Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models

Abstract

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising--like) models in equilibrium statistical mechanics, improving on the mean--field approximation and the Bethe--Peierls approximation, which can be regarded as the lowest level of the CVM. In recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models. The foundations of the CVM are briefly reviewed, and the relations with similar techniques are discussed. The main properties of the method are considered, with emphasis on its exactness for particular models and on its asymptotic properties. The problem of the minimization of the variational free energy, which arises in the CVM, is also addressed, and recent results about both provably convergent and message-passing algorithms are discussed.

36 pages, 17 figures

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Cited by in corpus (68)

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• Cluster structure of EU-15 countries derived from the correlation matrix analysis of macroeconomic index fluctuations
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• Partition function loop series for a general graphical model: free energy corrections and message-passing equations
• Replica Cluster Variational Method
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• Rigorous results on the local equilibrium kinetics of a protein folding model
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• Variational approximations for stationary states of Ising-like models
• On Cavity Approximations for Graphical Models
• A very fast inference algorithm for finite-dimensional spin glasses: Belief Propagation on the dual lattice
• A simple analytical description of the non-stationary dynamics in Ising spin systems
• Message passing and Monte Carlo algorithms: connecting fixed points with metastable states
• Multistability in an unusual phase diagram induced by the competition between antiferromagnetic-like short-range and ferromagnetic-like long-range interactions
• Statistical Analysis of Loopy Belief Propagation in Random Fields
• Cluster approximations for the TASEP: stationary state and dynamical transition
• The Stochastic complexity of spin models: Are pairwise models really simple?
• Entanglement Mean Field Theory and the Curie-Weiss Law
• Survey propagation at finite temperature: application to a Sourlas code as a toy model
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• Mean-field phase diagram and spin glass phase of the dipolar Kagome Ising antiferromagnet
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• Translation invariant extensions of finite volume measures
• Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models
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• Simplifying Generalized Belief Propagation on Redundant Region Graphs
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• Bethe free-energy approximations for disordered quantum systems
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• Bayesian image segmentations by Potts prior and loopy belief propagation
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• Statistical physics of loopy interactions: Independent-loop approximation and beyond
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• Correcting beliefs in the mean-field and Bethe approximations using linear response
• Multiparticle correlation expansion of relative entropy in lattice systems
• Loop-corrected belief propagation for lattice spin models
• Empirical Bayes Method for Boltzmann Machines
• On one-step replica symmetry breaking in the Edwards-Anderson spin glass model
• Adaptive Cluster Expansion for Ising spin models
• Finite-size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs
• Data-Driven Optimal Closures for Mean-Cluster Models: Beyond the Classical Pair Approximation
• Inefficiency of the block approximation in diploid Probabilistic Cellular Automata
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