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Loop calculus in statistical physics and information science.

Michael Chertkov1, Vladimir Y Chernyak

  • 1Theoretical Division and Center for Nonlinear Studies, LANL, Los Alamos, New Mexico 87545, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 16, 2006
PubMed
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We present an exact formula for the partition function in statistical models, revealing loop contributions beyond the Bethe-Peierls (belief propagation) approximation. This method explains the success of belief propagation by uncovering a small parameter in the series.

Area of Science:

  • Statistical Physics
  • Information Theory
  • Computational Mathematics

Background:

  • The Bethe-Peierls (belief propagation) approximation is widely used in statistical physics and information science.
  • Its success is often empirical, lacking a clear theoretical explanation.
  • Exact calculation of partition functions in discrete models can be computationally challenging.

Purpose of the Study:

  • To derive an exact expression for the partition function of discrete statistical models.
  • To theoretically explain the success of the Bethe-Peierls (belief propagation) approximation.
  • To introduce a framework for calculating loop corrections to the belief propagation approximation.

Main Methods:

  • Developed an exact expression for the partition function as a finite series.

Related Experiment Videos

  • Identified the leading term as the Bethe-Peierls (belief propagation) contribution.
  • Quantified the remaining terms as loop contributions on the factor graph, calculated using the Bethe-Peierls solution.

    • Main Results:

    • The partition function is expressed as a finite series including Bethe-Peierls and loop contributions.
    • A small parameter is identified, explaining the accuracy of the Bethe-Peierls approximation in many cases.
    • The loop calculus provides a systematic way to improve upon the Bethe-Peierls approximation.

    Conclusions:

    • The derived series offers a more complete understanding of statistical models beyond the Bethe-Peierls approximation.
    • The loop calculus framework has potential applications in statistical physics and information science.
    • This work provides theoretical justification for the widespread use and success of belief propagation methods.