I talked to an Anglican priest at a drinks reception. He mentioned that theologians have a debate about the applicability of what he called “Bayes Theory” (meaning, I think, the Bayesian interpretation of this famous Theorem) to the existence of God. I haven’t verified whether this debate is, in fact, taking place, but it did remind me of the following Bayesian paradox.
Suppose you are a risk neutral (without any loss of generality) bidder. I show you two boxes, call them L and R, and tell you that the prize inside one is twice as big as the prize inside the other. You open a box, let’s say it’s box L. You find £10 inside. I tell you that you can either switch to box R or stick with £10 from box L.
What do you do?
Well, you’re a risk neutral bidder. With probability one half box R contains £20 and with probability one half box R contains £5. In expectation it contains £12.50, which is more that £10 you’re getting from box L. So you choose to switch.
But there is something strange going on.
Suppose, you open box R/L first and find X in it. Then box L/R must contain X+X/4, which is higher than X. So you will always want to pick box L/R.
But if you know that, regardless of the outcome in your box, you will want to switch, why don’t you just pick the OTHER box in the first place? Using another example, if you take an umbrella when it rains and you take an umbrella when it shines (and it can only rains or shine), then you always take an umbrella.
Bayes’ Theorem doesn’t work in this case because you had an improper prior (prior which integrates to infinity over its range e.g. uniform prior over all real numbers): you were only told that one cup contains twice as much as the other; either of them could contain any amount.
Belief in the existence of God is an improper prior. The reason is that no one know exactly what God is like. There is essentially an infinite number of possibilities of what God could be (due to omnipotence, omniscience etc.), all of which must have a positive probability. So using Bayesian updating in the debate about God is simply bad logic.
So, if you see a miracle, don’t update you belief about the existence of God using Bayes’ Rule.