The Astrometry.net system works by generating likely hypotheses about each input image's astrometric calibration, and then testing those hypotheses with a verification process, which we have cast as a well-posed statistics problem in the formalism of Bayesian hypothesis testing. If we have this right, then the false-positive rate (which we want to have vanishingly small) is an explicit parameter in our system, not implicitly set by ad-hoc cuts or thresholds.
Unfortunately, our false-positive rate is orders of magnitude higher than it should be. This is because our well-posed statistics problem is an approximation to the problem we really need to solve. In detail, to do it right, we would need a generative model of the USNO-B Catalog (our basis for truth), the sky, and all sources of astronomical imaging. We are far from this, so we have to make approximations. Today Lang and I spent all day trying to improve our approximation. We didn't finish, but we learned a lot.