I'm going to start by taking Metacrock's side here, for once. As a working numerical scientist, I feel like I have a bit of weight to throw around on this subject, too
I think Metacrock is right in that the claim "ECREE" with regards to the existence of God
has little or nothing to do with Bayes' Theorem. The reason for this is actually clear when you look at the examples in this thread, cocaine use and cancer. If you give a random US citizen a cocaine test, the low incidence of cocaine use means that you'll get a lot of false positives. Cocaine use is an extraordinary claim, and you need a very accurate test (or several tests) in order to convincingly show cocaine use. If you go to a crackhouse, you'll probably be able to tell pretty reliably who is high at the moment without any drug test. Similarly, any particular cancer diagnosis is a rare and extraordinary claim, and the tests have limited power. But if one test is positive, you're in a cancer crackhouse, and more testing will be much more reliable. Bottom line: You need to know the prior odds of an outcome to know the reliability of a test
We don't know the prior odds of the existence of God. We cannot apply Bayes' Theorem to this question, because we don't have any data about the existence of God, period. I think this is the heart of Metacrock's point, and in this he is correct.
A caveat: I think the existence of God is an extraordinary claim, and my intuition says that such claims require extra convincing evidence. I agree with ECREE with regards to the existence of God, but will not use Bayes' Theorem as support for this opinion.
Finally, there's the whole miracles issue, too. Here, we do
know the prior odds of spontaneous recovery from various illnesses, or we can at least calculate a reliable estimate. Therefore, when someone prays and is healed, we can
apply Bayes' Theorem and we do
have a mathematical basis for ECREE in this context. Every argument for miracles (at Lourdes or elsewhere) that I've seen has failed to address this fundamental problem. Can Metacrock dig up a counterexample?