief Propagation and Bethe Approximations