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Is there a way to project the F1-score on the precision-recall curve for a such binary classifier? Is there a relationship between the area under the precision-recall curve and F1-score?

Mathematically, I know that F1-score is the double of the multiplication of recall and precision divided by the summation of recall and precision. But I need to know if there is a clear representation of F1-score on the precision-recall curve?

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You can calculate $F_{1}$-score for each point on precision-recall plane and for each value of $F_{1}$-score you will have multiple satisfying points. Therefore, you can plot level curves of $F_{1}$-score - in the following example yellow lines represent level curves of $F_{1}$-score from $0.1$ to $0.9$: example

As you can see from the shape of level curves, there is no trivial relation between area under precision-recall curve and $F_{1}$-score. If you need such a relation, you may look into using presicion-recall-gain curves instead: summary with link to paper

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