The mammogram problem is different because you’re only trying to determine whether a specific woman has cancer, not whether cancer exists at all as a phenomenon. If Bob was abducted by aliens, it implies that alien abduction is real, but the converse isn’t true. You either need to do two separate Bayesian updates (what’s the probability that Bob was abducted given his experience, and then what’s the probability of aliens given the new probability that Bob was abducted), or you need a joint distribution covering all possibilities (Bob not abducted, aliens not real; Bob not abducted, aliens real; Bob abducted, aliens real).
The mammogram problem is different because you’re only trying to determine whether a specific woman has cancer, not whether cancer exists at all as a phenomenon. If Bob was abducted by aliens, it implies that alien abduction is real, but the converse isn’t true. You either need to do two separate Bayesian updates (what’s the probability that Bob was abducted given his experience, and then what’s the probability of aliens given the new probability that Bob was abducted), or you need a joint distribution covering all possibilities (Bob not abducted, aliens not real; Bob not abducted, aliens real; Bob abducted, aliens real).