  We introduce novel results for approximate inference on planar graphical
models using the loop calculus framework. The loop calculus (Chertkov and
Chernyak, 2006b) allows to express the exact partition function Z of a
graphical model as a finite sum of terms that can be evaluated once the belief
propagation (BP) solution is known. In general, full summation over all
correction terms is intractable. We develop an algorithm for the approach
presented in Chertkov et al. (2008) which represents an efficient truncation
scheme on planar graphs and a new representation of the series in terms of
Pfaffians of matrices. We analyze in detail both the loop series and the
Pfaffian series for models with binary variables and pairwise interactions, and
show that the first term of the Pfaffian series can provide very accurate
approximations. The algorithm outperforms previous truncation schemes of the
loop series and is competitive with other state-of-the-art methods for
approximate inference.
