Belief Propagation and Revision in Networks with Loops

Local belief propagation rules of the sort proposed by P earl (1988) are guaranteed to converge to the optimal beliefs for singly connected networks. Recently, a n umber of researchers have empirically demonstrated good performance of these same algorithms on networks with loops, but a theoretical understanding of this performance has yet to be achieved. Here we l a y a foundation for an understanding of belief propagation in networks with loops. For networks with a single loop, we derive an analytical relationship between the steady state beliefs in the loopy network and the true posterior probability. Using this relationship we show a category of networks for which the MAP estimate obtained by belief update and by belief revision can be proven to be optimal (although the beliefs will be incorrect). We s h o w h o w nodes can use local information in the messages they receive in order to correct the steady state beliefs. Furthermore we p r o ve that for all networks with a single loop, the MAP estimate obtained by belief revision at convergence is guaranteed to give the globally optimal sequence of states. The result is independent of the length of the cycle and the size of the state space. For networks with multiple loops, we i n troduce the concept of a \balanced network" and show simulation results comparing belief revision and update in such networks. We show t h a t t h e T urbo code structure is balanced and present simulations on a toy T urbo code problem indicating the decoding obtained by belief revision at convergence is signiicantly more likely to be correct. A a b Figure 1: a. An example of the types of problems typically solved using belief propagation. Observed nodes are denoted by lled circles. A link between any t wo nodes implies a probabilistic compatability constraint. b. A simple network with a loop. Although belief propagation rules can be generalized to this network, a theoretical understanding of the algorithms behavior in such a n e t work has yet to be achieved.