1. 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

2. Chapter 2 (Part 1): Bayesian Decision Theory (Sections 1-4) 1. Introduction – Bayesian Decision Theory Pure statistics, probabilities known, optimal decision 2. Bayesian Decision Theory–Continuous Features 3. Minimum-Error-Rate Classification 4. Classifiers, Discriminant Functions and Decision Surfaces

3. Pattern Classification, Chapter 2 (Part 1) 2 1. Introduction The sea bass/salmon example State of nature, prior State of nature is a random variable The catch of salmon and sea bass is equiprobable P(  1 ) = P(  2 ) (uniform priors) P(  1 ) + P(  2 ) = 1 (exclusivity and exhaustivity)

4. Pattern Classification, Chapter 2 (Part 1) 3 Decision rule with only the prior information Decide  1 if P(  1 ) > P(  2 ) otherwise decide  2 Use of the class –conditional information P(x |  1 ) and P(x |  2 ) describe the difference in lightness between populations of sea and salmon

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6. 5 Posterior, likelihood, evidence P(  j | x) = P(x |  j ) P (  j ) / P(x) (Bayes Rule) Where in case of two categories Posterior = (Likelihood. Prior) / Evidence

7. Pattern Classification, Chapter 2 (Part 1) 6

8. 7 Decision given the posterior probabilities X is an observation for which: if P(  1 | x) > P(  2 | x) True state of nature =  1 if P(  1 | x) < P(  2 | x) True state of nature =  2 Therefore: whenever we observe a particular x, the probability of error is : P(error | x) = P(  1 | x) if we decide  2 P(error | x) = P(  2 | x) if we decide  1

9. Pattern Classification, Chapter 2 (Part 1) 8 Minimizing the probability of error Decide  1 if P(  1 | x) > P(  2 | x); otherwise decide  2 Therefore: P(error | x) = min [P(  1 | x), P(  2 | x)] (Bayes decision)

10. Pattern Classification, Chapter 2 (Part 1) 9 2. Bayesian Decision Theory – Continuous Features Generalization of the preceding ideas 1. Use of more than one feature 2. Use more than two states of nature 3. Allowing actions other than deciding the state of nature 4. Introduce a loss of function which is more general than the probability of error

11. Pattern Classification, Chapter 2 (Part 1) 10 Allowing actions other than classification primarily allows the possibility of rejection Refusing to make a decision in close or bad cases! The loss function states how costly each action taken is

12. Pattern Classification, Chapter 2 (Part 1) 11 Let {  1,  2,…,  c } be the set of c states of nature (or “categories”) Let {  1,  2,…,  a } be the set of possible actions Let (  i |  j ) be the loss incurred for taking action  i when the state of nature is  j

13. Pattern Classification, Chapter 2 (Part 1) 12 Overall risk R = Sum of all R(  i | x) for i = 1,…,a Minimizing R Minimizing R(  i | x) for i = 1,…, a for i = 1,…,a Conditional risk

14. Pattern Classification, Chapter 2 (Part 1) 13 Select the action  i for which R(  i | x) is minimum R is minimum and R in this case is called the Bayes risk = best performance that can be achieved!

15. Pattern Classification, Chapter 2 (Part 1) 14 Two-category classification  1 : deciding  1  2 : deciding  2 ij = (  i |  j ) loss incurred for deciding  i when the true state of nature is  j Conditional risk: R(  1 | x) =  11 P(  1 | x) + 12 P(  2 | x) R(  2 | x) =  21 P(  1 | x) + 22 P(  2 | x)

16. Pattern Classification, Chapter 2 (Part 1) 15 Our rule is the following: if R(  1 | x) < R(  2 | x) action  1 : “decide  1 ” is taken This results in the equivalent rule : decide  1 if: ( 21 - 11 ) P(x |  1 ) P(  1 ) > ( 12 - 22 ) P(x |  2 ) P(  2 ) and decide  2 otherwise

17. Pattern Classification, Chapter 2 (Part 1) 16 Likelihood ratio: The preceding rule is equivalent to the following rule: Then take action  1 (decide  1 ) Otherwise take action  2 (decide  2 )

18. Pattern Classification, Chapter 2 (Part 1) 17 Optimal decision property “If the likelihood ratio exceeds a threshold value independent of the input pattern x, we can take optimal actions”

19. Pattern Classification, Chapter 2 (Part 2) 18 3. Minimum-Error-Rate Classification Actions are decisions on classes If action  i is taken and the true state of nature is  j then: the decision is correct if i = j and in error if i  j Seek a decision rule that minimizes the probability of error which is the error rate

20. Pattern Classification, Chapter 2 (Part 2) 19 Introduction of the zero-one loss function: Therefore, the conditional risk is: “The risk corresponding to this loss function is the average probability error” 

21. Pattern Classification, Chapter 2 (Part 2) 20 Minimize the risk requires maximize P(  i | x) (since R(  i | x) = 1 – P(  i | x)) For Minimum error rate Decide  i if P (  i | x) > P(  j | x)  j  i

22. Pattern Classification, Chapter 2 (Part 2) 21 Regions of decision and zero-one loss function, therefore: If is the zero-one loss function wich means:

23. Pattern Classification, Chapter 2 (Part 2) 22

24. Pattern Classification, Chapter 2 (Part 2) 23 4. Classifiers, Discriminant Functions and Decision Surfaces The multi-category case Set of discriminant functions g i (x), i = 1,…, c The classifier assigns a feature vector x to class  i if: g i (x) > g j (x)  j  i

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26. Pattern Classification, Chapter 2 (Part 2) 25 Let g i (x) = - R(  i | x) (max. discriminant corresponds to min. risk!) For the minimum error rate, we take g i (x) = P(  i | x) (max. discrimination corresponds to max. posterior!) g i (x)  P(x |  i ) P(  i ) g i (x) = ln P(x |  i ) + ln P(  i ) (ln: natural logarithm!)

27. Pattern Classification, Chapter 2 (Part 2) 26 Feature space divided into c decision regions if g i (x) > g j (x)  j  i then x is in R i ( R i means assign x to  i ) The two-category case A classifier is a “dichotomizer” that has two discriminant functions g 1 and g 2 Let g(x)  g 1 (x) – g 2 (x) Decide  1 if g(x) > 0 ; Otherwise decide  2

28. Pattern Classification, Chapter 2 (Part 2) 27 The computation of g(x)

29. Pattern Classification, Chapter 2 (Part 2) 28