# F-measure with $\beta > 1$

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 $$F_{inf} = recall$$.

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

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 $$\beta > 1$$.