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CHAPTER 3: Bayesian Decision Theory Souce: Alpaypin with modifications by Christoph F. Eick; Remark: Belief Networks will be covered in April. Utility theory will be covered as part 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 see: http://en.wikipedia.org/wiki/Bayes'_theorem 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 K P Ci 0 and P Ci 1 i 1 c hoose Ci if P Ci | x maxk P Ck | x Remember: The disease/symptom example 6 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 Ck | x k 1 c hoosei if R i | x mink R k | x Remark: λik is the cost of choosing i when k is correct! If we use accuracy/error, then λik := If i=k then 0 else 1! 7 Losses and Risks: 0/1 Loss 0 if i k ik 1 if i k K Ri | 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 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 9 Example and Homework! C1=has cancer C2=has not cancer 12=9 21=72 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: … Ungraded Homework: to be discussed Feb. 6! 10 Homework: a) Determine the optimal decision making strategy Input: P(C1|x), R(1|x)=9xP(C2) R(2|x)=72xP(C1) R(reject|x)=3 Setting those equal receive: 9xP(C2)=72xP(C1) (P(C2)/P(C1))=8; additionally using P(C1)+P(C2)=1 we receive: P(C1)=1/9 and P(C2)=8/9 and the risk-minimizing decision rule becomes: IF P(C1)>1/9 THEN choose C1 ELSE choose C2 b) Now assume we also have a reject option and the cost for making no decision are 3: reject,2=3 reject, 1=3 Input: P(C1|x) First we find equating R(reject|x) with R(1|x) and R(2|x): If P(C2)≥1/3 P(C1) ≤2/3 reject should be preferred over class1 and P(C1)≥1/24 reject should be preferred over class2. Combining this knowledge with the previous decision rule we receive: IF P(C1)[0,1/24] THEN choose class2 ELSE IF P(C1)[2/3,1] THEN choose class1 ELSE choose reject 11 Discriminant Functions c hooseCi if gi x maxk gk x gi x , i 1,, K R i | x gi x P C i | x p x | C P C i i K decision regions R1,...,RK Ri x | gi x maxk gk x 12 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)