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On the Optimality of Solutions of the MaxProduct Belief Propagation Algorithm in Arbitrary Graphs
, 2001
"... Graphical models, suchasBayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. The maxproduct "belief propagation" algorithm is a localmessage passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tr ..."
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Cited by 185 (15 self)
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Graphical models, suchasBayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. The maxproduct "belief propagation" algorithm is a localmessage passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the fixedpoint yields the most probable a posteriori (MAP) values of the unobserved variables given the observed ones. Recently, good
MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies
, 2007
"... Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with maxproduct belief propagation (BP) and its variants. In particular, it is known that using BP on a singlecycle graph or tree reweighted BP on an arbitrary ..."
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Cited by 45 (4 self)
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Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with maxproduct belief propagation (BP) and its variants. In particular, it is known that using BP on a singlecycle graph or tree reweighted BP on an arbitrary graph will give the MAP solution if the beliefs have no ties. In this paper we extend the setting under which BP can be used to provably extract the MAP. We define Convex BP as BP algorithms based on a convex free energy approximation and show that this class includes ordinary BP with singlecycle, tree reweighted BP and many other BP variants. We show that when there are no ties, fixedpoints of convex maxproduct BP will provably give the MAP solution. We also show that convex sumproduct BP at sufficiently small temperatures can be used to solve linear programs that arise from relaxing the MAP problem. Finally, we derive a novel condition that allows us to derive the MAP solution even if some of the convex BP beliefs have ties. In experiments, we show that our theorems allow us to find the MAP in many realworld instances of graphical models where exact inference using junctiontree is impossible.
Correctnes of belief propagation in Gaussian graphical models of arbitrary topology
 NEURAL COMPUTATION
, 1999
"... Local "belief propagation" rules of the sort proposed byPearl [12] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"  using these ..."
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Cited by 1 (0 self)
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Local "belief propagation" rules of the sort proposed byPearl [12] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation"  using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannonlimit performance of "Turbo codes", whose decoding algorithm is equivalentto loopy belief propagation. Except for the