  We introduce novel results for approximate inference on planar graphical
models using the loop calculus framework. The loop calculus (Chertkov and
Chernyak, 2006) allows to express the exact partition function 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 (Certkov 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 the performance of the algorithm for the
partition function approximation for models with binary variables and pairwise
interactions on grids and other planar graphs. We study in detail both the loop
series and the equivalent Pfaffian series and show that the first term of the
Pfaffian series for the general, intractable planar model, 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.
