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We show precisely that message passing for inference in Gaussian graphical models on singly connected graphs is just a distributed implementation of Gaussian elimination without any need for backsubstitution. This observation allows us to generalize the procedure to arbitrary loopy Gaussian graphical models. We thus construct a message passing algorithm that is guaranteed to converge in finite time, and solve the inference problem exactly. The complexity of this algorithm grows gradually with the "distance" of the graph to a tree. This algorithm can be implemented in a distributed environment as, for example, in sensor networks.