1. Introduction to Bayesian Networks A Tutorial for the 66th MORS Symposium 23 - 25 June 1998 Naval Postgraduate School Monterey, California Dennis M. Buede Joseph A. Tatman Terry A. Bresnick
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14. Sample Factored Joint Distribution X 1 X 3 X 2 X 5 X 4 X 6 p(x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = p(x 6 | x 5 ) p(x 5 | x 3 , x 2 ) p(x 4 | x 2 , x 1 ) p(x 3 | x 1 ) p(x 2 | x 1 ) p(x 1 )
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16. Arc Reversal - Bayes Rule p(x 1 , x 2 , x 3 ) = p(x 3 | x 1 ) p(x 2 | x 1 ) p(x 1 ) p(x 1 , x 2 , x 3 ) = p(x 3 | x 2 , x 1 ) p(x 2 ) p( x 1 ) p(x 1 , x 2 , x 3 ) = p(x 3 | x 1 ) p(x 2 , x 1 ) = p(x 3 | x 1 ) p(x 1 | x 2 ) p( x 2 ) p(x 1 , x 2 , x 3 ) = p(x 3 , x 2 | x 1 ) p( x 1 ) = p(x 2 | x 3 , x 1 ) p(x 3 | x 1 ) p( x 1 ) is equivalent to is equivalent to X 1 X 3 X 2 X 1 X 3 X 2 X 1 X 3 X 2 X 1 X 3 X 2
17. Inference Using Bayes Theorem Tuber- culosis Lung Cancer Tuberculosis or Cancer Dyspnea Bronchitis Lung Cancer Tuberculosis or Cancer Dyspnea Bronchitis Lung Cancer Tuberculosis or Cancer Dyspnea Lung Cancer Dyspnea Lung Cancer Dyspnea The general probabilistic inference problem is to find the probability of an event given a set of evidence This can be done in Bayesian nets with sequential applications of Bayes Theorem
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20. Introduction to Bayesian Networks A Tutorial for the 66th MORS Symposium 23 - 25 June 1998 Naval Postgraduate School Monterey, California Dennis M. Buede Joseph A. Tatman Terry A. Bresnick
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24. Singly Connected Networks (or Polytrees) Definition : A directed acyclic graph (DAG) in which only one semipath (sequence of connected nodes ignoring direction of the arcs) exists between any two nodes. Do not satisfy definition Polytree structure satisfies definition Multiple parents and/or multiple children
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27. An Example: Simple Chain p(Paris) = 0.9 p(Med.) = 0.1 M TO|SM = M AA |TO = Ch Di Pa Me No Ce So Ch Di Paris Med. Chalons Dijon North Central South Strategic Mission Tactical Objective Avenue of Approach .8 .2 .1 .9 [ ] .5 .4 .1 .1 .3 .6 [ ]
28. Sample Chain - Setup (1) Set all lambdas to be a vector of 1’s; Bel(SM) = (SM) (SM) (SM) Bel(SM) (SM) Paris 0.9 0.9 1.0 Med. 0.1 0.1 1.0 (2) (TO) = (SM) M TO|SM ; Bel(TO) = (TO) (TO) (TO) Bel(TO) (TO) Chalons 0.73 0.73 1.0 Dijon 0.27 0.27 1.0 (3) (AA) = (TO) M AA|TO ; Bel(AA) = (AA) (AA) (AA) Bel(AA) (AA) North 0.39 0.40 1.0 Central 0.35 0.36 1.0 South 0.24 0.24 1.0 M AA|TO = M TO|SM = Strategic Mission Tactical Objective Avenue of Approach .8 .2 .1 .9 [ ] .5 .4 .1 .1 .3 .6 [ ]
30. Sample Chain - 2nd Propagation t TR T 0 5 ( ) . 1 .6 t = 0 (lR) = .8 .2 (SM) Bel(SM) (SM) Paris 0.8 0.8 1.0 Med. 0.2 0.2 1.0 t = 2 (TO) = (SM) M TO|SM (TO) Bel(TO) (TO) Chalons 0.66 0.66 0.71 Dijon 0.34 0.34 0.71 t = 2 (TO) = M AA|TO (SM) (AA) Bel(AA) (AA) North 0.39 0.3 0.5 Central 0.35 0.5 1.0 South 0.24 0.2 0.6 Intel. Rpt. Troop Rpt. Strategic Mission Tactical Objective Avenue of Approach
31. Sample Chain - 3rd Propagation (SM) Bel(SM) (SM) Paris 0.8 0.8 0.71 Med. 0.2 0.2 0.71 t = 3 (SM) = M TO|SM (TO) (TO) Bel(TO) (TO) Chalons 0.66 0.66 0.71 Dijon 0.34 0.34 0.71 t = 3 (AA) = (TO) M AA|TO (AA) Bel(AA) (AA) North 0.36 0.25 0.5 Central 0.37 0.52 1.0 South 0.27 0.23 0.6 Intel. Rpt. Troop Rpt. Strategic Mission Tactical Objective Avenue of Approach
32. Internal Structure of a Single Node Processor Processor for Node X Message to Parent U Message from Parent U M X|U M X|U k k (X) BEL = Message to Children of X Message from Children of X BEL(X) 1 (X) BEL(X) N (X) ... ... (X) (X) X (U) 1 (X) N (X) X (U) N (X) 1 (X)
43. Introduction to Bayesian Networks A Tutorial for the 66th MORS Symposium 23 - 25 June 1998 Naval Postgraduate School Monterey, California Dennis M. Buede, dbuede@gmu.edu Joseph A. Tatman, jatatman@aol.com Terry A. Bresnick, bresnick@ix.netcom.com http://www.gmu.edu - Depts (Info Tech & Eng) - Sys. Eng. - Buede
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45. Building BN Structures Bayesian Network Bayesian Network Bayesian Network Problem Domain Problem Domain Problem Domain Expert Knowledge Expert Knowledge Training Data Training Data Probability Elicitor Learning Algorithm Learning Algorithm
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47. Beta Distribution 1) n(n m/n) m(1 variance n m mean x) (1 x m) (n (m) (n) n) m, | (x p 1 m n 1 m Beta
48. Multivariate Dirichlet Distribution 1) m ( m ) m / m (1 m state i the of variance m m state i the of mean ...x x x ) (m )... (m ) (m ) m ( ) m ,..., m , m | (x p N 1 i i N 1 i i N 1 i i i i th N 1 i i i th 1 m 1 - m 1 m N 2 1 N 1 i i N 2 1 Dirichlet N 2 1
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54. Enemy Intent Trafficability of S. AA Trafficability of C. AA Trafficability of N. AA Troops @ NAI 1 Troops @ NAI 2 AA - Avenue of Approach NAI - Named Area of Interest Intelligence Reports Observations on Troop Movements Weather Forecast & Feasibility Analysis Strategic Mission Weather Tactical Objective Enemy’s Intelligence on Friendly Forces Avenue of Approach Deception Plan
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