Concerning the goal of the McGrews’ argument, Carrier said:
…the article in the Companion to Natural Theology does not come to any conclusion. One of the conspicuously missing things is the prior probability, one of the key premises of the entire argument. All they talk about are what we call two of the four premises of Bayes’ Theorem and they make an argument from those two premises.
But you can’t reach a conclusion without answering the other two premises and they never do, which I find disturbing because it suggests… and they don’t really explain this very well, I mean they kind of hint at it…
All they argue is that certain evidence makes the resurrection more probable… Again, that’s completely useless information. We could go from 1% to 10%, that’s 10 times more probable. Yeah, that makes it more probable. It’s still 90% chance it’s false… So, it’s a useless argument. Why would they publish in a companion to natural theology an incomplete argument that doesn’t even argue for the resurrection? What’s the point of that? And not even to explain in a closing paragraph as you would in a science paper, for example. If you did this in a science journal, believe me, the peer review would mandate that you have this closing paragraph explaining that you haven’t actually proved your conclusion you have just done one step of two essentially to do that…
…It’s such a crappy article… it has all these fancy calculations and stuff that make it look very impressive. It seems to me like it is hoodwinking the public in a way.
During the interview, I said “right” and “yeah” often, which was meant as acknowledgment, not agreement. But I did say this: “Right. I think it is something a little slippery going on.” I later apologized to Lydia for that, saying:
I wanted to apologize to you for saying “Yeah, it does seem like something slippery is going on.” That was me getting caught up in the “yeahs” instead of taking the time to thoroughly compare Carrier’s claims [to] the words in your article! I hope you’ll forgive that and harbor no hard feelings!
Luckily, Lydia said “shake and pax on it!” Which I translated to mean, “It’s all good, yo.”
But now, what of Carrier’s claim that the McGrews are being slippery about the intended conclusion of their argument?
Actually, the McGrews are quite explicit about the goal of their article:
At the outset, we need to make it clear what argument we are making and how we propose to do it. The phrase “the argument from miracles” implies that this is an argument to some other conclusion, and that conclusion is most naturally understood to be theism (T), the existence of a God at least roughly similar to the one believed in by Jews and Christians. It is, however, not our purpose to argue that the probability of T is high. Nor do we propose to argue that the probability of Christianity (C ) is high. Nor, despite the plural ‘miracles,’ do we propose to discuss more than one putative miracle. We intend to focus on a single claim for a miraculous event – the bodily resurrection of Jesus of Nazareth circa A.D. 33 (R). We shall argue that there is significant positive evidence for R, evidence that cannot be ignored and that must be taken into account in any evaluation of the total evidence for Christianity and for theism. [emphasis added]
More specifically, they write:
Even as we focus on the resurrection of Jesus [R], our aim is limited. To show that the probability of R given all evidence relevant to it is high would require us to examine other evidence bearing on the existence of God, since such other evidence – both positive and negative – is indirectly relevant to the occurrence of the resurrection. Examining every piece of data relevant to R more directly – including, for example, the many issues in textual scholarship and archeology which we shall discuss only briefly – would require many volumes. Our intent, rather, is to examine a small set of salient public facts that strongly support R. The historical facts in question are, we believe, those most pertinent to the argument. Our aim is to show that this evidence, taken cumulatively, provides a strong argument 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.
Furthermore, Lydia said in my interview with her:
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? …So we estimate Bayes factors for these various separate pieces of evidence, then we argue for the legitimacy of multiplying these 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… 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.
In a recent blog post, Lydia explains this more slowly:
The odds form of Bayes’s Theorem works like multiplying a fraction by a fraction…
The first fraction is the ratio of the prior probabilities. So, let’s take an example. Suppose that, to begin with (that is, before you get some specific evidence) some proposition H is ten times less probable than its negation. The odds are ten to one against it. Then the ratio of the prior probabilities is 1/10.
Now, the second fraction we’re going to multiply is the ratio of the likelihoods. So, for our simple example, suppose that the evidence is ten times more probable if H is true than if H is false. The evidence favors H by odds of 10/1. Then the ratio of the likelihoods (which is also called a Bayes factor) is 10/1.
If you multiply 1/10 × 10/1 you get 10/10.
The odds form of Bayes’s Theorem says that the ratio of the posterior probabilities equals the ratio of the priors times the ratio of the likelihoods. What this means is that in this imaginary case, after taking that evidence into account, the probability that the event happened is equal to the probability that it didn’t: what we would call colloquially 50/50. (You’ll notice that the ratio 50/50 has the same value as the ratio 10/10. In this case, that’s no accident.)
Okay, now, suppose, on the other hand, that the second fraction, the ratio of the likelihoods, is 1000/1. That is, the evidence is 1000 times more probable if H is true than if H is false. So the evidence favors H by odds of 1000 to 1.
Then, the ratio of the posteriors is 1/10 x 1000/1 = 1000/10 = 100/1, which means that after taking that evidence into account (evidence that is a thousand times more probable if H is true than if it is false), we should think of the event itself as a hundred times more probable 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 exactly what we say in the second quotation from our paper that I gave above… We say that we have argued for “a weight of evidence that would be sufficient to overcome a prior probability (or rather improbability) of 10^–40 for R and leave us with a posterior probability in excess of 0.9999.”
In our paper, we concentrate on the Bayes factor. The Bayes factor shows the direction of the evidence and measures its force. We argue that it is staggeringly high in favor of R for the evidence we adduce. Naturally, the skeptics will not be likely to agree with us on that. My point here and now, however, is that neither in the paper nor in my interview was there a mistake about probability, any insignificance or triviality in our intended conclusion, nor any deception. We are clear that we are not specifying a prior probability (to do so and to argue for it in any detail would require us to evaluate all the other evidence for and against the existence of God, since that is highly relevant to the prior probability of the resurrection, which obviously would lie beyond the scope of a single paper). Nonetheless, what we do argue is, if we are successful, of great epistemic significance concerning the resurrection, because it means that this evidence is so good that it can overcome even an incredibly low prior probability.
(This should all be familiar to those who worked through my tutorial on Bayes’ Theorem.)
So what can we conclude?
I think Carrier was wrong to suggest that the McGrews were slippery in the way they presented their argument, because the McGrews were in fact quite explicit on several occasions in their article (and in my interview with Lydia) about the form of their argument. I also think Carrier was wrong to suggest that arriving at an exceedingly high likelihood ratio pointing toward the Resurrection is “completely useless information,” for the reasons Lydia explained above.
I’m also embarrassed to have agreed with Carrier that “it does seem like something slippery is going on.” (Hence my apology.) I had skim-read the McGrews’ paper about a year before my interview with Carrier, but had forgotten enough about it that I simply went along with Carrier rather than checking his accusations against the actual content of the McGrews’ paper.
Oh. Carrier also said Lydia’s “facts are all wrong.” That will have to be another debate, as we didn’t cover that in my interview with Carrier.
Finally, the community at Less Wrong may have some trenchant criticisms of the McGrews’ article.
Update: Richard Carrier, too, has apologized:
I apologize for my remarks in my interview with Luke Muehlhauser. I badly overstated my impressions, and drew the wrong inference from what I took to be the opacity of your paper’s explanations. I agree with everything Luke has said in his latest blog on this question.
Your conclusion is useful if it were based on correct facts. Part of my point in the interview was that it was not, hence my conclusion was actually based on that statement (that your declarations regarding the facts were incorrect), not the actual mathematical result you produced which, if it were correctly derived from the facts,would be a strong result that would warrant more serious attention to the prior probability of divine intervention in the universe in general. And whether the facts are correct is a wholly separate issue Luke and I did not delve into in that interview (I do so in chapter eleven of The Christian Delusion, with further support in my book Not the Impossible Faith and chapters in The Empty Tomb: Jesus Beyond the Grave–none of which address you specifically, only the evidence). I should have corrected myself on these points.
And you do explain the absence of prior probability calculations (I said you “hinted” at it, which is inaccurate hyperbole). What I should have said was that this explanation is too opaque to lay readers and most don’t understand this caveat. Which is why I keep having people come up to me saying your article gives a Bayesian proof that the resurrection occurred or that Lydia McGrew “proved” the resurrection accounts are true (which even you would agree is not an accurate description of what your article does). I was reacting to those claims, not yours. I shouldn’t have assumed this was your design, but only an accidental effect.
Again, I regret my hyperbolic remarks, and I have asked Luke to make public my corrections in these regards. I apologize for misrepresenting you and I hope I can mend fences on this score.