Mark Frank: You are quite right about abusing Bayes' theorem. Bayes also says we have to take into account the a priori probability of a natural or supernatural explanation. You have only considered the likelihood side of the equation. This is the famous gremlins in the attic paradox. Given gremlins in the attic it is very likely that they caused a noise. But that doesn't mean that Gremlins in the attic are a likely explanation of a noise in the attic.
Quite correct. However, we don't have any good way of measuring what the "prior" side of Bayes' theorem ought to be, and I presume that it can differ amongst reasonable people. That was the whole point of the anti-frequentism argument that I gave a few posts ago. I don't think there's a good way to go back into one's belief system and discount everything that is the product of a non-truth-conducive belief-producing mechanism, a la the Outsider Test. This is partly because it is not transparent to any of us why we think as we do.