Transcription of Logistic Regression
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Logistic Regression
Logistic RegressionLogistic RegressionJia LiDepartment of StatisticsThe Pennsylvania State UniversityEmail: jialiJia Li jialiLogistic RegressionLogistic RegressionPreserve linear classification the Bayes rule: G(x) = arg maxkPr(G=k|X=x).IDecision boundary between classkandlis determined by theequation:Pr(G=k|X=x) =Pr(G=l|X=x).IDivide both sides byPr(G=l|X=x) and take log. Theabove equation is equivalent tologPr(G=k|X=x)Pr(G=l|X=x)= Li jialiLogistic RegressionISince we enforce linear boundary, we can assumelogPr(G=k|X=x)Pr(G=l|X=x)=a(k,l)0+ p j=1a(k,l) Logistic Regression , there are restrictive relations betweena(k,l)for different pairs of (k,l).Jia Li jialiLogistic RegressionAssumptionslogPr(G= 1|X=x)Pr(G=K|X=x)= 10+ T1xlogPr(G= 2|X=x)Pr(G=K|X=x)= 20+ (G=K 1|X=x)Pr(G=K|X=x)= (K 1)0+ TK 1xJia Li jialiLogistic RegressionIFor any pair (k,l):logPr(G=k|X=x)Pr(G=l|X=x)= k0 l0+ ( k l) of parameters: (K 1)(p+ 1).
Logistic Regression Fitting Logistic Regression Models I Criteria: find parameters that maximize the conditional likelihood of G given X using the training data. I Denote p k(x i;θ) = Pr(G = k |X = x i;θ). I Given the first input x 1, the posterior probability of its class being g 1 is Pr(G = g 1 |X = x 1). I Since samples in the training data set are independent, the
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