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Harnessing the Bethe Free Energy

Authors Victor Bapst, Amin Coja-Oghlan

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Victor Bapst
Amin Coja-Oghlan

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Victor Bapst and Amin Coja-Oghlan. Harnessing the Bethe Free Energy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 467-480, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)


Gibbs measures induced by random factor graphs play a prominent role in computer science, combinatorics and physics. A key problem is to calculate the typical value of the partition function. According to the "replica symmetric cavity method", a heuristic that rests on non-rigorous considerations from statistical mechanics, in many cases this problem can be tackled by way of maximising a functional called the "Bethe free energy". In this paper we prove that the Bethe free energy upper-bounds the partition function in a broad class of models. Additionally, we provide a sufficient condition for this upper bound to be tight.
  • Belief Propagation
  • free energy
  • Gibbs measure
  • partition function


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