Physics Fluctuomatics (Tohoku University) 1 Physical Fluctuomatics 12th Bayesian network and belief propagation in statistical inference Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University

Physics Fluctuomatics (Tohoku University) 2 Textbooks Kazuyuki Tanaka: Mathematics of Statistical Inference by Bayesian Network, Corona Publishing Co., Ltd., October 2009 (in Japanese).

Physics Fluctuomatics (Tohoku University) 3 Joint Probability and Conditional Probability A B Conditional Probability of Event B =b when Event A=a has happened. Probability of Event A=a Joint Probability of Events A=a and B=b Fundamental Probabilistic Theory for Image Processing

Physics Fluctuomatics (Tohoku University) 4 Fundamental Probabilistic Theory for Image Processing Marginal Probability of Event B AB C D Marginalization

Physics Fluctuomatics (Tohoku University) 5 Fundamental Probabilistic Theory for Image Processing Causal Independence C AB C AB

Physics Fluctuomatics (Tohoku University) 6 Fundamental Probabilistic Theory for Image Processing Causal Independence D ABC D ABC

Physics Fluctuomatics (Tohoku University) 7 Fundamental Probabilistic Theory for Image Processing A B C Causal Independence

Physics Fluctuomatics (Tohoku University) 8 Fundamental Probabilistic Theory for Image Processing A B C A B C Directed GraphUndirected Graph

Physics Fluctuomatics (Tohoku University) 9 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 10 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 11 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 12 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 13 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 14 Simple Example of Bayesian Networks C SR W C SR W Directed Graph Undirected Graph

Physics Fluctuomatics (Tohoku University) 15 Belief Propagation for Bayesian Networks Belief propagation cannot give us exact computations in Bayesian networks on cycle graphs. Applications of belief propagation to Bayesian networks on cycle graphs provide us many powerful approximate computational models and practical algorithms for probabilistic information processing.

Physics Fluctuomatics (Tohoku University) 16 Simple Example of Bayesian Networks

Physics Fluctuomatics (Tohoku University) 17 Joint Probability of Probabilistic Model with Graphical Representation including Cycles Directed Graph Undirected Hypergraph

Physics Fluctuomatics (Tohoku University) 18 Marginal Probability Distributions

Physics Fluctuomatics (Tohoku University) 19 Approximate Representations of Marginal Probability Distributions in terms of Messages

Physics Fluctuomatics (Tohoku University) 20 Approximate Representations of Marginal Probability Distributions in terms of Messages

Physics Fluctuomatics (Tohoku University) 21 Basic Strategies of Belief Propagations in Probabilistic Model with Graphical Representation including Cycles Approximate Expressions of Marginal Probabilities

Physics Fluctuomatics (Tohoku University) 22 Simultaneous Fixed Pint Equations for Belief Propagations in Hypergraph Representations Belief Propagation Algorithm

Physics Fluctuomatics (Tohoku University) 23 Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method

Physics Fluctuomatics (Tohoku University) 24 Belief Propagation for Bayesian Networks Belief propagation can be applied to Bayesian networks also on hypergraphs as powerful approximate algorithms.

Physics Fluctuomatics (Tohoku University) 25 Numerical Experiments Belief Propagation Exact

Physics Fluctuomatics (Tohoku University) 26 Numerical Experiments Belief Propagation

Physics Fluctuomatics (Tohoku University) 27 Linear Response Theory

Physics Fluctuomatics (Tohoku University) 28 Numerical Experiments

Physics Fluctuomatics (Tohoku University) 29 Summary Bayesian Network for Probabilistic Inference Belief Propagation for Bayesian Networks

Physics Fluctuomatics (Tohoku University) 30 Practice 11-1 Compute the exact values of the marginal probability Pr{X i } for every nodes i(=1,2,…,8), numerically, in the Bayesian network defined by the joint probability distribution Pr{X 1,X 2,…,X 8 } as follows: Each conditional probability table and probability table is given in Figure 3.12 and Table 3.11 in Kazuyuki Tanaka: Mathematics of Statistical Inference by Bayesian Network, Corona Publishing Co., Ltd., October 2009.

Physics Fluctuomatics (Tohoku University) 31 Practice 11-2 Each conditional probability table and probability table is given in Figure 3.12 and Table 3.11 in Kazuyuki Tanaka: Mathematics of Statistical Inference by Bayesian Network, Corona Publishing Co., Ltd., October The algorithm has appeared explicitly in the above textbook. Make a program to compute the approximate values of the marginal probability Pr{X i } for every nodes i(=1,2,…,8) by using the belief propagation method in the Bayesian network defined by the joint probability distribution Pr{X 1,X 2,…,X 8 } as follows: