We are now ready to focus on D, the facts about the witness of the disciples, and the independence assumptions which the McGrews make in arguing that this multiplicatively contributes 1039 to a final Bayes factor. We have already reduced the cumulative Bayes factor from 1044 to 1041 by arguing that Paul's conversion does not significantly contribute to the likelihood of the Resurrection. As noted in the last past, we are also accepting the characterization of the factual record which the McGrews make, leaving dispute of that question to a more general analysis, the outlines of which are to be found in part 2. The relevant facts, and some of the argument, which I assume in this post are to be found in part 5. You will not understand this post if you have not closely followed the previous discussion.
For simplification, we may assume - and safely - that p(D|R)=1. If you think that this compromises accuracy or is unfair, try ascribing to it a value of 1/2 or 1/10 or even 1/100, and see how much it affects my conclusions. It won't prove to be crucial. There may be room to dispute it, but that discussion falls outside of the matter at hand, which is to question the conclusions of the McGrews while sharing their premises. This done, we may focus entirely on p(D|~R). This single value is the most important that occurs in their analysis, so it merits the closest focus.
Recall that there are 13 disciples in D; I denote them as D1,...,D13. If we assume conditional independence under ~R, then
Recall that the McGrews do not argue that this assumption is plausible. Instead, they feel that the failures of this assumption tend to favor their argument, that is,
Since the quantity on the left hand side represents the denominator of relevant Bayes factor - where the numerator is 1 by assumption - increasing it decreases the Bayes factor. Given the general likelihood formula (p.26), the McGrews argue in my notation and with our shared assumptions the following (p.42):
where p(D1|~R) is an approximate average of the 13 p(Di|~R). From the messiness of this formula, one can see the allure of independence. But remember the crucial detail is the inequality, not the equality, of the left hand side and right hand side in the above formula. And note that regardless of independence, one must explain at least one disciple's testimony without relying on the others.
Here is where we turn to their defense of this thesis (pp.40-46), which, just as in our previous discussion of the strength of a disciple, goes wrong for the following reason: the McGrews assume that the only significant, explanatory alternatives to the Resurrection are those which claim that the disciples were frauds, pranksters, cynical, or the victims of hallucination or delusion. As I mentioned, I am quite willing to agree that the disciples were devout, sincere, and dedicated followers of Jesus, and that they were not likely the victims of elaborate pranks, and that they were not likely to forge conspiracies without good reason. But just as in the discussion of Paul's conversion, the McGrews expend little effort imagining how it is that Jesus' followers may have realistically reacted to Jesus' death. They never say, or do not closely pursue, something like, "given that the Resurrection did not occur, here are several likely ways that the disciples may have reacted..." They instead assume that no such plausible reaction would at all explain their later behavior.
I think it plausible that the Crucifixion occurred and that the Crucifixion surprised the disciples. I doubt they would expect their Messiah to die the humiliating death of a criminal. But I think this for a good reason: I do not think that anyone there would have expected that, except for those who already condemned him as a heretic. Why is this important? Because this is exactly how skeptics argue. They do not need to invent vivid hallucinations to explain the disciples' commitment. Instead, they ask how well-intentioned followers would likely react and argue from there.
For example, suppose that you are one of Jesus' disciples. He has been dead for some time now, and you are frustrated. You are frustrated because you still believe that Jesus was the Messiah, and because you believe that the Messiah has appointed you to spread his message. The end of the world is coming soon, and Jesus will be the judge. It is vitally important that others in these troubled times understand and believe. But unfortunately, the audience is not receptive. What good, after all, is a dead Messiah for liberating the Jews from the Romans and restoring the Law? There are rumors among followers and admirers of Jesus, especially in distant regions, that he could not have died as he did. Why shouldn't there be? Nobody expected this, of all horrors, to happen. But you did not waste your life by abandoning your family to follow Jesus. His moral message is still sound, and the brotherhood of your Christians is still precious to you. The communalistic message of Christianity can provide for the poor and the beggars and others who live as you have lived.
You are faced with a choice: (1) lie, exaggerate, fail to discredit rumors, or otherwise insist that Jesus better fulfilled the expectations of doubters, or (2) quash the rumers, alienate the faithful, and suffer virtually no success against the skepticism of your peers. Pace the McGrews, you have every incentive to choose (1), as do the other Appointed. You are already willing to risk persecution and death for Jesus' teachings. Why risk dying without successfully guiding others to Salvation? The Messiah did not do as you expected, but he will one day return to rule the world, and he had always explained why your mortal expectations were presumptuous before. Yes, Jesus is currently dead, but why should his teachings die before he returns? You've been tested before; now you are ready to pass the test.
Even this scenario, which I do not think too implausible, is too simplistic. The dilemma may have been the same, but the psychology leading to the preaching of the Resurrection was probably nowhere near this crude. If you like, you can imagine other, similar scenarios which you find more plausible. The question you must address is this: were Jesus to die and remain dead, how likely is it that the disciples would have gone quiet, abandoned Jesus' teachings, and renounced their appointed status? To answer this is to partly answer the important question: what do you expect to have happened if the Resurrection did not occur? Do you expect that the disciples might have rationalized the Crucifixion? If so, how? Is a rationalization like the Resurrection story deeply unlikely, say p(`rationalization'|~R)=10-10?
It is hypotheses like these, unconsidered by the McGrews, that make the difference and undermine their defense of independence, because we need not dispute that the disciples were courageous, committed men in order to maintain it. In fact, these hypotheses are the product of their dedication. To ask how strongly this affects the Bayes factor is to ask how likely it is that the disciples would not have agreed to `lie in the service of the Truth' or would not have been able to rationalize this testimony in some other manner, or at the very least, would not have initially struck down the rumors among the faithful, and eventually come to believe them, or believe that their `essence' was true, or some other such thing. I do not think I need to list precedents of such behavior.
I think that this consideration alone, pursued in detail, is enough to push the Bayes factor way down; I think that others cleverer than myself can explore other plausible alternatives as well. But if we assume that this hypothesis does not share the bulk of p(~R), it is worth asking what the rest of the elements in ~R might look like. The McGrews appear to assume that it is dominated by completely un-explanatory hypotheses, i.e. H:p(H|~R)=0, but I do not think that this is the case. At the very least, I think that this is a very poor approximation whenever we start tossing around numbers like 1039.
How exactly should this affect p(D|~R)? Suppose that the hypothesis I outlined with respect to a single disciple gives us something like p(D1|~R)=0.01, an estimate which I think rather conservative. Now we ask how this should affect the other disciples: here, look at the same dilemma again, adding that other disciples are already claiming R. By the time we reach 7 or 8 disciples, the others follow very naturally. In this way, we can be sure that the independence assumption is overly hard on skeptical alternatives. More formally, we can give a tentative estimate of p(D|~R)=10-4 or p(D|~R)=10-5, and plausibly something higher.
So a tentative readjustment of the multiplicative contribution of the disciples to the cumulative Bayes factor is 105. I think that this is generous, and I think that skeptics can ground this or a lower value. But I am still admitting the following: given the characterization of the texts and the history of the McGrews, the testimony of the disciples is best explained by the Resurrection. What I dispute is how significant this detail must be.
Taking stock, our reduction of the Bayes factor is from 1044 to 107. The last fact which the McGrews discuss is the testimony of the women (W). That will be the topic of my next post. After that detail - perhaps unimportant relative to the other aspects of the argument - I can round all of this noise up and see where it leaves us.