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Suppose that a classifier A classifies a data-point $x$ in class LA1 with probability PA1 and class LA2 with probability PA2 (with LA1 != LA2 and PA1>PA2); and that a classifier B classifies a data-point $x$ in class LB1 with probability PB1 and class LB2 with probability PB2 (with LB1 != LB2 and PB1>PB2).

Note that LA1 may be equal to LB1 or to LB2, and that LB1 may be equal to LA1 or to LA2.

Question: given these results, how can one predict the class of $x$ ?

Example: - A classifies $x$ in class 1 (i.e. LA1) with probability 0.8, and in class 2 (i.e. LA2) with probability 0.1. - B classifies $x$ in class 1 (i.e. LB1) with probability 0.6 and in class 3 (i.e. LB2) with probability 0.2.

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  • $\begingroup$ I dont understand why LA1 and LA2 and LB1 and LB2 MAY be respectively equal. Why don't you know? $\endgroup$
    – jerad
    Nov 29, 2012 at 17:48
  • $\begingroup$ @jerad LA and LB are just notations, that means Label given by classifier A, or Label given by classifier B. I'm just saying that for a given point x, it will be classified in a given class by classifier A, and in another or same class by classifier B. Example: A classify x in class 1 (i.e. LA1) with probability 0.8, and in class 2 (i.e. LA2) with probability 0.1. B classify x in class 1 (i.e. LB1) with probability 0.6 and in class 3 (i.e. LB2) with probability 0.2. Here LA1 = LB1 = class 1. $\endgroup$
    – shn
    Nov 29, 2012 at 18:09
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    $\begingroup$ The question is ill stated. Without any measure of the accuracy/reliability of the classifiers (such as true positive fraction), we can't be sure that even if A classified x as LA1 100% of the time, x is ever an element of LA1. You should also specify whether these classes are mutually exclusive (like died/living) $\endgroup$
    – AdamO
    Jan 28, 2013 at 20:37
  • $\begingroup$ @AdamO what do you mean by "whether these classes are mutually exclusive (like died/living)" ? If you make a detailed answer about that would be awesome. $\endgroup$
    – shn
    Jan 29, 2013 at 10:20
  • $\begingroup$ Is it possible for x to simultaneously fall in both LA1 and LA2? $\endgroup$
    – AdamO
    Jan 29, 2013 at 15:50

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If you dont know whether or not LA1 = LB1 and LA2 = LB2 then you have no way of knowing if your classifiers are commensurate. If however you do know that the two classes are the same for both classifiers, then there's a broad class of methods known as Ensemble Learning available for combining the their outputs to improve classification performance.

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  • $\begingroup$ We know if LA1 = LB1 or not by just comparing LA1 with LB1 ! I've never said that we dont know whether or not two labels are equals ! $\endgroup$
    – shn
    Nov 29, 2012 at 18:30
  • $\begingroup$ Okay, I guess misinterpreted this comment: "Note that LA1 may be equal to LB1 or to LB2, and that LB1 may be equal to LA1 or to LA2." $\endgroup$
    – jerad
    Nov 29, 2012 at 18:41

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