# Why to downweight Precision in nominator of F-beta when I actually want to upweight Precision?

F-beta score's formula calculates like this:

$$F_{\beta} = (1+ \beta^2) \frac{PR}{\beta^2P + R}$$

However, according to some sources, in case I want to add more emphasis to Precision I should use beta < 1, and complementary, in case I want to add less emphasis to Precision than Recall, I should use beta > 1. This somehow seems me contrary to the purpose; why to downweight beta in the denominator when I actually want to assign more weight to Precision in the caluculation?

*Bonus question: either way the answer is for the question above, is there any particular formula to define beta if I want to assign different weights to Precision and Recall? Maybe in proportion to the cost difference in false negatives and false positives? Or simply just use beta=0.5 and beta=2 as a rule of thumb?

• I made some changes in the $F_\beta$ formula mentioned as it was misleading (missing parenthesis as well as having an addition when it should have been a multiplication). Commented Aug 15, 2020 at 7:49

By setting $$\beta$$ to $$0$$ we are effectively getting "just Precision" that is because the Recall multipliers cancel each other out and we are left with Precision only. By using a lower $$\beta$$ we do not down-weight Precision in any way, we are up-weighting as we allow it to dominate the fraction's final value through the numerator.
If we already know the cost of our FP/FP we can directly use them. $$\beta$$ itself reflects our trade-off between Recall and Precision in the sense of $$\beta=\frac{\text{Recall}}{\text{Precision}}$$; therefore the values of $$0.5$$ and $$2$$ merely reflect that we hypotheises that: "we value Precision twice as much as Recall" (for $$\beta=0.5$$) or "we value Recall twice as much as Precision" (for $$\beta=2$$). Obviously if we value them equally $$\beta=1$$ and we get our standard $$F_1$$ score.