  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 for attractive binary pairwise MRFs which is guaranteed to
return an $\epsilon$-approximation to the global minimum of $F$ in polynomial
time provided the maximum degree $\Delta=O(\log n)$, where $n$ is the number of
variables. Here we significantly improve this algorithm and derive several
results including a new approach based on analyzing first derivatives of $F$,
which leads to performance that is typically far superior and yields a fully
polynomial-time approximation scheme (FPTAS) for attractive models without any
degree restriction. Further, the method applies to general (non-attractive)
models, though with no polynomial time guarantee in this case, leading to the
important result that approximating $\log$ of the Bethe partition function,
$\log Z_B=-\min F$, for a general model to additive $\epsilon$-accuracy may be
reduced to a discrete MAP inference problem. We explore an application to
predicting equipment failure on an urban power network and demonstrate that the
Bethe approximation can perform well even when BP fails to converge.
