Suppose you have really good evidence for a miracle. Let's say that given the evidence, the probability of the miracle having occurred is about 80%.
Now, you also know that miracles can only occur if the supernatural exists. Moreover, the prior probability of the supernatural, according to your best estimates, is very low—say, 25%.
Does Bayes' Theorem in this case justify rejecting the supernatural despite the compelling evidence for the miracle in question?
My thought is that the probability would be set up in this way:
Let's suppose that
P(M|S) is very high for the sake of argument (or just 1). Does the prior probability of the supernatural that you've established outweigh the incorporation of new data (the well-supported miracle) such that belief in the supernatural is still not warranted?
The above use of Bayes' Theorem definitely seems wrong. What would be the correct way to set up the probabilities under Bayes' Theorem in this case? I'm assuming something like the following: that the prior probability of the supernatural excluding the miracle in question is 25%, that the probability of the miracle having occurred given the evidence is 80%, and that the probability of the miracle given the supernatural is very high, say 99%.
EDIT: Here is an updated version of how I'm guessing you might use Bayes' Theorem in this case. E is the evidence that we have available and S is supernaturalism is true. Does this work?