I've never understood why ML folks insist on using precision, which depends on the base rate. Why do they not use probability of false alarm as in traditional detection theory?
It doesn't really depend on the base rate though. The idea is that a high precision model will have all the targets at one end of the distribution, regardless of base rate.
Precision is P(T|R), where R is the event that a thing is reported, and T is the event that the thing is true. By Bayes' rule, P(T|R) = P(T)/(1+Pfa/Pd). So in terms of the traditional statistics Pd and Pfa, it is proportional to the base rate.