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Is the F1 score sensitive or indifferent to the threshold (for defining positive or negative)?

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Yes, it is. The formula:

$$F_1 = \frac{2TP}{2TP + FP + FN}$$

There is no symmetry. You have only $TP$ in the nominator which means that $F_1=0$ if $threshold=1$.

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Yes it is.

For an evaluation measure independent from the threshold, look at the Area Under the Curve (AUC).

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