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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:

P(S|M)=P(M|S)xP(S)/P(M)

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?

Updated Bayes'

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Not really.

You have

P(Miracle occurred) = 0.8

P(supernatural exists | Miracle occurred) = 1 (because you said a miracle can only happen if the supernatural exists)

P(supernatural exists | No Miracle occurred) = 0.25 (that no miracle occured doesn't give new information, I'd argue? Nothing interesting happened, so you take your prior assumption of 25% that the supernatural exists)

What happens is that your new prior assumption about the existence of the supernatural becomes something higher in light of the new evidence:

P(supernatural exists) = P(supernatural exists | Miracle occured) * P (Miracle occurred) + P(supernatural exists | No miracle occured) * P(No miracle occured) = 1 * 0.8 + 0.25 * 0.2 = 0.85

So, if there is a 80% chance a miracle happened, then your prior estimate of how likely it is that the supernatural changes - to 85%.

What's interesting about this calculation is that I don't see how the prior could become less? I am assuming because nothing really can falsify the supernatural I guess?

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  • What if the miracle and the supernatural are not independent? Take, for example, the Resurrection and Christianity. P(C|R) is probably high (0.99) and let's just say we have good evidence for R so P(R) is 0.8. The problem for your formula is that P(C|~R) is 0 (Christianity is false if no resurrection), and that isn't the same thing as your prior probability for C before considering the evidence for R. How would you adjust the calculation in this case?
    – natojato
    Commented Sep 21, 2020 at 15:16
  • @natojato Well, the evidence is the evidence. If, somehow, you knew R is true, then that becomes sure knowledge and P(R) = 1. The only thing that can change is the belief about the unobserved variables (C). So, in your example, then P(C) is 0.99*0.8 + 0 * 0.2 which I don't bother to put in a calculator but which is slightly lower than 0.8. Btw. of course the miracle and the supernatural are not independent, otherwise the miracle wouldn't give information regarding the supernatural.
    – kutschkem
    Commented Sep 22, 2020 at 6:07

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