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CHAPTER 3: Bayesian Decision Theory Souce: Alpaypin with modifications by Christoph F. Eick; last changed on February 4, 2009. Remark: Belief Networks will be covered Early April 2011. Utility theory will be covered as port of reinforcement learning. Probability and Inference Result of tossing a coin is {Heads,Tails} Random var X {1,0} Bernoulli: P {X=1} = po Sample: X = {xt }Nt =1 Estimation: po = # {Heads}/#{Tosses} = ∑t xt / N Prediction of next toss: Heads if po > ½, Tails otherwise In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". P(X=k)= 2 Binomial Distribution 3 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) Classification Credit scoring: Inputs are income and savings. Output is low-risk vs high-risk Input: x = [x1,x2]T ,Output: C {0,1} Prediction: C 1 if P (C 1 | x 1,x 2 ) 0.5 choose C 0 otherwise or equivalent ly C 1 if P (C 1 | x 1,x 2 ) P (C 0 | x 1,x 2 ) choose C 0 otherwise 4 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) Bayes’ Rule prior posterior likelihood P C p x | C P C | x p x evidence P C 0 P C 1 1 p x p x | C 1P C 1 p x | C 0P C 0 p C 0 | x P C 1 | x 1 5 Bayes’ Rule: K>2 Classes p x | Ci P Ci P Ci | x p x p x | Ci P Ci K p x | Ck P Ck k 1 P Ci 0 and K P C 1 i 1 i choose Ci if P Ci | x max k P C k | x 6 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) Losses and Risks Actions: αi Loss of αi when the state is Ck : λik Expected risk (Duda and Hart, 1973) K R i | x ik P C k | x k 1 choose i if R i | x min k R k | x 7 Losses and Risks: 0/1 Loss 0 if i k ik 1 if i k K R i | x ik P Ck | x k 1 P Ck | x k i 1 P Ci | x For minimum risk, choose the most probable class Remark: This strategy is not optimal in other cases 8 Example and Homework! C1=has cancer C2=has not cancer 12=9 21=90 Homework: a) Determine the optimal decision making strategy Inputs: P(C1|x), P(C2|x) Decision Making Strategy:… b) Now assume we also have a reject option and the cost for making no decision are 3: reject,2=3 reject, 1=3 Inputs: P(C1|x), P(C2|x) Decision Making Strategy: … Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 9 Losses and Risks: Reject 0 if i k ik if i K 1, 0 1 1 otherwise Risk for reject K R K 1 | x P Ck | x k 1 R i | x P Ck | x 1 P Ci | x k i choose Ci if P Ci | x P Ck | x k i and P Ci | x 1 reject otherwise 10 Discriminant Functions choose Ci if gi x max k gk x gi x , i 1,, K R i | x gi x P Ci | x p x | C P C i i K decision regions R1,...,RK Ri x | gi x max k gk x 11 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)