Yusuke Watanabe and Kenji Fukumizu 2009 J. Phys. A: Math. Theor. 42 045001 doi:10.1088/1751-8113/42/4/045001
Loop series expansion with propagation diagrams
Yusuke Watanabe and Kenji Fukumizu
Show affiliationsThe Bethe approximation is a successful method for approximating partition functions of probabilistic models associated with a graph. Recently, Chertkov and Chernyak derived an interesting formula called 'loop series expansion', which is an expansion of the partition function. The main term of the series is the Bethe approximation while other terms are labelled by subgraphs called generalized loops. In this paper, we derive a loop series expansion of binary pairwise Markov random fields with 'propagation diagrams', which describes rules as to how 'first messages' and 'secondary messages' propagate. Our approach allows us to express the loop series in the form of a polynomial with coefficients positive integers. Using the propagation diagrams, we establish a new formula that shows a relation between the exact marginal probabilities and their Bethe approximations.
- PACS
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
- MSC
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62M40 Random fields; image analysis
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
- Subjects
- Dates
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Issue 4 (30 January 2009)
Received 8 August 2008, in final form 7 November 2008
Published 17 December 2008
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Loop series expansion with propagation diagrams
Yusuke Watanabe and Kenji Fukumizu 2009 J. Phys. A: Math. Theor. 42 045001
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