Error Bounds Between Marginal Probabilities and Beliefs of Loopy Belief Propagation Algorithm

Belief propagation (BP) algorithm has been becoming increasingly a popular method for probabilstic inference on general graphical models. When networks have loops, it may not converge and, even if converges, beliefs, Le., the result of the algorithm, may not be equal to exact marginal probabilties. When networks have loops, the algorithm is called Loopy BP (LBP). Tatikonda and Jordan applied Gibbs measures theory to LBP algorithm and derived a suffcient convergence condition. In this paper, we utilize Gibbs measure theory to investigate the discrepancy between a marginal probabilty and the corresponding belief. Consequently, in particular, we obtain an error bound if the algorithm converges under a certain condition. It is a general result for the accuracy of the algorithm. We also perform numerical experiments to see the effectiveness of the result.