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Math Help - Classification problem with 0-1 loss

  1. #1
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    Classification problem with 0-1 loss

    Given training data  (X_1,Y_1),...,(X_n,Y_n) where  X_i \in \mathbb {R}^d,Y_i \in \{ 0,1 \} under 0-1 loss.

    Let p=P(Y=1) , Let R_{Bayes} denote the Bayes risk.

    a) Show that R_{Bayes} \leq min(p,1-p)

    b) Show that equality holds above if X and Y are independent.

    c) Exhibit a join distribution where X and Y are not independent but R_{Bayes} = min(p,1-p)

    Solution so far:

    a) Suppose that  f_{Bayes} is the Bayes learner, then:

    R_{Bayes} = min EL[Y_k,f(X_k)]= min \sum _{k=1}^n P[f(X_k) \neq Y_k ]
     \leq \sum _{k=1}^n P(1 \neq Y_k) = 1-P(Y=1)=1-p

    I konw that mine must be wrong because it doesn't help me with the next two problems, please help thanks!
    Last edited by tttcomrader; October 10th 2012 at 10:12 AM.
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  2. #2
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    Re: Classification problem with 0-1 loss

    Hey tttcomrader.

    Recall that p >= 0 so the minimum of (p,1-p) will always be less than 1 - p.

    As a consideration of all possibilities let p < 1 - p this means p < 0.5. Now consider 1 - p < p This means p > 0.5.

    So if 1 - p is not the minimum then p is the minimum, but this is always less than 1 - p so the inequality still holds.
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