Your conclusion doesn’t seem to follow from your proof
You claim that the optimal threshold should be 10%. Assuming this is true, this implies that, for example, Alice should publish any poem which has at least a 10% probability of being good. This doesn’t imply that 90% of her poems are bad, actually this seems to imply that strictly less than 10% of her poems are bad. Looking at the second graphe, for example (the one with the four rectangles), you chose a 10% threshold, but about 15% of the poems are good
You are correct that precision is (in general) higher than the threshold. So if Alice publishes anything with at least 10% likelihood of being good, then more than 10% of her poems will be good. Whereas, if Alice aims for a precision of 10% then her promising threshold will be less than 10%.
Unless I’ve made a typo somewhere (and please let me know if I have), I don’t claim the optimal promising threshold τ⋆ = 10%. You can see in Graph 5 that I propose a promising threshold of 3.5%, which gives a precision of 10%.
I’ll edit the article to dispel any confusion. I was wary of giving exact values for the promising threshold, because τ⋆=3.5 yields 10% precision only for these graphs, which are of course invented for illustrative purposes.
Your conclusion doesn’t seem to follow from your proof
You claim that the optimal threshold should be 10%. Assuming this is true, this implies that, for example, Alice should publish any poem which has at least a 10% probability of being good. This doesn’t imply that 90% of her poems are bad, actually this seems to imply that strictly less than 10% of her poems are bad. Looking at the second graphe, for example (the one with the four rectangles), you chose a 10% threshold, but about 15% of the poems are good
You are correct that precision is (in general) higher than the threshold. So if Alice publishes anything with at least 10% likelihood of being good, then more than 10% of her poems will be good. Whereas, if Alice aims for a precision of 10% then her promising threshold will be less than 10%.
Unless I’ve made a typo somewhere (and please let me know if I have), I don’t claim the optimal promising threshold τ⋆ = 10%. You can see in Graph 5 that I propose a promising threshold of 3.5%, which gives a precision of 10%.
I’ll edit the article to dispel any confusion. I was wary of giving exact values for the promising threshold, because τ⋆=3.5 yields 10% precision only for these graphs, which are of course invented for illustrative purposes.
You’re right, my mistake