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It is common to define the F-measure as a function of precision and recall, as mentioned in [1]:

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

However I came across some other cases, another definition is used [2] (without weights):

$F = H(sensitivity, 1- specificity)$

Where H is harmonic mean.

Reference:

  1. F - measure derivation (harmonic mean of precision and recall)

  2. https://link.springer.com/chapter/10.1007/978-3-540-68947-8_133.

  3. https://stackoverflow.com/a/52892413/2243842
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The one is general formula the other you get for Beta=1:

The beta value greater than 1 means we want our model to pay more attention to the model Recall as compared to Precision. On the other, a value of less than 1 puts more emphasis on Precision. So you just want to generalise, and punish certain mistakes more.

So to conclude: Correct in mathematical sense is always to generalise and derive special cases, in that sense the first one is preferable since setting beta to one you get the 'standard' F-1-harmonic-mean-formula.

http://scikit-learn.org/stable/modules/generated/sklearn.metrics.fbeta_score.html

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Yes, as they are effectively synonyms of one another. See for instance this link

If you pay attention, the first formula is the (weighted) harmonic mean of the recall and precision.

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