Suppose that we are training a nave Bayes classifier and a logistic regression classifier: f f :

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Suppose that we are training a naïve Bayes classifier and a logistic regression classifier: ff : XYXY, which maps a dd-dimensional real-valued feature vector XRdXRd to a binary class label Y{0,1}Y{0,1}. In the naïve Bayes classifier, we assume that all XiXi where i=1,,ni=1,,n are conditionally independent given the class label YY and the class prior P(Y)P(Y) follow the Bernoulli distribution with P(Y=1)=θP(Y=1)=θ. Now, prove the equivalence of logistic regression and naïve Bayes under these two assumptions.

a. For each XiXi, we assume it is drawn from the Gaussian distribution P(XiY=k)P(XiY=k) N(μik,σik)N(μik,σik) where k=0,1k=0,1. We also assume that σi0=σi1=σiσi0=σi1=σi.

b. For each XiXi, we assume it is drawn from the Bernoulli distribution P(Xi=1Y=k)=pkP(Xi=1Y=k)=pk where k=0,1k=0,1.

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Data Mining Concepts And Techniques

ISBN: 9780128117613

4th Edition

Authors: Jiawei Han, Jian Pei, Hanghang Tong

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