Average precision computes the area under the recall-precision curve by the trapezoidal rule (or midpoint rule). However, we could also compute the F1 score for every threshold and then take the average. Is there a benefit in considering the area under the curve and not the average F1?
In other words: The average precision is given by $\int_{-\infty}^{\infty} f(x) dx \approx \sum_{x}\frac{f(x+1) + f(x)}{2}\Delta x$ where $x$ is the recall at all thresholds and $f(x)$ returns the precision.
I am wondering why we do not consider $\mathbb{E}[\text{F1}(X)] = \int_{-\infty}^{\infty} \text{F1}(x)p(x) dx \approx \frac{1}{T}\sum_x \text{F1}(x)$ where $x$ is the threshold.
