I understand that the current atheist meme on this, which shows a rather striking lack of understanding of probability, is to say that if one does not argue for a particular prior probability for some proposition, one literally can say nothing meaningful about the confirmation provided by evidence beyond the statement that there is some confirmation or other.
This is flatly false, as both the second of the quotations above from the paper and my rather detailed explanation to Luke M. show.
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Note this quotation, transcribed from the podcast of Luke's interview with Lydia:
In Bayesian terms, what we do in the article is that we try to separate what ... one might call ... the indirect evidence, which would be relevant to that prior probability, from the direct evidence. So the things that would be relevant to the prior probability would be things like evidence for and against theism, for example, evidence for and against the existence specifically of the God of Israel, the God of the Jews, or other evidence prior to Jesus' purported resurrection regarding who Jesus was, and so forth. That would all be relevant to the prior. And what we focus on in the article instead is what we might call the direct evidence, the evidence that supposedly tells you what happened, what you might call reports...You might call it evidence after the fact. So what we focus on are the testimony of the disciples and of certain women that said that they saw and spoke with Jesus, the evidence of the disciples' willingness to die for that testimony, and the evidence of the conversion of the Apostle Paul. And what we try to do is we use a modeling device known as a Bayes factor. Roughly speaking, a Bayes factor tries to model, number one, which way the evidence is pointing and, number two, how strongly the evidence is pointing that way. And what you're trying to do at that point is you're trying to look at explanatory resources of the hypothesis, in this case, the resurrection, and the negation of the hypothesis. How well does each of these explain the evidence, and is there a big difference between how well each of these explains the evidence? I should clarify that when I say a difference, too, it's actually a ratio ... it's very important that you measure it by the ratio, not by the difference. But you need to look at those two hypotheses and see which one gives you a better expectation of that evidence and how much better is that expectation. So we estimate Bayes factors for these various separate pieces of evidence, then we argue for the legitimacy of multiplying these Bayes factors, because that gives you a lot of kick, and you have to discuss that issue, and we do, of independence, and whether it's legitimate to multiply them in order to combine those Bayes factors, and that ends up with this very high, high combined Bayes factor in our estimate ... And so what we estimate is that you could have this overwhelmingly low prior probability (and I don't actually think that the prior probability is this low. I think it's low, but I don't think it's this low) of 10^-40 and still give a probability to the resurrection in excess of .9999. And we don't get to that by saying in fact the evidence gives us a posterior probability in excess of .9999. We just say, well this is the power of the ... combined Bayes factor, and a combined power that great could overcome this great of a prior improbability and would give you this high of a posterior probability. So that's the basic method.
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