Download presentation
Presentation is loading. Please wait.
Published byKenneth Dennis Modified over 9 years ago
1
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 1 Physical Fluctuomatics Applied Stochastic Process 9th Belief propagation Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/
2
Physical Fuctuomatics (Tohoku University) 2 Textbooks Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese), Chapter 8. Kazuyuki Tanaka: Mathematics of Statistical Inference by Bayesian Network, Corona Publishing Co., Ltd., October 2009 (in Japanese), Chapters 6-9.
3
Physical Fuctuomatics (Tohoku University) 3 What is an important point in computational complexity? How should we treat the calculation of the summation over 2 N configuration? N fold loops If it takes 1 second in the case of N=10, it takes 17 minutes in N=20, 12 days in N=30 and 34 years in N=40. Markov Chain Monte Carlo Method Belief Propagation Method This Talk Previous Talk
4
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 4 Probabilistic Model and Belief Propagation Probabilistic Information Processing Probabilistic Models Bayes Formulas Belief Propagation J. Pearl: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, 1988). C. Berrou and A. Glavieux: Near optimum error correcting coding and decoding: Turbo-codes, IEEE Trans. Comm., 44 (1996). Bayesian Networks
5
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 5 Mathematical Formulation of Belief Propagation Similarity of Mathematical Structures between Mean Field Theory and Bepief Propagation Y. Kabashima and D. Saad, Belief propagation vs. TAP for decoding corrupted messages, Europhys. Lett. 44 (1998). M. Opper and D. Saad (eds), Advanced Mean Field Methods ---Theory and Practice (MIT Press, 2001). Generalization of Belief Propagation S. Yedidia, W. T. Freeman and Y. Weiss: Constructing free-energy approximations and generalized belief propagation algorithms, IEEE Transactions on Information Theory, 51 (2005). Interpretations of Belief Propagation based on Information Geometry S. Ikeda, T. Tanaka and S. Amari: Stochastic reasoning, free energy, and information geometry, Neural Computation, 16 (2004).
6
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 6 Generalized Extensions of Belief Propagation based on Cluster Variation Method Generalized Belief Propagation J. S. Yedidia, W. T. Freeman and Y. Weiss: Constructing free- energy approximations and generalized belief propagation algorithms, IEEE Transactions on Information Theory, 51 (2005). Key Technology is the cluster variation method in Statistical Physics 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).
7
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 7 Belief Propagation in Statistical Physics In graphical models with tree graphical structures, Bethe approximation is equivalent to Transfer Matrix Method in Statistical Physics and give us exact results for computations of statistical quantities. In Graphical Models with Cycles, Belief Propagation is equivalent to Bethe approximation or Cluster Variation Method. Bethe Approximation Trandfer Matrix Method (Tree Structures) Belief Propagation Cluster Variation Method (Kikuchi Approximation) Generalized Belief Propagation
8
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 8 Applications of Belief Propagations Image Processing K. Tanaka: Statistical-mechanical approach to image processing (Topical Review), J. Phys. A, 35 (2002). A. S. Willsky: Multiresolution Markov Models for Signal and Image Processing, Proceedings of IEEE, 90 (2002). Low Density Parity Check Codes Y. Kabashima and D. Saad: Statistical mechanics of low-density parity-check codes (Topical Review), J. Phys. A, 37 (2004). S. Ikeda, T. Tanaka and S. Amari: Information geometry of turbo and low-density parity-check codes, IEEE Transactions on Information Theory, 50 (2004). CDMA Multiuser Detection Algorithm Y. Kabashima: A CDMA multiuser detection algorithm on the basis of belief propagation, J. Phys. A, 36 (2003). T. Tanaka and M. Okada: Approximate Belief propagation, density evolution, and statistical neurodynamics for CDMA multiuser detection, IEEE Transactions on Information Theory, 51 (2005). Satisfability Problem O. C. Martin, R. Monasson, R. Zecchina: Statistical mechanics methods and phase transitions in optimization problems, Theoretical Computer Science, 265 (2001). M. Mezard, G. Parisi, R. Zecchina: Analytic and algorithmic solution of random satisfability problems, Science, 297 (2002).
9
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 9 Strategy of Approximate Algorithm in Probabilistic Information Processing It is very hard to compute marginal probabilities exactly except some tractable cases. What is the tractable cases in which marginal probabilities can be computed exactly? Is it possible to use such algorithms for tractable cases to compute marginal probabilities in intractable cases?
10
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 10 Graphical Representations of Tractable Probabilistic Models ABCDE ABCDE BCD X XX = =
11
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 11 Graphical Representations of Tractable Probabilistic Models ABCDE AB BCDE X
12
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 12 Graphical Representations of Tractable Probabilistic Models ABCDE AB BCDE X AB BCDE
13
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 13 Graphical Representations of Tractable Probabilistic Models ABCDE AB BCDE X AB BCDE A B
14
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 14 Graphical Representations of Tractable Probabilistic Models ABCDE AB BCDE X AB BCDE A B A BCDE
15
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 15 Graphical Representations of Tractable Probabilistic Models A BCDE
16
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 16 Graphical Representations of Tractable Probabilistic Models CDE X A BCDE A BC
17
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 17 Graphical Representations of Tractable Probabilistic Models CDE X CDE A BCDE A BC A BC X
18
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 18 Graphical Representations of Tractable Probabilistic Models CDE X CDE B C A BCDE A BC A BC X
19
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 19 Graphical Representations of Tractable Probabilistic Models CDE X CDE B C A BCDE A BC A BC B CDE X
20
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 20 Graphical Representations of Tractable Probabilistic Models ABCDE
21
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 21 Graphical Representations of Tractable Probabilistic Models A BCDE ABCDE
22
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 22 Graphical Representations of Tractable Probabilistic Models A BCDE B CDE ABCDE
23
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 23 Graphical Representations of Tractable Probabilistic Models A BCDE B CDE ABCDE C DE
24
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 24 Graphical Representations of Tractable Probabilistic Models A BCDE B CDE ABCDE C DE D E
25
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 25 Graphical Representations of Tractable Probabilistic Models ABCEE CCD X XX = = F E X A B E C D F
26
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 26 Graphical Representations of Tractable Probabilistic Models A B E C D F
27
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 27 Graphical Representations of Tractable Probabilistic Models A B E C D F A B E C D F A C A C
28
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 28 Graphical Representations of Tractable Probabilistic Models A B E C D F A B E C D F A B E C D F B C B C
29
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 29 Graphical Representations of Tractable Probabilistic Models A B E C D F A B E C D F A B E C D F E C D F
30
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 30 Graphical Representations of Tractable Probabilistic Models A B E C D F A B E C D F A B E C D F E C D F E C D F
31
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 31 Graphical Representations of Tractable Probabilistic Models A B E C D F A B E C D F A B E C D F E C D F E C D F E F
32
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 32 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F
33
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 33 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F A B E C E D E F =
34
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 34 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F A B E C E D E F = = E C E D E F
35
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 35 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F A B E C E D E F = = E C E D E F E C D F =
36
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 36 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F A C E D E F = = E C E D E F = E C B C A C B C A B E C D F
37
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 37 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C D F = E C D F = E C A B E C E C D F A B E C D F Recursion Formulas for Messages
38
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 38 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages A B E C E C E C D F E F E C D F E D E C D F E C A B E C A C A B E C B C A C A C B C B C E D E DE F E F A B E C D F Step 1 Step 2 Step 3
39
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 39 Graphical Representations of Tractable Probabilistic Models Graphical Representation of Marginal Probability in terms of Messages Step 1 Step 2 Step 3 A B E C D F A B E C D F A B E C D F A B E C D F = E C D F = B C = A B E C =
40
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 40 Belief Propagation 1 2 3 4 5 6 Probabilistic Models with no Cycles
41
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 41 Belief Propagation 1 2 4 1 1 3 2 6 5 2 1 2 3 4 5 6 Probabilistic Model on Tree Graph
42
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 42 Probabilistic Model on Tree Graph 1 2 3 4 5 6
43
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 43 Belief Propagation 1 2 3 4 5 6 Probabilistic Model on Tree Graph
44
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 44 Belief Propagation for Probabilistic Model on Tree Graph No Cycles!!
45
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 45 Belief Propagation for Probabilistic Model on Square Grid Graph E : Set of all the links
46
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 46 Belief Propagation for Probabilistic Model on Square Grid Graph
47
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 47 Belief Propagation for Probabilistic Model on Square Grid Graph
48
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 48 Marginal Probability
49
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 49 Marginal Probability 2
50
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 50 Marginal Probability 2 2
51
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 51 Marginal Probability
52
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 52 Marginal Probability 1 2
53
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 53 Marginal Probability 1 2 1 2
54
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 54 Belief Propagation for Probabilistic Model on Square Grid Graph
55
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 55 Belief Propagation for Probabilistic Model on Square Grid Graph 1 4 53 2 6 8 7
56
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 56 Belief Propagation for Probabilistic Model on Square Grid Graph 2 17 6 8 1 4 53 2 6 8 7
57
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 57 Belief Propagation for Probabilistic Model on Square Grid Graph 2 17 6 8 Message Update Rule 1 4 53 2 6 8 7 3 2 1 5 4 1 2
58
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 58 Belief Propagation for Probabilistic Model on Square Grid Graph 2 1 3 4 5 3 2 1 5 4 1 2 Fixed Point Equations for Messages
59
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 59 Fixed Point Equation and Iterative Method Fixed Point Equation
60
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 60 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
61
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 61 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
62
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 62 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
63
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 63 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
64
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 64 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
65
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 65 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method
66
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 66 Belief Propagation for Probabilistic Model on Square Grid Graph Four Kinds of Update Rule with Three Inputs and One Output
67
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 67 Interpretation of Belief Propagation based on Information Theory Free Energy Kullback-Leibler Divergence
68
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 68 Interpretation of Belief Propagation based on Information Theory Free Energy KL Divergence
69
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 69 Interpretation of Belief Propagation based on Information Theory Bethe Free Energy Free Energy KL Divergence
70
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 70 Interpretation of Belief Propagation based on Information Theory
71
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 71 Interpretation of Belief Propagation based on Information Theory Lagrange Multipliers to ensure the constraints
72
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 72 Interpretation of Belief Propagation based on Information Theory Extremum Condition
73
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 73 Interpretation of Belief Propagation based on Information Theory 1 42 5 3 1 4 53 2 6 8 7 Extremum Condition
74
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 74 Interpretation of Belief Propagation based on Information Theory 1 42 5 3 1 4 53 2 6 8 7 Message Update Rule
75
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 75 Interpretation of Belief Propagation based on Information Theory 1 3 42 5 1 4 5 3 2 6 8 7 1 42 53 = Message Passing Rule of Belief Propagation
76
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 76 Summary Belief Propagation Transfer Matrix Method Bethe Approximation and Cluster Variation Method Iterative Algorithm Future Talks 10th Probabilistic image processing by means of physical models 11th Bayesian network and belief propagation in statistical inference
77
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 77 Practice 9-1 We consider a probability distribution P(a,b,c,d,x,y) defined by Show that marginal Probability is expressed by
78
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 78 Practice 9-2 By substituting to, derive the following equation.
79
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 79 Practice 9-3 Make a program to solve the nonlinear equation x=tanh(Cx) for various values of C. Obtain the solutions for C=0.5, 1.0, 2.0 numerically. Discuss how the iterative procedures converge to the fixed points of the equations in the cases of C=0.5, 1.0, 2.0 by drawing the graphs of y=tanh(Cx) and y=x.
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.