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

la1noxz 2022-09-03 Answered
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
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Answers (1)

Demarion Thornton
Answered 2022-09-04 Author has 11 answers
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).
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