Approximating the Bethe Partition Function

When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy F , and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced by Weller and Jebara for attractive binary pairwise MRFs which is guaranteed to return an !-approximation to the global minimum of F in polynomial time provided the maximum degree ∆ = O(log n), where n is the number of variables. Here we extend their approach and derive a new method based on analyzing first derivatives of F , which leads to much better performance and, for attractive models, yields a fully polynomial-time approximation scheme (FPTAS) without any degree restriction. Further, our methods apply to general (nonattractive) models, though with no polynomial time guarantee in this case, demonstrating that approximating log of the Bethe partition function, logZB = −minF , for a general model to additive !-accuracy may be reduced to a discrete MAP inference problem. This allows the merits of the global Bethe optimum to be tested.