## Abstract

We discuss a generic model of Bayesian inference with binary variables defined on edges of a planar graph. The Loop Calculus approach of Chertkov and Chernyak (2006 Phys. Rev. E 73 065102(R) [cond-mat/0601487]; 2006 J. Stat. Mech. P06009 [cond-mat/0603189]) is used to evaluate the resulting series expansion for the partition function. We show that, for planar graphs, truncating the series at single-connected loops reduces, via a map reminiscent of the Fisher transformation (Fisher 1961 Phys. Rev. 124 1664), to evaluating the partition function of the dimer-matching model on an auxiliary planar graph. Thus, the truncated series can be easily re-summed, using the Pfaffian formula of Kasteleyn (1961 Physics 27 1209). This allows us to identify a big class of computationally tractable planar models reducible to a dimer model via the Belief Propagation (gauge) transformation. The Pfaffian representation can also be extended to the full Loop Series, in which case the expansion becomes a sum of Pfaffian contributions, each associated with dimer matchings on an extension to a subgraph of the original graph. Algorithmic consequences of the Pfaffian representation, as well as relations to quantum and non-planar models, are discussed.

Original language | English (US) |
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Article number | P05003 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2008 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2008 |

Externally published | Yes |

## Keywords

- Analysis of algorithms
- Error correcting codes
- Heuristics

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty