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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 Utility theory will be covered as port of reinforcement learning.
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Probability and Inference
Result of tossing a coin is Î {Heads,Tails} Random var X Î{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)=
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Binomial Distribution
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
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Classification Credit scoring: Inputs are income and savings.
Output is low-risk vs high-risk Input: x = [x1,x2]T ,Output: C Î {0,1} Prediction: Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
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Bayes’ Rule prior likelihood posterior evidence
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Bayes’ Rule: K>2 Classes
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
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Losses and Risks Actions: αi Loss of αi when the state is Ck : λik
Expected risk (Duda and Hart, 1973)
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Losses and Risks: 0/1 Loss
For minimum risk, choose the most probable class Remark: This strategy is not optimal in other cases
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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)
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Losses and Risks: Reject
Risk for reject
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Discriminant Functions
K decision regions R1,...,RK Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
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