Prove that the F-measure of any binary classifier is ≤ p r e c i s i o n + r e c a l l 2 Let P = p r e c i s i o n and R = r e c a l l F measure = 2 P R P + R Note that p r e c i s i o n = t p t p + f p and r e c a l l = t p t p + f n where t p = true positive, f p = false positive, t n = true negative and f n = false negative

Question

Prove that the   F-measure of any binary classifier is   ≤                    p        r        e        c        i        s        i        o        n        +        r        e        c        a        l        l            2      Let   P  =  p  r  e  c  i  s  i  o  n and   R  =  r  e  c  a  l  l  F measure   =                    2        P        R                    P        +        R            Note that   p  r  e  c  i  s  i  o  n  =                    t        p                    t        p        +        f        p             and   r  e  c  a  l  l  =                    t        p                    t        p        +        f        n            where   t  p  = true positive,   f  p  = false positive,   t  n  = true negative and   f  n  = false negative

Demarion Thornton · Accepted Answer

Note that   0  ≤  P  ≤  1;   0  ≤  R  ≤  1                    2        P        R                    P        +        R              ≤                    P        +        R            2        4  P  R  ≤  (  P  +  R      )    2    4  P  R  ≤      P    2    +      R    2    +  2  P  R  0  ≤      P    2    +      R    2    −  2  P  R  0  ≤  (  P  −  R      )    2  From here you can plug in your definitions for   P and   R, and show that the result is at minimum   0 in the required range (more specifically when   P  =  R, i.e.   t  p  =  f  n).

Prove that the F-measure of any binary classifier is <= (precision+recal)/2. Let P=precision and R=recall. F measure =(2PR)/(P+R). Note that precision=(tp)/(tp+fp) and recall=(tp)/(tp+fn) where tp = true positive, fp = false positive, tn = true negative and fn = false negative

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