1. Graduate School of Information Sciences Tohoku University, Japan
Mathematical Structures of Loopy Belief Propagation and Cluster Variation Method
Kazuyuki Tanaka
Graduate School of Information Sciences
Tohoku University, Japan
16 September, 2008
IW-SMI2008

2. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Concluding Remarks
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3. Abstract
We review of conventional generalized loopy belief propagation based on cluster variation method.
Some interpretations and extensions of the adaptive TAP method are given by using the cluster variation method.
Conventional formalisms and extensions of the quantum belief propagation are also summarized.
16 September, 2008
IW-SMI2008

4. Belief Propagation
Belief Propagation (BP) has been proposed in order to achieve probabilistic inference systems (Pearl, 1988).
It has been suggested that BP has a closed relationship to mean field methods in the statistical mechanics (Kabashima and Saad 1998).
Generalizations of the BP have been proposed based on the cluster variation method which is one of the advanced mean field methods in the statistical mechanics (Yedidia, Freeman and Weiss, 2000).
16 September, 2008
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5. Cluster Variation Method
Cluster Variation Method for Classical System
R. Kikuchi: A theory of cooperative phenomena, Phys. Rev., 81 (1951).
T. Morita: Cluster variation method of cooperative phenomena and its generalization I, J. Phys. Soc. Jpn, 12 (1957).
Cluster Variation Method for Quantum System
T. Morita: Cluster variation method of cooperative phenomena and its generalization II, Quantum Statistics, J. Phys. Soc. Jpn, 12 (1957).
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6. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Summary
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7. V: Set of all the nodes E: Set of all the edges State Variable
Classical Spin System
Probability Distribution in the present talk
V: Set of all the nodes
E: Set of all the edges
State Variable 2 3 1 4 5 7 6
State Vector
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8. Variational Principle of Free Energy Minimum
Gibbs distribution
satisfies the minimization of free energy
Free Energy for
Trial Probability
Q(x)
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9. Tractable Probabilistic Model
In the case of J=0 for all the edges ij
Factorizable Form 2 3 1 4 5 7 6
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10. Conventional Loopy Belief Propagation- - i i j i j
Example 1 2 3 1 2 - 1 2 3 = + + + 1 + 2 3 - 2 - 3 - + 2 - 3 3
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11. Conventional Loopy Belief Propagation
In the cluster variation method, the free energy is approximately expressed in terms of the free energies of all the edges and nodes.
Bethe Free Energy i j - i j i
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12. Conventional Loopy Belief Propagation
Marginal probabilities of edges and nodes are determined so as to minimize the Bethe free energy under some constraint conditions.
Lagrange Multipliers to ensure the constraints
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13. Conventional Loopy Belief Propagation
Extremum
Condition
Approximate Forms of Marginals
Message Passing Rules j i
Reducibility Conditions
Output
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14. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Summary
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15. Tractable Probabilistic Model
Gaussian graphical model
Averages, variances and covariances are exactly calculated by using multidimensional Gaussian integral formulas.
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16. Tractable Probabilistic Model
Gaussian graphical model
For the given probability distribution P(x), we introduce the Gaussian graphical model so as to satisfy the following relations:
Marginal Probability
Approximate form of Pi(xi) should be determined by using the cluster variation method.
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17. Interpretation of Adaptive TAP
Adaptive TAP Free energy can be given in terms of the free energies of the Gaussian graphical model and of all the nodes in the present binary spin system as follows:
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18. Interpretation of Adaptive TAP
Marginal probabilities of edges and nodes are determined so as to minimize the adaptive TAP free energy under some constraint conditions.
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19. Interpretation of Adaptive TAP
By taking the first deviations of the adaptive TAP free energy, we derive the approximate form of marginal probabilities as follows:
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20. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Summary
16 September, 2008
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21. Extension of Adaptive TAP
In the similar arguments, we can construct the following approximate free energy: i i i j i j i i j j
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22. Extension of Adaptive TAP
Approximate forms of marginal probabilities are derived by considering the following Lagrange multipliers for the constraint conditions.
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23. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Summary
16 September, 2008
IW-SMI2008

24. Density Matrix and Reduced Density Matrixj i
If we extend these procedure by the conventional Gaussian mixture model
and Potts model to a quantum-statistical-mechanical model,
we have to consider a quantum loopy belief propagation
for not probability distribution but density matrix.
The reducibility conditions from two-body marginal to one-body marginal
is similar to the conventional BP.
Reducibility
Condition
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25. Reduced Density Matrix and Effective Fields
All effective field
are matrices j
By considering a Bethe free energy for given density matrix and by taking the first variation,
we derive the approximate form of marginal density matrices. i
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26. Conventional Belief Propagation for Quantum Statistical Systems
Propagating Rule of Effective Fields j i
By substituting these approximate marginal density matrices to the reducibility conditions,
we can derive the message passing rules for density matrix.
Output
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27. Quantum Heisenberg Model
2Nx2N Matrix
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28. Simple Cubic Lattice Approximation Curie Point Bethe 0.9102 Kikuchi
High Temperature Expansion
+ Pade Approximation: (J>0)
Approximation
Curie Point
Bethe
0.9102
Kikuchi
0.9108
Approximation
Neel Point
Anti Neel Point
Bethe
1.0153
0.4298
Kikuchi
1.0232
0.2056
Ground state is always paramagnetic state in both Bethe and Kikuchi approximations.
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29. Random Heisenberg Magnets by using Bethe Approximation
S. Katsura and K. Shimada:
Phys. Stat. Sol. 97 (1980). 1 F
Para
Square Lattice 1 F
Para AF
Simple Cubic Lattice
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30. Quantum Adaptive TAP approach
Bose
Lattice Gas
Trial Reduced Density Matrices
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31. Quantum Heisenberg Model
Holstain-Primakof Transformation
Tractable Part
Pair Approximation of QCVM
T. Morita: A Bose Lattice Gas Equivalent to Heisenberg Model and Its QCVM Study,
Journal of the Physical Society of Japan, Vol.64, No.4, pp , 1995.
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32. Contents Introduction Conventional Belief Propagation
Interpretation of Adaptive TAP method by Cluster Variation Method
Extension of Adaptive TAP method
Quantum Belief Propagation
Summary
16 September, 2008
IW-SMI2008

33. Summary
We reviewed the conventional loopy belief propagation based on cluster variation method.
Some interpretations and extensions of the adaptive TAP method have been given by using the cluster variation method.
Conventional formalisms and extensions of the quantum belief propagation have been also summarized.
16 September, 2008
IW-SMI2008