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Prove that the F-measure of any binary classifier is $\leq\dfrac{precision+recall}{2}$

Let $P=precision$ and $R=recall$

I have that the F measure $=\dfrac{2PR}{P+R}$

Note that $precision=\dfrac{tp}{tp+fp}$ and $recall=\dfrac{tp}{tp + fn}$

where tp = true positive, fp = false positive, tn = true negative and fn = false negative

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$F$-measure is merely a harmonic mean of precision and recall. It is generally known that harmonic mean of nonnegative numbers is not more than their arithmetic mean. In particular, it is true for two numbers: precision and recall.

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I'll help get you started. Note that $0\le P \le 1$; $0\le R \le 1$.

$\dfrac{2PR}{P+R} \le \dfrac{P+R}{2}$

$4PR \le (P+R)^2$

$4PR \le P^2+R^2+2PR$

$0 \le P^2+R^2-2PR$

$0 \le (P-R)^2$

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. tp=fn).

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