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Chapter 2 (part 3) Bayesian Decision Theory (Sections 2-6,2-9)

Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, 2000 with the permission of the authors and the publisher. Chapter 2 (part 3) Bayesian Decision Theory (Sections 2-6,2-9).

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Chapter 2 (part 3) Bayesian Decision Theory (Sections 2-6,2-9)

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  1. Pattern ClassificationAll materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, 2000with the permission of the authors and the publisher

  2. Chapter 2 (part 3)Bayesian Decision Theory (Sections 2-6,2-9) Discriminant Functions for the Normal Density Bayes Decision Theory – Discrete Features

  3. Discriminant Functions for the Normal Density • We saw that the minimum error-rate classification can be achieved by the discriminant function gi(x) = ln P(x | i) + ln P(i) • Case of multivariate normal Pattern Classification, Chapter 2 (Part 3)

  4. Case i = 2.I(I stands for the identity matrix) Pattern Classification, Chapter 2 (Part 3)

  5. A classifier that uses linear discriminant functions is called “a linear machine” • The decision surfaces for a linear machine are pieces of hyperplanes defined by: gi(x) = gj(x) Pattern Classification, Chapter 2 (Part 3)

  6. Pattern Classification, Chapter 2 (Part 3)

  7. The hyperplane separatingRiand Rj always orthogonal to the line linking the means! Pattern Classification, Chapter 2 (Part 3)

  8. Pattern Classification, Chapter 2 (Part 3)

  9. Pattern Classification, Chapter 2 (Part 3)

  10. Case i =  (covariance of all classes are identical but arbitrary!) • Hyperplane separating Ri and Rj (the hyperplane separating Ri and Rj is generally not orthogonal to the line between the means!) Pattern Classification, Chapter 2 (Part 3)

  11. Pattern Classification, Chapter 2 (Part 3)

  12. Pattern Classification, Chapter 2 (Part 3)

  13. Case i = arbitrary • The covariance matrices are different for each category (Hyperquadrics which are: hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoids, hyperparaboloids, hyperhyperboloids) Pattern Classification, Chapter 2 (Part 3)

  14. Pattern Classification, Chapter 2 (Part 3)

  15. Pattern Classification, Chapter 2 (Part 3)

  16. Bayes Decision Theory – Discrete Features • Components of x are binary or integer valued, x can take only one of m discrete values v1, v2, …, vm • Case of independent binary features in 2 category problem Let x = [x1, x2, …, xd ]twhere each xiis either 0 or 1, with probabilities: pi = P(xi = 1 | 1) qi = P(xi = 1 | 2) Pattern Classification, Chapter 2 (Part 3)

  17. The discriminant function in this case is: Pattern Classification, Chapter 2 (Part 3)

  18. Bayesian Belief Network • Features • Causal relationships • Statistically independent • Bayesian belief nets • Causal networks • Belief nets Pattern Classification, Chapter 2 (Part 3)

  19. x1 and x3 are independent Pattern Classification, Chapter 2 (Part 3)

  20. Structure • Node • Discrete variables • Parent, Child Nodes • Direct influence • Conditional Probability Table • Set by expert or by learning from training set • (Sorry, learning is not discussed here) Pattern Classification, Chapter 2 (Part 3)

  21. Pattern Classification, Chapter 2 (Part 3)

  22. Examples Pattern Classification, Chapter 2 (Part 3)

  23. Pattern Classification, Chapter 2 (Part 3)

  24. Pattern Classification, Chapter 2 (Part 3)

  25. Evidence e Pattern Classification, Chapter 2 (Part 3)

  26. Ex. 4. Belief Network for Fish P(a) a1=winter, 0.25a2=spring, 0.25 a3=summer,0.25 a4=autumn, 0.25 P(b) A B b1=north Atlantic, 0.6b2=south Atlantic, 0.4 P(x|a,b) X x1=salmonx2=sea bass P(d|x) C D P(c|x) d1=wide, d2=thinx1 0.3, 0.7x2 0.6, 0.4 c1=light,c2=medium, c3=darkx1 0.6, 0.2, 0.2x2 0.2, 0.3, 0.5 Pattern Classification, Chapter 2 (Part 3)

  27. Belief Network for Fish • Fish was caught in the summer in the north Atlantic and is a see bass that is dark and thin • P(a3,b1,x2,c3,d2)= P(a3)P(b1)P(x2|a3,b1)P(c3|x2)P(d2|x2)=0.25*0.6*0.4*0.5*0.4=0.012 Pattern Classification, Chapter 2 (Part 3)

  28. Light, south Atlantic, fish? Pattern Classification, Chapter 2 (Part 3)

  29. Normalize Pattern Classification, Chapter 2 (Part 3)

  30. Conditionally Independent Pattern Classification, Chapter 2 (Part 3)

  31. Medical Application • Medical diagnosis • Uppermost nodes: biological agent • (virus or bacteria) • Intermediate nodes: diseases • (flu or emphysema) • Lowermost nodes: symptoms • (high temperature or coughing) • Finds the most likely disease or cause • By entering measured values Pattern Classification, Chapter 2 (Part 3)

  32. Exercise 50 (based on Ex. 4) • (a) • December 20, north Atlantic, thin • P(a1)=P(a4)=0.5, P(b1)=1, P(d2)=1 • Fish? Error rate? • (b) • Thin, medium lightness • Season? Probability? • (c) • Thin, medium lightness, north atlantic • Season?, probability? Pattern Classification, Chapter 2 (Part 3)

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