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COMPE 467 - Pattern Recognition

Bayesian Decision Theory. COMPE 467 - Pattern Recognition. Covariance matrices for all of the classes are identical, But covariance matrices are arbitrary. and are independent of i. So,. Covariance matrices are different for each category.

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COMPE 467 - Pattern Recognition

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  1. Bayesian Decision Theory COMPE 467 - Pattern Recognition

  2. Covariance matrices for all of the classes are identical, • But covariance matrices are arbitrary.

  3. and are independent of i. So,

  4. Covariance matrices are different for each category.

  5. only is independent of i. So,

  6. EXAMPLE:

  7. EXAMPLE (cont.):

  8. EXAMPLE (cont.): Find the discriminant function for the first class.

  9. EXAMPLE (cont.): Find the discriminant function for the first class.

  10. EXAMPLE (cont.): Similarly, find the discriminant function for the second class.

  11. EXAMPLE (cont.): The decision boundary:

  12. EXAMPLE (cont.): The decision boundary:

  13. EXAMPLE (cont.): Using MATLAB we can draw the decision boundary: (to draw the decision boundary in MATLAB) >> s = 'x^2-10*x-4*x*y+8*y-1+2*log(2)'; >> ezplot(s)

  14. EXAMPLE (cont.): Using MATLAB we can draw the decision boundary:

  15. EXAMPLE (cont.):

  16. Error Probabilities and Integrals

  17. Error Probabilities and Integrals

  18. Error Probabilities and Integrals

  19. Receiver Operating Characteristics • Another measure of distance between two Gaussian distributions. • found a great use in medicine, radar detection and other fields.

  20. Receiver Operating Characteristics

  21. Receiver Operating Characteristics

  22. Receiver Operating Characteristics

  23. Receiver Operating Characteristics • If both diagnosis and test are positive,it is called a true positive. The probability of a TP to occur is estimated by counting thetrue positives in the sample and divide by the sample size. • If the diagnosis is positive andthe test is negative it is called a false negative (FN). • False positive (FP) and true negative(TN) are defined similarly.

  24. Receiver Operating Characteristics • The values described are used to calculate different measurements of the quality of thetest. • The first one is sensitivity, SE, which is the probability of having a positive test among the patients who have a positive diagnosis.

  25. Receiver Operating Characteristics • Specificity, SP, is the probability of having a negative test among the patients who have anegative diagnosis.

  26. Receiver Operating Characteristics • Example:

  27. Receiver Operating Characteristics • Example (cont.):

  28. Receiver Operating Characteristics • Overlap in distributions:

  29. Bayes Decision Theory – Discrete Features • Assume x = (x1.......xd) takes only m discrete values • Components of x are binary or integer valued, x can take only one of m discrete values v1, v2, …, vm

  30. Bayes Decision Theory – Discrete Features (Decision Strategy) • Maximize class posterior using bayes decision rule P(1 | x) = P(x | j) . P (j) S P(x | j) . P (j) Decide i if P(i | x) > P(i | x)for all i ≠ j or • Minimize the overall risk by selecting the action i = *: deciding 2 * = arg min R(i | x)

  31. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  32. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  33. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  34. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  35. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  36. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem)

  37. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem) EXAMPLE

  38. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem) EXAMPLE

  39. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem) EXAMPLE

  40. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem) EXAMPLE

  41. Bayes Decision Theory – Discrete Features (Independent binary features in 2-category problem) EXAMPLE

  42. APPLICATION EXAMPLE Bit – matrix for machine – printed characters a pixel Here, each pixel may be taken as a feature For above problem, we have is the probabilty that for letter A,B,… B A

  43. Summary • To minimize the overall risk, choose the action thatminimizes the conditional risk R (α |x). • To minimize the probability of error, choose the class thatmaximizes the posterior probability P (wj |x). • If there are different penalties for misclassifying patternsfrom different classes, the posteriors must be weightedaccording to such penalties before taking action. • Do not forget that these decisions are the optimal onesunder the assumption that the “true” values of theprobabilities are known.

  44. References • R.O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, New York: John Wiley, 2001. • Selim Aksoy, Pattern Recognition Course Materials, 2011. • M. Narasimha Murty, V. Susheela Devi, Pattern Recognition an Algorithmic Approach, Springer, 2011. • “Bayes Decision Rule Discrete Features”, http://www.cim.mcgill.ca/~friggi/bayes/decision/binaryfeatures.html, access: 2011.

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