Extra Thoughts: The odds form of Bayes's Theorem

It's been brought to my attention that an atheist styling himself some form of probability expert has been going about implying that Tim and I are deceptive or slippery in our demonstration of our argument for the resurrection, that I was deceptive in my consultation with Luke Muehlhauser on Common Sense Atheism, that our line for the resurrection in our Blackwell anthology paper is worthless, and heaven knows what else.

he trouble seems to halt from the fact that our article focuses on the likelihood ratio (known as the Bayes factor) for the grounds we adduce rather than on the prior probability of the resurrection. First of all, for the record: No, we were not remotely deceptive or misleading about this in the article. We were painfully explicit about it. (The entire preprint text of the clause is available on-line here. Viz.Even as we concentrate on the resurrection of Jesus, our aim is limited. To indicate that the chance of R given all evidence relevant to it is high would take us to try other evidence bearing on the creation of God, since such other evidence - both prescribed and negative - is indirectly relevant to the occurrence of the resurrection. Examining every part of information relevant to R more directly - including, for example, the many issues in textual scholarship and archeology which we shall discuss only briefly - would take many volumes. Our intent, rather, is to see a little set of prominent public facts that strongly support R. The historic facts in motion are, we believe, those most apt to the argument. Our aim is to indicate that this evidence, taken cumulatively, provides a solid line of the sort Richard Swinburne calls "C-inductive" - that is, whether or not P(R) is greater than some specified value such as 0.5 or 0.9 given all evidence, this evidence itself heavily favors R over ~R.andBut our estimated Bayes factors for these pieces of evidence were, respectively, 10^2, 10^39, and 10^3. Sheer multiplication through gives a Bayes factor of 10^44, a burden of show that would be sufficient to defeat a prior probability (or rather improbability) of 10^-40 for R and give us with a posterior probability in surplus of 0.9999.In my question with Luke M. I said this (transcribed from the podcast, available here, at approximately 16:10 to 19:30):In Bayesian terms, what we do in the clause is that we try to tell 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 world 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 concentrate on in the article instead is what we might claim the direct evidence, the show that supposedly tells you what happened, what you might call reports.You might call it evidence later the fact. So what we concentrate on are the testimonial of the disciples and of certain women that said that they saw and rung with Jesus, the grounds of the disciples' willingness to die for that testimony, and the grounds of the rebirth 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 show is pointing and, number two, how powerfully the show is pointing that way. And what you're trying to do at that place is you're trying to appear at explanatory resources of the hypothesis, in this case, the resurrection, and the negation of the hypothesis. How good does each of these explain the evidence, and is there a big difference between how well each of these explains the show? I should clarify that when I say a difference, too, it's really a ratio.it's really significant that you evaluate it by the ratio, not by the difference. But you want to appear at those two hypotheses and see which one gives you a better prospect of that show and how much better is that expectation. So we estimate Bayes factors for these various separate pieces of evidence, then we contend for the authenticity of multiplying these Bayes factors, because that gives you a lot of kick, and you get to discuss that issue, and we do, of independence, and whether it's logical to reproduce them in place to unite 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 make this overwhelmingly low prior probability (and I don't really imagine that the prior probability is this low. I imagine it's low, but I don't believe it's this low) of 10^-40 and even yield a chance to the resurrection in surplus of .9999. And we don't get to that by saying in fact the evidence gives us a posterior probability in surplus of .9999. We simply say, well this is the force of the.combined Bayes factor, and a combined power that great could master this capital of a prior improbability and would make you this high of a posterior probability. So that's the canonical method.This is all exceedingly clear: We were arguing for a certain magnitude of verification of the resurrection by the grounds we adduce.I understand that the current atheist meme on this, which shows a rather striking deficiency of apprehension of probability, is to say that if one does not contend for a particular prior probability for some proposition, one literally can say nothing meaningful about the check provided by evidence beyond the argument that there is some confirmation or other.This is flatly false, as both the back of the quotations above from the composition and my rather detailed account to Luke M. show.Let me try to lay this out, step by step, for those who are interested:The odds form of Bayes's Theorem works like multiplying a divide by a fraction-a pretty simple mathematical operation we all conditioned to do in grammar school (hopefully). The first fraction is the proportion of the prior probabilities. So, let's assume an example. Suppose that, to start with (that is, before you get some specific evidence) some proposition H is ten times less likely than its negation. The odds are ten to one against it. Then the proportion of the prior probabilities is1/10.Now, the second fraction we're passing to reproduce is the proportion of the likelihoods. So, for our simple example, suppose that the show is ten times more likely if H is true than if H is false. The evidence favors H by odds of 10/1. Then the proportion of the likelihoods (which is likewise called a Bayes factor) is10/1.If you multiply1/10 x 10/1you get10/10.The odds form of Bayes's Theorem says that the proportion of the posterior probabilities equals the proportion of the priors times the proportion of the likelihoods. What this way is that in this imaginary case, after taking that show into account, the chance that the event happened is adequate to the chance that it didn't: what we would call colloquially 50/50. (You'll discover that the ratio 50/50 has the same rate as the ratio 10/10. In this case, that's no accident.Okay, now, suppose, on the other hand, that the second fraction, the proportion of the likelihoods, is1000/1. That is, the show is m times more likely if H is true than if H is false. So the evidence favors H by odds of m to 1. Then, the ratio of the posteriors is 1/10 x 1000/1 = 1000/10 = 100/1,which way that after taking that show into account (evidence that is a 1000 times more likely if H is true than if it is false), we should think of the outcome itself as a 100 times more likely than its negation.See how this works?What this amounts to is that if we can argue for a high Bayes factor (that second fraction), even if we don't say what the prior odds are, we can say something very significant-namely, how low of a prior probability this evidence can overcome. That is just what we say in the second quotation from our report that I gave above. It is precisely what I explain to Luke M. We say that we have argued for "a burden of show that would be sufficient to defeat a prior probability (or rather improbability) of 10^-40 for R and give us with a posterior probability in surplus of 0.9999."In our paper, we focus on the Bayes factor. The Bayes factor shows the direction of the grounds and measures its force. We contend that it is staggeringly high in favour of R for the grounds we adduce. Naturally, the skeptics will not be likely to tally with us on that. My place here and now, however, is that neither in the report nor in my question was there a misunderstanding about probability, any insignificance or triviality in our intended conclusion, nor any deception. We are cleared that we are not specifying a prior probability (to do so and to contend for it in any detail would take us to measure all the early grounds for and against the existence of God, since that is extremely relevant to the prior probability of the resurrection, which apparently would lie beyond the range of a single paper). Nonetheless, what we do indicate is, if we are successful, of great epistemic significance concerning the resurrection, because it means that this show is so near that it can overcome even an incredibly low prior probability.I trust that this is now clear up.Update: See also this discourse of Bayesian probability and Richard Carrier at Victor Reppert's blog, here.