Lect3_Bayes_error_ROC

Lect3_Bayes_error_ROC - Lecture 3: Bayesian Decision Theory...

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1 Lecture 3: Bayesian Decision Theory II Outline: 1. Bayes risk, Bayes error, and empirical error. 2. Two-state case. 3. ROC curve and PR curve. Lecture notes Stat 231-CS276A, © S.C. Zhu Bayes Risk Feature X Class w Action α Three key variables in Bayesian decision theory and their causal relations p(x|w) p(w) λ(α |w) We are given three functions The Bayes risk is Lecture notes Stat 231-CS276A, © S.C. Zhu
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2 Empirical Risk The Bayes decision theory assumes that there is underlying probability p(x,w) , and the Bayes risk is averaged w.r.t. to p(x,w) . In practice, we only have a set of labeled data (testing) drawn from p(x,w) , We can estimate the Bayes risk by accumulating all the risks Lecture notes Stat 231-CS276A, © S.C. Zhu i.e. wrong decisions over the testing set. In case of 0-1 loss: = = M j j j c x M 1 ) ) ( ( 1 1 α = = M j j j c x M R 1 ) | ) ( ( 1 ) ( ˆ λ Bayesian error In a special case, like fish classification, the action is classification, we assume a 0/1 error. j i j i w if w = = 0 ) | ( j i j i w if w = 1 ) | ( ) x | ( p 1 ) x | p(w ) x | ( i w j i i j = = R The risk for classifying x to class α i is, The optimal decision is to choose the class that has maximum posterior probability ) x | ( p max arg )) x | ( p 1 ( min arg ) x ( = = Lecture notes Stat 231-CS276A, © S.C. Zhu Ω Ω The total risk for a decision rule, in this case, is called the Bayesian error x ) x ( ) ) x | ) x ( ( 1 ( x ) x ( ) x | ( ) ( d p p d p error p error p R = = =
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3 Bayes Risk and Bayes Error One can minimize the Bayes risk for each x Lecture notes Stat 231-CS276A, © S.C. Zhu
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This note was uploaded on 11/24/2010 for the course STAT 201a taught by Professor Wu during the Spring '10 term at Pasadena City College.

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Lect3_Bayes_error_ROC - Lecture 3: Bayesian Decision Theory...

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