Having some familiarity with this theorem from a mathematical perspective, I think I can shed some light on how it might or might not apply to religious beliefs. The basic statement of the theorem is:
P(A|B) = P(B|A) x P(A) / P(B)
I will explain this and try to apply it to a religious belief all at the same time.
Let P(A) stand for a person’s assessment of the probability that God exists. Say that probability was at the time the person woke up today. (“A” is the statement that God exists and P means “the probability of”.)
Then suppose that person observed a genuine miracle today. Let P(B) stand for the probability that such that miracles are witnessed any given day. Depending on what kind of miracle this is, P(B) might be higher or lower. For example, many consider the birth of a baby to be a miracle. This “miracle” has a very high probability of occurring. On the other hand, feeding 5000 people with a few fish and loaves is very rare indeed. Let us assume we are talking about the later type of miracles - those with a very low empirical probability of happening on any given day.
Now for those strange vertical lines in the formula ("|"). That means “given that…”. So P(A|B) means the person’s view of the probability that God exists given that he witnessed a miracle today. Intuitively we would expect P(A|B) to be higher than P(A). That is, someone’s belief that God exists should be helped out by witnessing a miracle. But let’s see how Bayes Theorem says that.
The only term in Bayes Theorem we have not talked about yet is P(B|A), that is, the probability that a miracle is witnessed, given that God does exist. Well, if God exists, and central to God’s nature is occasionally interacting with Man through miracles, we would expect the probability of witnessing a miracle to be a little higher if God does exist than if He doesn’t. In fact, someone who does not believe that God exists most likely puts the probability of witnessing a miracle at zero. So we would expect P(B|A) to be significantly higher than P(B). Any by looking at where these terms appear in Bayes Theorem, we see that the ratio of these two terms is exactly the ratio by which the probability that God exists goes up in someone’s mind after witnessing a miracle.
So, how applicable is this to religious belief? Not very. If you read the Wikipedia example of “Bayes Theorem” you will see where it really does help. In that case we wanted to know the probability that a certain person was a woman, given the fact that the person had long hair. This seems difficult to know directly. But Bayes Theorem helps you find this using the probability that a person has long hair, given that the person is a woman. That statistic is supposedly easier to look up than the first one. So Bayes helps out there.
In our example, we want to know how much the belief that God exists is affected by witnessing a miracle. But to use Bayes Theorem we would have to first know how much the probability that a miracle is witnessed is affected by the existence of God, and that is no easier to know, so Bayes does not help us here, and that is why I think Bayes Theorem has very little applicability to questions about religious beliefs.
You will notice that I took care to say “a person’s view of the probability” rather than just “the probability”. The reason for that was to emphasize that there is no such thing as an absolute probability. Probability depends on knowledge. Different people have different knowledge. Therefore probability for one person is not the same as probability for another. For example someone who knows that the dice are loaded has one probability. Someone who does not know the dice are loaded calculates a different probability.
