Consider the F-measure $F_{\beta}$. I know that with $\beta < 1$ precision is given more importance, ending with $F_{0} = precision$. With $\beta > 1$ means recall gets the upper hand, with the other extreme at $F_{inf} = recall$.

\begin{equation} F_{\beta} = (1+ \beta^2) \cdot \frac{Precision \cdot Recall}{(\beta^2 \cdot Precision) + Recall} \end{equation}

I understand why for $\beta < 1$ precision is given more importance, but I don't understand for $\beta > 1$


Define $\gamma = 1/\beta$, $P=\textrm{precision}$, and $R=\textrm{recall}$.

Then $$F_{\beta} = (1 + 1/\gamma^2)\cdot \frac{PR}{P/\gamma^2 + R} = (\gamma^2 + 1) \cdot \frac{PR}{\gamma^2R + P}$$, which is the $F_\gamma$ measure with the roles of $P$ and $R$ switched.

So the same reasoning that shows precision is preferred for $\beta < 1$ can be used to show that recall is preferred when $\gamma = 1/\beta < 1$, i.e. when $\beta > 1$.


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