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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 ...


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