Bayesian Networks                Unit 5 Probabilistic              Graphical Models (PGM)                    Wang, Yuan-Ka...
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05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
05 probabilistic graphical models
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  1. 1. Bayesian Networks Unit 5 Probabilistic Graphical Models (PGM) Wang, Yuan-Kai, 王元凱 ykwang@mails.fju.edu.tw http://www.ykwang.tw Department of Electrical Engineering, Fu Jen Univ. 輔仁大學電機工程系 2006~2011 Reference this document as: Wang, Yuan-Kai, “Probabilistic Graphical Models," Lecture Notes of Wang, Yuan-Kai, Fu Jen University, Taiwan, 2011.Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  2. 2. Bayesian Networks Unit : Probabilistic Graphical Models p. 2 Goal of This Unit • Learn how to – Build graphical model (network model) by graph theory – Inference under uncertainty according to probability theory • Theory of Bayesian networks – Conditional independence – D-Separation – Basic algorithm: • Variable Elimination • Introduce some BN models – MRF, HMM, DBN, Naïve Bayes, … Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  3. 3. Bayesian Networks Unit : Probabilistic Graphical Models p. 3 Related Units • Background – Statistical inference – Graph theory • Next units – Exact inference algorithms – Approximate inference algorithms Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  4. 4. Bayesian Networks Unit : Probabilistic Graphical Models p. 4 References for Self-Study • Chapter 14, Artificial Intelligence-a modern approach, 2nd, by S. Russel & P. Norvig, Prentice Hall, 2003 • E. Charniak, Bayesian networks without tears, AI Magazine • T. A. Stephenson, An introduction to Bayesian network theory and usage, IDIAP research report, IDIAP-RR-00-03, 2000 • B. D’Ambrosio, Inference in Bayesian networks, AI Magazine, 1999 • M. I. Jordan & Y. Weiss, Probabilistic Inference in graphical models, Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  5. 5. Bayesian Networks Unit : Probabilistic Graphical Models p. 5 Contents 1. Representing Uncertain Knowledge .............. 18 2. Various PGM Models ..................................... 52 3. Conditional Independence …………………. 66 4. Inference .......................................................... 88 5. Applications on Computer Vision ................. 136 6. Summary ……………………………………. 146 7. References …………………………………… 152 Fu Jen University Fu Jen University Department of Electrical Engineering Department of Electrical Engineering Yuan-Kai Wang Copyright Wang, Yuan-Kai Copyright
  6. 6. Bayesian Networks Unit : Probabilistic Graphical Models p. 6 Example – Car Diagnosis Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  7. 7. Bayesian Networks Unit : Probabilistic Graphical Models p. 7 Examples on Computer Vision Hand Upper Head Torso Upper Hand Anthropological Forearm Size Forearm Size Arm Size Size Arm Size Size Measurements Size Sf St Size Sf Sh Sa Shd Sa Sh A Left Left Left Right Right Right Joints Neck Wrist Elbow Shoulder Shoulder Elbow Wrist J N Wl El Sl Sr Er Wr Left Left Left Head Torso Right Right Right Components Hand Forearm Upper Arm H T Upper Arm Forearm Hand C Hl Fl Ul Ur Fr Hl Observations Observations Oij O Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  8. 8. Bayesian Networks Unit : Probabilistic Graphical Models p. 8 Where do PGMs come from ? • Common problems in real life : – Complex, Uncertain Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  9. 9. Bayesian Networks Unit : Probabilistic Graphical Models p. 9 Graph + Probability • Graph has P(X,Y) – Node + Edge X Y • Two kinds of graph – Directed graph – Undirected graph P(X|Y) • Probability has X Y – Random variable  Node – Probability  Edge • Directed graph : conditional probability • Undirected graph: joint probability Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  10. 10. Bayesian Networks Unit : Probabilistic Graphical Models p. 10 Probabilistic Modeling of Problems (1/2) • Usually node has Burglary Earthquake two semantics P(A|B,E) – Cause Alarm – Effect P(J|A) P(M|A) • Causal relationships John Calls Mary Calls between nodes – Probabilistic – Conditional probability P(Y|X): P(Effect|Cause) – X and Y are not independent – Directed graph Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  11. 11. Bayesian Networks Unit : Probabilistic Graphical Models p. 11 Probabilistic Modeling of Problems (2/2) • If node has no causal semantics • But happens together Student X (influence each other) – Probabilistic P(X,Y) – Joint probability P(X,Y) Student Y – X and Y are not independent – Undirected graph Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  12. 12. Bayesian Networks Unit : Probabilistic Graphical Models p. 12 Cause/Effect  Class/Feature (1/2) • In pattern recognition Face Expression /computer vision P(f |class) P(f2|class) – Cause  class 1 – Effect  feature Eyebrow Mouth Motion Motion Facial expression image Base image (neutral expression) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  13. 13. Bayesian Networks Unit : Probabilistic Graphical Models p. 13 Cause/Effect  Class/Feature (2/2) • Face detection: Face 2-class classification object P(f1|class) P(f2|class) Skin Eye Color pattern Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  14. 14. Bayesian Networks Unit : Probabilistic Graphical Models p. 14 Cause/Effect  State/Observation P(xt|xt-1) xt+1 • In video analysis Real Real Real location x location x location (Tracking) t-1 t – Cause  State P(zt-1|xt-1) zt-1 zt – Effect  Observation Observed Observed location location Real position : xt Predicted position Detected position : zt x-t+1 P ( z t | xt ) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  15. 15. Bayesian Networks Unit : Probabilistic Graphical Models p. 15 What Are PGMs Good For? Medicine Speech Bio- Computer informatics recognition Vision Text Classification Computer Stock market troubleshooting  Classification: P(class|feature)  Prediction: P(Effect|Cause)=?  Diagnosis: P(Cause|Effect)=? Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  16. 16. Bayesian Networks Unit : Probabilistic Graphical Models p. 16 Three Problems in PGM Real Real Real • Representation location location location – Given a problem – Build its graphical model Observed location Observed location (Construction of Bayesian network) xt-1 x x • Inference Real location Real t Real t+1 location location – Given a set of evidences nodes z – Get probabilities of node(s) Observedzt-1 Observedt location location • Learning – Learn the CPT of a BN x z – Learn the graphical structure 1 3 P(xt|xt-1) of a BN 2 6 P(zt-1|xt-1) 3 9 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  17. 17. Bayesian Networks Unit : Probabilistic Graphical Models p. 17 Structure of Related Lecture Notes Problem Structure Data Learning PGM B E Representation Learning A Unit 5 : Probabilistic Graphical Units 16~ : MLE, EM Unit 9 : Hybrid BN J M Units 10~15: Naïve Bayes, MRF, HMM, DBN, Kalman filter P(B) Parameter P(E) Learning P(A|B,E) P(J|A) Query Inference P(M|A) Unit 6: Exact inference Unit 7: Approximate inference Unit 8: Temporal inference Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  18. 18. Bayesian Networks Unit : Probabilistic Graphical Models p. 18 1. Representing Uncertain Knowledge Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  19. 19. Bayesian Networks Unit : Probabilistic Graphical Models p. 19 Review (1/3) Bayes’ Theorem Likelihood Prior P (e | h ) P ( h ) P (h | e)  P (e) Probability Posterior of Evidence • Probability of an hypothesis, h, can be updated when evidence, e, has been obtained • It is usually not necessary to calculate P(e) directly •As it can be obtained by normalizing the posterior probabilities, P(hi | e) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  20. 20. Bayesian Networks Unit : Probabilistic Graphical Models p. 20 Review (2/3) Marginalization P ( X )   P ( X , h) hH Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  21. 21. Bayesian Networks Unit : Probabilistic Graphical Models p. 21 Review (3/3) • Full joint probability distribution FJD – Can answer any question P(X|E=e) P(X|E=e) = hP(X, e, h) – But become intractably large as the number of variables grows • Independence and conditional CPT independence among random variables – CPTs = FJD – But can greatly reduce the number of probabilities that need to specified Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  22. 22. Bayesian Networks Unit : Probabilistic Graphical Models p. 22 A Simple Bayesian Network • 1 FJD = 2 CPTs P(C) – P(Cavity, Toothache) 0.002 = P(Toothache|Cavity) * P(Cavity) Cavity – P(X,Y)=P(X|Y)P(Y) Causal Relationship =P(Y|X)P(X) • Graphical model Toothache can represent – Causal relationship T P(T|C) – Joint relationship T 0.70 F 0.01 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  23. 23. Bayesian Networks Unit : Probabilistic Graphical Models p. 23 A Burglary Network P(E) (random) The graph Burglary P(B) 0.002 variables Earthquake is directed 0.001 and acyclic B E P(A|B,E) T T 0.95 A P(J|A) Alarm T F 0.95 T 0.90 F T 0.29 F 0.05 F F 0.001 A P(M|A) John Calls Mary Calls T 0.70 F 0.01 A conditional probability distribution quantifies the effects of the parents on node Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  24. 24. Bayesian Networks Unit : Probabilistic Graphical Models p. 24 Compact Representation • If all n nodes have  k parents •  O(2k n) vs. O(2n) parameters P(E) Burglary P(B) 0.002 Earthquake 0.001 B E P(A|B,E) T T 0.95 A P(J|A) Alarm T F 0.95 T 0.90 F T 0.29 F 0.05 F F 0.001 A P(M|A) John Calls Mary Calls T 0.70 F 0.01 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  25. 25. Bayesian Networks Unit : Probabilistic Graphical Models p. 25 Formal Definition of a BN • Directed Acyclic Graph (DAG) –Nodes : Random variables –Edges : Direct influence between 2 variables • CPTs : Quantifies the dependency of two variables A B  P(X|Parent(X)) –Ex : P(C|A,B), P(D|A) • A priori distribution : D C for each node with no parents –Ex : P(A) and P(B) E Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  26. 26. Bayesian Networks Unit : Probabilistic Graphical Models p. 26 Conditional Independence in the Directed Acyclic Graph • Topology of network encodes dependency/independence • Weather is independent of the other variables • Cavity has direct influence on Tooth and Catch • Toothache and Catch are conditionally independent given Cavity Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  27. 27. Bayesian Networks Unit : Probabilistic Graphical Models p. 27 Conditional Probability Table (CPT) P(W) P(C) 0.001 0.02 C P(T|C) C P(Catch|C) T 0.90 T 0.70 F 0.05 F 0.01 P(Xi|Parent(Xi)) or P(Xi|Pa(Xi)) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  28. 28. Bayesian Networks Unit : Probabilistic Graphical Models p. 28 Causality and Bayesian Networks • Not every BN describes causal relationships between the variables • Consider the dependence between Lung Cancer, L, and the X-ray test, X. • A BN with causality L X P(x|l)=0.6 P(l)=0.001 P(x|l)=0.02 • Another BN represents the same distribution and independencies without causality P(l1|x1)=0.02915 L X P(x1)=0.02058 P(l1|x2)=0.00041 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  29. 29. Bayesian Networks Unit : Probabilistic Graphical Models p. 29 Example - Construction of BN (1/3) • I have a burglar alarm installed at home • I am at work • Neighbor John calls to say my alarm is ringing • But neighbor Mary doesnt call • Sometimes its set off by minor earthquakes • Is there a burglar? Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  30. 30. Bayesian Networks Unit : Probabilistic Graphical Models p. 30 Example - Construction of BN (2/3) • Step 1: Find Random variables – Burglar, Earthquake, Alarm, JohnCalls, MaryCalls • Step 2: Represent the causal relationships among random variables – A burglar can set the alarm off – An earthquake can set the alarm off – The alarm can cause Mary to call – The alarm can cause John to call • Step 3: Use network topology with probability Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  31. 31. Bayesian Networks Unit : Probabilistic Graphical Models p. 31 Example - Construction of BN (3/3) • 5 Boolean random variables + 5 CPTs P(E) Burglary Earthquake 0.002 P(B) 0.001 B E P(A|B,E) T T 0.95 Alarm T F 0.95 A P(J|A) F T 0.29 T 0.90 F F 0.001 F 0.05 A P(M|A) John Calls Mary Calls T 0.70 F 0.01 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  32. 32. Bayesian Networks Unit : Probabilistic Graphical Models p. 32 Marginalization in Bayesian Network P (b, e, a, j )   P(b, e, a, j , h)   P(b, e, a, j, M ) hH M  m , m P (b, e)   P(b, e, h)     P(b, e, A, J , M ) hH M  m , m A  a , a J  j ,  j Burglary Earthquake Alarm John Calls Mary Calls Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  33. 33. Bayesian Networks Unit : Probabilistic Graphical Models p. 33 Markov Chain, Conditional Probability, Independence, and Directed Edge • Markov chain P(X|L) L X – L and X are dependent, not independent • Markov chain  Has conditional prob.  Not independent  Has directed edge Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  34. 34. Bayesian Networks Unit : Probabilistic Graphical Models p. 34 Common Causes Smoking It is a DAG Bronchitis Lung Cancer • Markov condition: I(B, L | S), i.e. P(b | l, s) = P(b | s) • If SB and SL, and “Joe is a smoker” • “Joe has Bronchitis” v.s. “Joe has Lung Cancer” ? • “Joe has Bronchitis” will not give us any more information about the probability of “Joe has Lung Cancer” Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  35. 35. Bayesian Networks Unit : Probabilistic Graphical Models p. 35 Common Effects Burglary Earthquake Alarm It is a DAG • Markov condition: I(B, E), i.e. P(b | e) = P(b) • Burglary and Earthquake are independent of each other • However they are conditionally dependent given Alarm • If the alarm has gone off, news that there had been an earthquake would ‘explain away’ the idea that a burglary had taken place Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  36. 36. Bayesian Networks Unit : Probabilistic Graphical Models p. 36 Markov Assumption Ancestor • Markov chain v.s. independence Parent • Random variable X Y1 Y2 is independent of its non-descendents, X given its parents Pa(X) – Formally, I (X, NonDesc(X) | Pa(X)) Non-descendent Descendent Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  37. 37. Bayesian Networks Unit : Probabilistic Graphical Models p. 37 Markov Assumption Example • In this example: Earthquake Burglary – I ( E, B ) – I ( B, {E, R} ) – I ( R, {A, B, C} | E ) Radio Alarm – I ( A, R | B,E ) – I ( C, {B, E, R} | A) Call Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  38. 38. Bayesian Networks Unit : Probabilistic Graphical Models p. 38 Joint Probability Distribution • Note that our joint distribution with 5 variables can be represented as: P(e, b, r , a, c)  P(e) P(b | e) P(r | e, b) P(a | e, b, r ) P(c | e, b, r , a) But due to the Markov condition we have, for example, P (c | e, b, r , a )  P (c | a ) The joint probability distribution can be expressed as P(e, b, r , a, c)  P(e) P(b | e) P(r | e) P(a | e, b) P(c | a) • Ex: the probability that someone has a smoking history, lung cancer but not bronchitis, suffers from fatigue and tests positive in an X-ray test is P ( s, b, l , f , x )  0.2  0.75  0.003  0.5  0.6  0.000135 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  39. 39. Bayesian Networks Unit : Probabilistic Graphical Models p. 39 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  40. 40. Bayesian Networks Unit : Probabilistic Graphical Models p. 40 Representing the Joint Distribution • For a BN with nodes X1, X2, …, Xn n P( x1 , x2 ,..., xn )   P( xi | pa( xi )) FJD i 1 n CPTs • An enormous saving can be made regarding the number of values required for the joint distribution • For n binary variables •2n – 1 values are required for FJD • For a BN with n binary variables and •Each node has at most k parents •Less than 2kn values are required for CPTs Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  41. 41. Bayesian Networks Unit : Probabilistic Graphical Models p. 41 Exercise (1/2) S D G U E H P(s, d, g, u, e  A, h  C)  P(s)P(d)P(g | s)P(u | s, d)P(e  A| g, u)P(h  C | u) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  42. 42. Bayesian Networks Unit : Probabilistic Graphical Models p. 42 Exercise (2/2) • P(a, b, c, d, e) a = P(e | a, b, c, d) P(a, b, c, d) by the product rule b c = P(e | c) P(a, b, c, d) by cond. indep. assumption d e = P(e | c) P(d | a, b, c) P(a, b, c) = P(e | c) P(d | b, c) P(c | a, b) P(a, b) = P(e | c) P(d | b, c) P(c | a) P(b | a) P(a) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  43. 43. Bayesian Networks Unit : Probabilistic Graphical Models p. 43 Exercises • Facial Expression Recognition • Face Detection • Face Tracking Using GeNIe • Body Segmentation http://genie.sis.pitt.edu/ Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  44. 44. Bayesian Networks Unit : Probabilistic Graphical Models p. 44 Another Example : Water-Sprinkler P(C) Cloudy 0.5 C P(S|C) T 0.1 C P(R|C) F 0.5 T 0.8 F 0.2 Sprinkler Rain S R P(W|S,R) T T 0.99 WetGrass T F 0.9 F T 0.9 F F 0.0 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  45. 45. Bayesian Networks Unit : Probabilistic Graphical Models p. 45 Inference in Water-Sprinkler (1/2) • If the grass is wet (WetGrass=True) – Two possible explanations : rain or sprinkler – Which is the more likely? Pr( S  T ,W  T ) Sprinkler Pr( S  T | W  T )  Pr(W  T )  c,r Pr(C , R, S  T ,W  T )  0.2781  0.430 Pr(W  T ) 0.6471 Pr(R  T ,W  T ) Rain Pr(R  T | W  T )  Pr(W  T )  c,s Pr(C, S , R  T ,W  T )  0.4581  0.708 Pr(W  T ) 0.6471 The grass is more likely to be wet because of the rain Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  46. 46. Bayesian Networks Unit : Probabilistic Graphical Models p. 46 Inference in Water-Sprinkler (2/2) P(C) Cloudy 0.5 C P(S|C) T 0.1 C P(R|C) F 0.5 T 0.8 F 0.2 Sprinkler Rain S R P(W|S,R) T T 0.99 T F 0.9 WetGrass F T 0.9 Time needed F F 0.0 Using Bayes chain rule : for calculations Pr(C , R, S , W )  Pr(C )  Pr( R | C )  Pr( S | R, C )  Pr(W | R, C , S ) 2 x 4 x 8 x 16 = 1024 Using conditional independency properties : Pr(C , R, S , W )  Pr(C )  Pr( R | C )  Pr( S | C )  Pr(W | R, S ) 2 x 4 x 4 x 8 = 256 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  47. 47. Bayesian Networks Unit : Probabilistic Graphical Models p. 47 Inference (1/5) P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7 1 0.9 1 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.6 0.4 0.5 0.3 0.2 Earthquake Burglary 0.4 0.3 0.1 0 0.2 0.1 0 Radio Alarm E B P(A|E,B) e b 0.9 0.1 e b 0.2 0.8 Call e b 0.9 0.1 e b 0.01 0.99 C=t Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  48. 48. Bayesian Networks Unit : Probabilistic Graphical Models p. 48 Inference (2/5) P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.2 Earthquake Burglary 0.3 0.2 0.1 0.1 0 0 Radio Alarm R=t Call C=t Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  49. 49. Bayesian Networks Unit : Probabilistic Graphical Models p. 49 Inference (3/5) P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 Earthquake Burglary 0.3 0.2 0.2 0.1 0.1 0 0 P(E=t|C=t,R=t)=0.97 Radio Alarm P(B=t|C=t,R=t) = 0.1 1 1 0.9 0.9 0.8 R=t 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.1 Call 0.2 0.1 0 0 C=t Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  50. 50. Bayesian Networks Unit : Probabilistic Graphical Models p. 50 Inference (4/5) P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 Earthquake Burglary 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Radio Alarm P(E=t|C=t,R=t)=0.97 P(B=t|C=t,R=t) = 0.1 1 0.9 R=t 1 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.3 Call 0.4 0.3 0.2 0.2 0.1 0.1 0 0 C=t Explaining away effect Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  51. 51. Bayesian Networks Unit : Probabilistic Graphical Models p. 51 Inference (5/5) P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.5 Earthquake Burglary 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Radio Alarm P(E=t|C=t,R=t)=0.97 P(B=t|C=t,R=t) = 0.1 1 R=t 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.3 Call 0.4 0.3 0.2 0.2 0.1 0.1 0 0 C=t “Probability theory is nothing but common sense reduced to calculation” – Pierre Simon Laplace Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  52. 52. Bayesian Networks Unit : Probabilistic Graphical Models p. 52 2. Various PGM Models Taxonomy Factor Graph Naïve Bayes Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  53. 53. Bayesian Networks Unit : Probabilistic Graphical Models p. 53 Directional v.s. Undirectional Directed Undirected ( Bayesian networks) ( Markov networks) x1 x2 x1 x2 y1 y2 y1 y2 1 p(x, y)   p(xi | x pa(i ) ) p(y j | x pa( j ) ) p (x, y )   a (x, y ) i j Z a Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  54. 54. Bayesian Networks Unit : Probabilistic Graphical Models p. 54 Naive Bayes Model • Strong (Naive) assumption of problems – A single cause directly influences a number of effects – All effects are conditionally independent, given the cause n P( x1 , x2 ,..., xn )   P( xi | pa ( xi )) i 1 P(Cause, Effecti , Effectn )  P(Cause) P( Effecti | Cause) 2n+1 probabilities  O(n) i More details on another unit Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  55. 55. Bayesian Networks Unit : Probabilistic Graphical Models p. 55 Naïve Bayesian Classifier (NBC) • Use Naïve Bayes for classification P (Class | Feature1 ,  Featuren ) Class  P ( Feature1 ,  Featuren , Class) n  P (Class) P ( Featurei | Class) Feature 1  Feature n i 1 Face Face Expression object Skin Eye Eyebrow Mouth Color pattern Motion Motion Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  56. 56. Bayesian Networks Unit : Probabilistic Graphical Models p. 56 Temporal Causality Represented by Bayesian Networks • Temporal Causality – In many systems, data arrives sequentially – Dealing causality with time • Dynamic Bayes nets (DBNs) can be used to model such time-series (sequence) data • Special cases of DBNs include – State-space models (Kalman filter) – Hidden Markov models (HMMs) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  57. 57. Bayesian Networks Unit : Probabilistic Graphical Models p. 57 State Space Models (SSM) t = 1 2 3 • Hidden Markov Model X1 X2 X3 XT • Kalman Filter Y1 Y2 Y3 YT n P( x1 , x2 ,..., xn )   P( xi | pa( xi )) i 1 P ( X 1 ,..., X T , Y1 ,  , YT )  P ( X 1:T , Y1:T )  P( X 1 ) P(Y1 | X 1 ) P( X 2 | X 1 ) P (Y2 | X 2 )  P( X T | X T 1 ) P(YT | X T ) n   P( X i | X i 1 ) P(Yi | X i ), where P( X 1 | X 0 )  P( X 1 ) i 1 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  58. 58. Bayesian Networks Unit : Probabilistic Graphical Models p. 58 DBN (1/2) More complex temporal models than HMM & Kalman Slice 1 Slice 2 t=1 2 3 4 5 (DAG) (DAG) + Repeat Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  59. 59. Bayesian Networks Unit : Probabilistic Graphical Models p. 59 DBN (2/2) t=1 2 3 4 5 n P( x1 , x2 ,..., xn )   P( xi | pa( xi )) i 1 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  60. 60. Bayesian Networks Unit : Probabilistic Graphical Models p. 60 Bayesian SSM Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  61. 61. Bayesian Networks Unit : Probabilistic Graphical Models p. 61 Factorial SSM • Multiple hidden sequences • Avoid exponentially large hidden space Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  62. 62. Bayesian Networks Unit : Probabilistic Graphical Models p. 62 Example: Markov Random Field • Typical application: image region labelling yi xi Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  63. 63. Bayesian Networks Unit : Probabilistic Graphical Models p. 63 Example: Conditional Random Field y y y y xi Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  64. 64. Bayesian Networks Unit : Probabilistic Graphical Models p. 64 Markov Random Fields (1/2) Undirected graph Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  65. 65. Bayesian Networks Unit : Probabilistic Graphical Models p. 65 MRF (2/2) y   Parameter   tying    x  Local evidence Compatibility with neighbors (compatibility with image) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  66. 66. Bayesian Networks Unit : Probabilistic Graphical Models p. 66 3. Conditional Independencies • A Bayesian network/probabilistic graphical model G, represents a set of Markov Independencies P • There is a factorization theorem P ( X 1 ,..., X n )   P ( X i | Pai ) i • This section inspects deeper meanings of conditional independence for – The factorization theorem – Inference algorithms in later units Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  67. 67. Bayesian Networks Unit : Probabilistic Graphical Models p. 67 Conditional Independence • Dependencies – Two connected nodes influence each other • Independent – Example: I(B;E) • Conditional Independent – Example • I(J;M|A)? • I(B;E|A)? – d-seperation Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  68. 68. Bayesian Networks Unit : Probabilistic Graphical Models p. 68 D-Separation • It is a rule describing the influences between nodes Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  69. 69. Bayesian Networks Unit : Probabilistic Graphical Models p. 69 Serial (Intermediate Cause) • Indirect causal effect, no evidence B • Clearly burglary will effect Marry call A • Same situation for indirect evidence effect, M because independence is symmetric • If I(E;M|A) then I(M;E|A) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  70. 70. Bayesian Networks Unit : Probabilistic Graphical Models p. 70 Diverging (Common Cause) • Influence can flow A from John call to Mary call if we don‘t know whether or not J M there is alarm. • But I(J;M|A) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  71. 71. Bayesian Networks Unit : Probabilistic Graphical Models p. 71 Converging (Common Effect) • Influence can‘t flow from E B Earthquake to burglary if we don‘t know whether or not there is alarm • So I(E;B) A • Special structure which cause independence. • V-Structure Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  72. 72. Bayesian Networks Unit : Probabilistic Graphical Models p. 72 Independence of Two Events Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  73. 73. Bayesian Networks Unit : Probabilistic Graphical Models p. 73 D-Separation for 3 Nodes Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  74. 74. Bayesian Networks Unit : Probabilistic Graphical Models p. 74 Path Blockage (1/3) • Three cases: –Common cause Blocked Blocked Unblocked Active E E – Intermediate cause R A R A –Common Effect Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  75. 75. Bayesian Networks Unit : Probabilistic Graphical Models p. 75 Path Blockage (2/3) • Three cases: –Common cause Blocked Unblocked Active E E – Intermediate cause A A –Common Effect C C Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  76. 76. Bayesian Networks Unit : Probabilistic Graphical Models p. 76 Path Blockage (3/3) Blocked Unblocked Active Three cases: – Common cause E B – Intermediate cause E B A – Common Effect A C E B C A C Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  77. 77. Bayesian Networks Unit : Probabilistic Graphical Models p. 77 General Case Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  78. 78. Bayesian Networks Unit : Probabilistic Graphical Models p. 78 D-Separation in General • X is d-separated from Y, given Z, – If all paths from a node in X to a node in Y are blocked, given Z • Checking d-separation can be done efficiently (linear time in number of edges) – Bottom-up phase: Mark all nodes whose descendents are in Z – X to Y phase: Traverse (BFS) all edges on paths from X to Y and check if they are blocked Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  79. 79. Bayesian Networks Unit : Probabilistic Graphical Models p. 79 Paths (1/2) • Intuition: dependency must “flow” along paths in the graph • A path is a sequence of neighboring variables Earthquake Burglary Examples: • REAB Radio Alarm • CAER Call Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  80. 80. Bayesian Networks Unit : Probabilistic Graphical Models p. 80 Paths (2/2) • For a path between two end nodes X, Y • The path is a – Active path • If we can find dependency between X & Y – Blocked path • If we cannot find dependency between X & Y • X & Y are conditional independent • X & Y are D-Separated • We want to classify situations in which paths are active Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  81. 81. Bayesian Networks Unit : Probabilistic Graphical Models p. 81 D-Separation Example 1 (1/3) E B – d-sep(R,B)? R A C Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  82. 82. Bayesian Networks Unit : Probabilistic Graphical Models p. 82 D-Separation Example 1 (2/3) – d-sep(R,B) = yes E B – d-sep(R,B|A)? R A C Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  83. 83. Bayesian Networks Unit : Probabilistic Graphical Models p. 83 D-Separation Example 1 (3/3) – d-sep(R,B) = yes E B – d-sep(R,B|A) = no – d-sep(R,B|E,A)? R A C Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  84. 84. Bayesian Networks Unit : Probabilistic Graphical Models p. 84 D-Separation Example 2 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  85. 85. Bayesian Networks Unit : Probabilistic Graphical Models p. 85 D-Separation Example 3 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  86. 86. Bayesian Networks Unit : Probabilistic Graphical Models p. 86 d-separation: Car Start Problem• 1. ‘Start’ and ‘Fuel’ are dependent on each other.• 2. ‘Start’ and ‘Clean Spark Plugs’ are dependent on each other.• 3. ‘Fuel’ and ‘Fuel Meter Standing’ are dependent on each other.• 4. ‘Fuel’ and ‘Clean Spark Plugs’ are conditionally dependent on each other given the value of ‘Start’.• 5. ‘Fuel Meter Standing’ and ‘Start’ are conditionally independent given the value of ‘Fuel’. Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  87. 87. Bayesian Networks Unit : Probabilistic Graphical Models p. 87 Exercises P(xt|xt-1) xt+1 Face Real Real Real Expression location x location x location t-1 t P(zt-1|xt-1) zt-1 zt Observed Observed Eyebrow Mouth location location Motion Motion Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  88. 88. Bayesian Networks Unit : Probabilistic Graphical Models p. 88 4. Inference • 4.1 What Is Inference • 4.2 How Inference • 4.3 Inference Methods Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  89. 89. Bayesian Networks Unit : Probabilistic Graphical Models p. 89 4.1 What Is Inference Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  90. 90. Bayesian Networks Unit : Probabilistic Graphical Models p. 90 Exercises (1/2) • Face detection Facial Expression Recog. Face Face object Expression Skin Eye Eyebrow Mouth Color pattern Motion Motion Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  91. 91. Bayesian Networks Unit : Probabilistic Graphical Models p. 91 Exercises (2/2) P(xt|xt-1) xt+1 • Face tracking Real Real Real location x location x location t-1 t P(zt-1|xt-1) zt-1 zt Observed Observed location location Real position : xt Predicted position x-t+1 Detected position : zt P ( z t | xt ) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  92. 92. Bayesian Networks Unit : Probabilistic Graphical Models p. 92 3 Kinds of Variables in Inference • Remember the general inference procedure in previous unit (uncertainty inference unit) • Let P(X|E=e) be the query – X be the query variable – E be the set of evidence variables V S • e be the observed values of E – H be the remaining T L unobserved variables A B (Hidden variables) X D Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  93. 93. Bayesian Networks Unit : Probabilistic Graphical Models p. 93 The Burglary Example Query : P(Burglary|John Calls=true) Query variables: X Burglary Earthquake Burglary Evidence variables: E=e John Calls = true Alarm Hidden variables: H Earthquake, Alarm, John Calls Mary Calls Marry Calls Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  94. 94. Bayesian Networks Unit : Probabilistic Graphical Models p. 94 The Asia Example • Query P(L|v,s,d) V S – Query variables: L – Evidence variables: T L V=true, S=true, D=true A B – Hidden variables: T, X, A, B X D Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  95. 95. Bayesian Networks Unit : Probabilistic Graphical Models p. 95 arg max P(X|e) • For P(X | e), if X is a Boolean variable • P(X | e) will compute 2 probabilities P(X=true | e) = 0.8 P(X=false | e) = 0.2 • arg maxx P(X=x|e) will get a decision P(X=true | e) = 0.8 Max X = True P(X=false | e) = 0.2 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  96. 96. Bayesian Networks Unit : Probabilistic Graphical Models p. 96 Five Types of Queries in Inference • For a probabilistic graphical model G • Given a set of evidence E=e • Query the PGM with – P(e) : Likelihood query – arg max P(e) : Maximum likelihood query – P(X|e) : Posterior belief query – arg maxx P(X=x|e) : (Single query variable) Maximum a posterior (MAP) query – arg maxx …x P(X1=x1, …, Xt=xt|e) : 1 t Most probable explanation (MPE) query Also called Viterbi decoding Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  97. 97. Bayesian Networks Unit : Probabilistic Graphical Models p. 97 Likelihood Query P(e) (1/2) Input video Probability of Evidence X1 X2 Xt An HMM e1 for Surprise E1 E2 Et e2 e1:t P (E1:t=e1:t) … et Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  98. 98. Bayesian Networks Unit : Probabilistic Graphical Models p. 98 Likelihood Query P(e) (2/2) • Marginalization of all hidden variables    P( E  e, H  h) hH      P ( E1:t  e1:t , X 1 ,  , X t ) X1 X 2 Xt   P( E X 1 X t 1:t  e1:t , X 1 ,  , X t ) n    P( X X 1 X t i 1 i | X i 1 ) P( Ei | X i ), where P ( X 1 | X 0 )  P ( X 1 ) X1 X2 Xt E1 E2 Et Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  99. 99. Bayesian Networks Unit : Probabilistic Graphical Models p. 99 Maximum Likelihood Query arg max P(e) Input video An HMM X1 X2 Xt for Surprise e1 PS(Xt|Xt-1), E1 E2 Et PS(Ei|Xi) P Surprise(e1:t) e2 e1:t Max P Cry(e1:t) … X1 X2 Xt Cry HMM PC(Xt|Xt-1), et E1 E2 Et PC(Ei|Xi) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  100. 100. Bayesian Networks Unit : Probabilistic Graphical Models p. 100 Maximum Likelihood Query arg max P(e) • Likelihood query P(E=e) Step 1: Bayes theorem P ( E  e) Step 2: Marginalization    P ( E  e, H  h) of all hidden variables hH • Query arg max P(E=e) Step 1: Bayes theorem Step 2: Marginalization  arg max  P ( E  e, H  h)  of all hidden variables hH Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  101. 101. Bayesian Networks Unit : Probabilistic Graphical Models p. 101 Posteriori Belief Query P(X|e) • Usually applied on tracking – Use temporal models of PGM • 4 query types – Filtering: P(Xt | E1=e1,…, Et=et)=P(Xt |e1:t) – Prediction: P(Xt+1 | e1:t) – Smoothing: P(Xt-k | e1:t) (Fixed-lag smoothing) X1 X2 Xt Xt+1 E1 E2 Et Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  102. 102. Bayesian Networks Unit : Probabilistic Graphical Models p. 102 P(X|e) – Filtering (1/2) • P(Xt | e1:t) X1 X2 Xt E1 E2 Et Real position: xi Filtered position: x’t Detected position: ei P ( z t | xt ) Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  103. 103. Bayesian Networks Unit : Probabilistic Graphical Models p. 103 P(X|e) – Filtering (2/2) • Inference of the query P(Xt|e1:t) is P( X t , e1:t ) Step 1: P( X t | e1:t )  P(e1:t ) Bayes theorem  P( X t , e1:t ) Step 2: Marginalization    P ( X t , e1:t , X 1  X t 1 ) X 1 X t 1 of all hidden variables Step 3:     P ( X i | X i 1 )P (ei | X i ) Chaining by X  X i 1~ t 1 t 1 conditional independence Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  104. 104. Bayesian Networks Unit : Probabilistic Graphical Models p. 104 P(X|e) – Prediction (1/2) • P(Xt+k | e1:t) for k > 0 For k=1 X X Xt Xt+1 1 2 E1 E2 Et Real position : xi Predicted position Detected position : ei x’t+1 Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  105. 105. Bayesian Networks Unit : Probabilistic Graphical Models p. 105 P(X|e) – Prediction (2/2) • Inference of the query P(Xt+1|e1:t) is P( X t 1 , e1:t ) Step 1: P( X t 1 | e1:t )  P(e1:t ) Bayes theorem  P( X t 1 , e1:t ) Step 2: Marginalization    P ( X t 1 , e1:t , X 1  X t ) X 1 X t of all hidden variables Step 3:  P ( X t 1 | X t )   P ( X i | X i 1 )P (ei | X i ) Chaining by X  X i 1~ t1 t conditional independence Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  106. 106. Bayesian Networks Unit : Probabilistic Graphical Models p. 106 P(X|e) – Smoothing (1/3) • P(Xk | e1:t) for 1  k < t X1 X2 Xk Xt E1 E2 Ek Et Real position: xt Smoothed position: xt Detected position: zt Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  107. 107. Bayesian Networks Unit : Probabilistic Graphical Models p. 107 P(X|e) – Smoothing (2/3) • Inference of the query P(Xk|e1:t) is P( X k , e1:t ) Step 1: P( X k | e1:t )  P(e1:t ) Bayes theorem  P( X k , e1:t ) Step 2: Marginalization   P,e, 1X:t , X 1  X t ) X 1 X k 1 , X K 1 ( of all hidden variables t Step 3: Chaining by   ,, X it P( X i | X i 1 )P(ei | X i ) X X , X 1~ 1 k 1 K 1 t conditional independence Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  108. 108. Bayesian Networks Unit : Probabilistic Graphical Models p. 108 P(X|e) – Smoothing (3/3) • Fixed-lag smoothing Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  109. 109. Bayesian Networks Unit : Probabilistic Graphical Models p. 109 MAP Query (1/2) • arg maxx P(Xi=x|e) • Usually applied on Classification – Find most likely class X=x, given the evidence e (feature) P(X=Surprise|e) If P(X=Smile|e) is the max probability Smile = arg maxx P(Xi=x|e) P(X=Smile|e) Facial X={Surprise, Smile, …}  Expression Eyebrow Mouth  Motion Motion Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  110. 110. Bayesian Networks Unit : Probabilistic Graphical Models p. 110 MAP Query (2/2) • MAP query arg maxx P(X=x|E=e) Step 1: arg max P( X  x | e) x Bayes theorem P ( X  x, e)  arg max x P ( e) Step 2:   arg max P( X  x, e) x Marginalization of all hidden variables   arg max  P ( X  x, e, H  h) x hH Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  111. 111. Bayesian Networks Unit : Probabilistic Graphical Models p. 111 MPE Query • Also called Viterbi decoding • arg maxx P(X1=x1,…, Xt=xt|e1:t) • = arg maxx1:t P(X1:t|e1:t) • = Smoothing for X1:t-1 + Filtering X1 X2 Xt E1 E2 Et Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  112. 112. Bayesian Networks Unit : Probabilistic Graphical Models p. 112 Exercises • Face Detection • Facial Expression Recognition • Face Tracking • Body Segmentation X={Surprise, Smile, …} P(xt|xt-1) xt+1 Facial Expression Real Real Real location x location x location t-1 t P(zt-1|xt-1) zt-1 zt Eyebrow Mouth  Motion Observed Observed Motion location location Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
  113. 113. Bayesian Networks Unit : Probabilistic Graphical Models p. 113 4.2 How Inference • Inference of the query P(X|E=e) is P ( X , E  e) Step 1: P ( X | E  e)  P ( E  e) Bayes theorem  P ( X , E  e) Step 2: Marginalization    P ( X , E  e, H  h) of all hidden variables hH Step 3: Chaining by     P( X i | Pa ( X i )) hH i 1~ n conditional independence Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright
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