Explanation of the F beta formula
That's a great question, because on its face it seems like the weight should be $\beta$ alone, and, it should be in front of recall. The answer is in the text from which that reference is taken, on page 133: http://www.dcs.gla.ac.uk/Keith/pdf/Chapter7.pdf

The definition is designed to make the metric indifferent to a change in precision or recall when $P/R = \beta$. That is, $F_\beta$ increases by the same amount when either precision or recall increases, at the point where precision is already $\beta$ times bigger than recall.

The definition does indeed weight recall more highly as you can verify. Honestly on re-reading the text above, I was confused, because I don't see how it makes sense to think of "equilibrium" as the point where precision is much bigger, if recall matters more.

I plugged in the formula to Wolfram Alpha, and:


Hm. These are only equal if $R/P = \beta$! I think the paper may have misstated this then, or else I've really missed something. It's a formula whose value changes at the same rate with respect to precision or recall, when recall is already $\beta$ times larger, and in that sense it corresponds to treating recall as $\beta$ time more important.