learning classifiers from only positive and unlabeled data: theory ...

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16 Theoretical foundations (Liu et al 2002) (X, Y): X - input vector, Y {1, -1} - class label. f : classification function We rewrite the probability of error Pr[f(X) Y] = Pr[f(X) = 1 and Y = -1] + (1) Pr[f(X) = -1 and Y = 1] We have Pr[f(X) = 1 and Y = -1] = Pr[f(X) = 1] – Pr[f(X) = 1 and Y = 1] = Pr[f(X) = 1] – (Pr[Y = 1] – Pr[f(X) = -1 and Y = 1]). Plug this into (1), we obtain Pr[f(X) Y] = Pr[f(X) = 1] – Pr[Y = 1] (2) + 2Pr[f(X) = -1|Y = 1]Pr[Y = 1]
17 Theoretical foundations (cont) Pr[f(X) Y] = Pr[f(X) = 1] – Pr[Y = 1] (2) + 2Pr[f(X) = -1|Y = 1] Pr[Y = 1] Note that Pr[Y = 1] is constant. If we can hold Pr[f(X) = -1|Y = 1] small, then learning is approximately the same as minimizing Pr[f(X) = 1]. Holding Pr[f(X) = -1|Y = 1] small while minimizing Pr[f(X) = 1] is approximately the same as minimizing Pr u[f(X) = 1] while holding Pr P[f(X) = 1] ≥ r (where r is recall) if the set of positive examples P and the set of unlabeled examples U are large enough. Theorem 1 and Theorem 2 in [Liu et al 2002] state these formally in the noiseless case and in the noisy case.
Page 1 and 2: LEARNING CLASSIFIERS FROM ONLY POSI
Page 3 and 4: 3 Outline Statement of the Problem
Page 5 and 6: 5 Learning from Positive & Unlabele
Page 7 and 8: 7 Direct Marketing Company has dat
Page 9 and 10: 9 Related Works Denis (1998) shows
Page 11 and 12: 11 Theory: Define the Problem Let
Page 13 and 14: 13 Estimators of c = p(s=1|y=1) c
Page 15: 15 Application to Real-world Data
Page 19 and 20: 18 19 An Application: Text Classifi
Page 21 and 22: 21 U P Step 1 Step 2 positive negat
Page 23 and 24: 23 Step 1: The Spy technique
Page 25 and 26: 25 Step 1: Other methods 1-DNF met
Page 27 and 28: Step 2: EM 27
Page 29 and 30: 29 Can SVM be applied directly? Ca
Page 31 and 32: 31 Soft margin SVM To deal with ca
Page 33 and 34: 33 Biased SVM (noisy case) If we a
Page 35 and 36: 35 Empirical Evaluation Two-step s
Page 37 and 38: Table 1: Average F scores on Reuter
Page 39 and 40: 39 Negative Training Data can be Ha
Page 41 and 42: 41 Negative Training Data can be Ha
Page 43 and 44: 43 Some discussions All the curren

learning classifiers from only positive and unlabeled data: theory ...

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