Joe and Brandon seem to think if worked in 13th c Hungary so I am sure they would support it as well.

There are various formulas available for Bayesian calculations, but some not-very-advanced algebra shows them to be all mathematically equivalent. My preferred formula is bit more complicated than the others, but it has the advantage that all relevant assumptions must be explicitly accounted for. When used properly, it doesn’t allow any hidden assumptions to slip in. Here it is:

P(H|E) = P(H)*P(E|H) / [P(H)*P(E|H) + P(~H)*P(E|~H)].

Even this is a slightly abbreviated version. Every term is assumed to terminate with a “|B” or “.B”, meaning “on background knowledge.” Thus the numerator, written in full, would be

P(H|B)*P(E|H.B).

In other words, we’re not supposed to ignore, or to treat as irrelevant, everything else we think we know about how the world operates. Our prior probability is to be estimated, and evidence evaluated, according to everything we knew or thought we knew before we had a look at this evidence.

P(H|E), which I will call the consequent probability, is the probability that the hypothesis is true given the evidence. This is what we’re trying to calculate. The formula presents it as a function of four variables, but really only three because one of them is determined by another. P(H), the prior probability, is the probability we should have assigned to the hypothesis before discovering the evidence. P(~H) is the prior probability that the hypothesis is false, and this has to be 1 – P(H). This leaves just two more variables to estimate. P(E|H) is the probability of the evidence obtaining given a true hypothesis, and P(E|~H) is the probability of the evidence obtaining given a false hypothesis.

We can schematize this a bit to get a better idea of the relationships among the three estimated variables. Changing some letters and eliminating several keystrokes, we get

P = AB / (AB + CD).

But then we recall that C = 1 – A, and now we have

P = AB / [AB + (1 - A)D].

A mathematical consequence of this is that if B = D, then P = A. In other words, whenever the evidence is just as likely whether the hypothesis is true or false, then the consequent probability equals the prior probability: P(H|E) = P(H). In this case, the evidence is epistemically irrelevant. Whatever justification we had for believing or rejecting the hypothesis before seeing the evidence, we have neither more nor less justification afterward.

Whether the consequent probability is more or less than the prior depends strictly on the difference between B and D, i.e. the value of P(E|H) – P(E|~H). A positive value produces P(H|E) > P(H), a negative value yields P(H|E) < P(H), and larger values produce a larger difference between prior and consequent. (A negative value just means the evidence is against the hypothesis: It increases whatever justification we might have had for doubting it.)

It is also helps to consider some extreme cases. If we assign a prior probability of zero, then by mathematical necessity, the consequent probability will be zero. If our minds are made up that the hypothesis is impossible, then no evidence will convince us, and according to Bayes, it should not convince us. With a prior of zero, evidence is irrelevant. This means that Bayes forces us, if we claim that good enough evidence will change our minds, to admit that the hypothesis is not an impossibility: Its prior probability can’t be zero. Likewise, if the prior probability is 1, then the consequent probability is also 1, regardless of the evidence, because then P(~H) = 0, meaning the hypothesis cannot possibly be false.

With that background, let’s consider some numbers for the current problem. Our evidence, E, is certain documents reporting that some innocent people underwent trial by ordeal conducted by certain priests and, on subsequent examination, were found to be uninjured, resulting in their acquittal of the charges against them. The hypothesis, H, is that they were protected from injury by divine intervention, i.e. a miracle occurred. The contrary hypothesis, ~H, is that some natural occurrence prevented the defendants from being detectably injured. A thorough analysis would have to separately evaluate all possible naturalistic alternatives to divine intervention, but for simplicity we’ll assume only one is worth considering: priestly complicity in a sham ordeal, i.e. the trials were faked by those responsible for conducting them, presumably because those priests were antecedently convinced that those accused were innocent. We need estimates for: P(H), the prior probability that a miracle occurred; P(E|H), the probability that if the miracle had occurred, we would have this evidence; and P(E|~H), the probability that we would have the same evidence if something other than a miracle had occurred.

An atheist is going to think that P(H) is, if not zero, so close as to make no difference, but let’s try to really hard to avoid begging the question. Fairness might suggest that we ask a theist what he thinks the prior probability is, but it might be hard to find a theist who will give us a straight answer. For starters, though, let’s suppose we don’t have any better reason to doubt the report than to believe it. Almost by definition, then, this gives us a prior probability of 0.5. This is not realistic to any atheist, but bear with me.

What is the probability that we would have this evidence if the miracle had really occurred? We need first the probability that a record would have been produced. We can assume for the present discussion that they almost always were, and so let’s go with 0.99. But those records didn’t always survive, and the probability we’re looking for is the probability that we would have those records now, 700 years later. What percentage of church records from 13th century Hungary are still extant? I have no idea, but let’s guess half of them. As it turns out in this case, any guess will do, but when you’re doing Bayes, you have to ask these questions. The probabilities multiply, so we get P(E|H) = 0.5, rounding to one significant figure.

And what if there was no real miracle? Would that have made a difference in the probability that the records would have been produced, and would have survived into modern times so that we’d know about them? The authenticity of the ordeal, i.e. whether or not a miracle really happened, should have made no difference to the survival of the documents. Once they were produced, subsequent custodians would have had no reason to treat them differently from records of authentic ordeals. And ex hypothesi, the trial was to be conducted the same as any authentic trial, and so the record-producing process would have been identical for the sake of appearances. Thus we get P(E|~H) = P(E|H), and so we are just as likely to have this evidence whether the ordeal was authentic or faked.

This means, as noted above, that regardless of our estimate for P(H), we get the same value for P(H|E). Whatever antecedent reason we had for believing that a miracle happened on this occasion, we have no better reason for believing that it happened just because we have these documents saying that it happened.

Thus, to whatever extent I am justified in thinking that a miracle is more likely to be faked than to have actually occurred, to that extent I am justified in believing that this particular miracle was faked. Maybe I’m not justified at all. Maybe I should believe that most of the time, when someone says a miracle happened, it really did happen. But that is a separate issue and must be supported by its own argument. In this particular case, the evidence is just as likely to have existed whether or not the miracle really happened, and so it does nothing to enhance the credibility of this particular miracle, not matter what antecedent credibility I’m entitled to give it.

The OP argued that the probability of a fake was extremely low, but we’re considering only two hypotheses, H and ~H, and the sum their assigned probabilities must be 1.0. In that case, an argument for P(~H) = 0.1, let us say, must entail P(H) = 0.9. Such a high value cannot be defended with the claim that in this particular case, P(~H) must be very low. The value assigned to P(H) must be determined before the evidence is examined, or at least on the pretense that the evidence has never been examined, and only that value can determine P(~H).

Agreed.

and the dogmatic naturalism of Leeson, Levitt, and Dubner.

The dogmatism of some naturalists tells us nothing more about the validity of naturalism than the dogmatism of some theists tells us about the validity of believing in God.

we'd say he/she or 'it' doesn't control it but sustain it and intervenes sometimes via miracles and always via 'his' grace via the holy spirit, the idea of grace being something gratuitous something not deserved in the strict sense as in what happened during the fall when God 'took away' the graces that we were not by our nature entitled to.

i used to think of God as someone really lonely who wanted company.

It's not inconceivable to me that we live in a universe controlled by a being who is indifferent to our fate, or perhaps curious about it in the sense that a child is curious about flies with their wings pulled off.

Articles like this one help to grease the path, in a way.

"and these natures possess the ability to act directly"

would you say independently as well?

I'm not really sure. Most of what he wrote was substantive and fair. Apparently he was a little too harsh on Joe and Baron's latest piece. Here's a link to Phil's comment:

http://outshine-the-sun.blogspot.com/2015/10/estranged-notions-trial-by-fire.html#comment-2320081999