[derbOac]

I've worked clinically and taught this in professional training programs. It's a classical clinical decision making problem.

The solution is fairly straightforward I think: require the test companies to report the posterior probability of the illness given the test result. Probably with some kind of uncertainty interval.

[rscho]

The posterior depends on the prior. So not very feasible in practice.

[derbOac]

Generally speaking in clinical settings there are rough estimates of base rates of diseases available, either through biological theory (eg, mutation rates) or empirical studies or both. Not always to be sure, but often enough to mitigate against some of the problems discussed.