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Kazuyuki Tanaka Graduate School of Information Sciences

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Presentation on theme: "Kazuyuki Tanaka Graduate School of Information Sciences"— Presentation transcript:

1 Linear Response Formula and Generalized Belief Propagation for Probabilistic Inference
Kazuyuki Tanaka Graduate School of Information Sciences Tohoku University, Sendai , Japan Reference K. Tanaka: Probabilistic Inference by means of Cluster Variation Method and Linear Response Theory, IEICE Trans. on Inf. & Syst., vol.E86-D, no.7, pp , 2003. 28 November, 2005 CIMCA2005, Vienna

2 Probabilistic Inference and Belief Propagation
Introduction Probabilistic Inference and Belief Propagation Bayes Formula Probabilistic Inference Probabilistic Model Marginal Probability Probabilistic models on networks with some loops =>Good Approximation Probabilistic models on tree-like networks with no loops =>Exact Results Belief Propagation Generalization 28 November, 2005 CIMCA2005, Vienna

3 Contents Introduction Probabilistic Inference
Generalized Belief Propagation Linear Response Concluding Remarks 28 November, 2005 CIMCA2005, Vienna

4 Probabilistic Inference
28 November, 2005 CIMCA2005, Vienna

5 Probabilistic Inference
Probabilistic Inference and Probabilistic Model 28 November, 2005 CIMCA2005, Vienna

6 Probabilistic Inference
Probabilistic Inference and Probabilistic Model 28 November, 2005 CIMCA2005, Vienna

7 Probabilistic Inference
Probabilistic Model and Beliefs Statistics: Marginal Probability Statistical Mechanics: One-body distribution Probabilistic Inference: Belief 28 November, 2005 CIMCA2005, Vienna

8 Contents Introduction Probabilistic Inference
Generalized Belief Propagation Linear Response Concluding Remarks 28 November, 2005 CIMCA2005, Vienna

9 Tractable Model Probabilistic models on tree graph or cactus tree graph are tractable. Factorizable Probabilistic models with many loops are not tractable. Not Factorizable 28 November, 2005 CIMCA2005, Vienna

10 Generalized Belief Propagation
Cactus Tree Graph Cactus Tree Graph By solving the system of linear equations for deviation of average and by applying the linear response formula to the solutions, we obtain the correlation function as the inverse of matrix G. Original Graph 28 November, 2005 CIMCA2005, Vienna

11 Generalized Belief Propagation
Message Passing Algorithm 28 November, 2005 CIMCA2005, Vienna

12 Generalized Loopy Belief Propagation
Expression of Marginal Probability in terms of Messages 28 November, 2005 CIMCA2005, Vienna

13 Interpretation of Generalized Belief Propagation
28 November, 2005 CIMCA2005, Vienna

14 Fixed Point Equation and Iterative Method
28 November, 2005 CIMCA2005, Vienna

15 Numerical Experiments
GBP Exact 28 November, 2005 CIMCA2005, Vienna

16 Numerical Experiments
28 November, 2005 CIMCA2005, Vienna

17 Contents Introduction Probabilistic Inference
Generalized Belief Propagation Linear Response Concluding Remarks 28 November, 2005 CIMCA2005, Vienna

18 Linear Response 28 November, 2005 CIMCA2005, Vienna

19 Final Result 28 November, 2005 CIMCA2005, Vienna
This is a final result in the present talk. The correlation function is obtained by calculating the inverse matrix of G. The matrix G is constructed from the short range correlation. B is a set of basic clusters in the cluster variation method. Every basic cluster must not be a subcluster of another element in the set of basic clusters. C is a set of basic clusters and its subclusters. Mu is a Mobius function. The cluster variation method is specified by defining the set of basic clusters. The matrix A is constructed from the belief of probabilistic model P(x) and is calculated by employing a generalized belief propagation. 28 November, 2005 CIMCA2005, Vienna

20 Numerical Experiments
GBP+LR Exact 28 November, 2005 CIMCA2005, Vienna

21 Contents Introduction Probabilistic Inference
Generalized Belief Propagation Linear Response Concluding Remarks 28 November, 2005 CIMCA2005, Vienna

22 Concluding Remarks Future Problems
Generalized Belief Propagation + Linear Response Theory Future Problems Statistical Learning of Conditional Probability.  Maximum Likelihood Framework EM algorithm In this talk, we give the general formula for correlation by combining the cluster variation method with the linear response theory. The framework is one of extensions of Kappen et al 1998 and Tanaka 1998. As the other previous work, we have already given a general CVM approximate formula for Fourier transform of correlation of probabilistic model on any regular lattice. 28 November, 2005 CIMCA2005, Vienna

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