Why f beta score define beta like that?

This is the F beta score:
Fβ=(1+β2)precisionrecall(β2precision)+recall

The Wikipedia article states that Fβ "measures the effectiveness of retrieval with respect to a user who attaches β times as much importance to recall as precision".

I did not get the idea. Why define β like that? Can I define Fβ like this:

Fβ=(1+β)precisionrecall(βprecision)+recall

And how to show β times as much importance?

Answer

Letting β be the weight in the first definition you provide and ˜β the weight in the second, the two definitions are equivalent when you set ˜β=β2, so these two definitions represent only notational differences in the definition of the Fβ score. I have seen it defined both the first way (e.g. on the wikipedia page) and the second (e.g. here).

The F1 measure is obtained by taking the harmonic mean of precision and recall, namely the reciprocal of the average of the reciprocal of precision and the reciprocal of recall:

F1=1121precision+121recall=2precisionrecallprecision+recall

Instead of using weights in the denominator that are equal and sum to 1 (12 for recall and 12 for precision), we might instead assign weights that still sum to 1 but for which the weight on recall is β times as large as the weight on precision (ββ+1 for recall and 1β+1 for precision). This yields your second definition of the Fβ score:

Fβ=11β+11precision+ββ+11recall=(1+β)precisionrecallβprecision+recall

Again, if we had used β2 instead of β here we would have arrived at your first definition, so the differences between the two definitions are just notational.

Attribution
Source : Link , Question Author : tidy , Answer Author : josliber

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