Saturday, January 23, 2010

My Rebuttal to Tim and Lydia McGrew


This post has been completely re-written. The new version appears here.


Tim and Lydia McGrew have written a chapter in The Blackwell Companion to Natural Theology titled The argument from miracles: a cumulative case for the resurrection of Jesus of Nazareth. Understanding Bayes factors is an prerequisite to understanding the McGrews' argument and my rebuttal to it. Their article includes a description of what you need to know - this post does not.

The article covers a lot of ground, and I'm not responding to all of it. While you should read the article yourself, here's my brief description of the portion I'm responding to:

Scholars disagree as to the accuracy of the Gospels and New Testament. But what if we conclude them to be as accurate as any other historical document, not counting the times they refer to miraculous events? Which is to say, it's accurately reporting what Peter said, even if this isn't what Peter actually saw. This is still an interesting question even to people who disagree with the historical conclusions of conservative scholars.

If these assumptions can be used to make a solid case for the Resurrection, this means that the case against Christianity depends on the problems in the Gospels and Acts even as a mere historical documents. Reasons for discounting the Gospels as even history are within the last couple centuries. If the case against Christianity rests on this more recent scholarship, that means David Hume would have been a Christian if he had accurately evaluated the evidence available to him. (Talk about ultimately refuting Hume...) I guess one could technically hold this position and still not be a Christian. I have no problem saying that Hume would have been unjustified in believing in evolution if he had heard of the idea but not the evidence for it. But I don't hold that Hume should have believed in the Resurrection. To hold this position means I should be willing to argue against the Resurrection, even under the assumption of the Gospels' and Acts' historical reliability.

Moving on the argument itself, Paul, the disciples, and the women who went to his tomb all claimed to have seen Jesus. To what degree does this support the resurrection? Bayesian statistics give a language with which to communicate the answer. Let R be the Resurrection of Jesus, and P, D, and W be the events that each of Paul, the disciples, and the women claimed to have seen Jesus, and in many of these cases, died for this belief.

There are 13 disciples in the argument (the twelve minus Judas plus Matthias plus James the Just.) The events that they testified they had seen Jesus will be denoted
D1-D13. For each of these and for Paul, the McGrews estimate that their Bayes factor supporting R is 1000. What this would mean was that the martyrdom of a specific disciple given R was fairly likely, while roughly a 1 in 1000 chance given ~R. They estimate the factor for W to be 100.

First suppose these are independent. If so, the cumulative Bayes factor is found by multiplication, which gives 10^3 * 10^(3 * 13) * 10^2 = 10^44. This would be strong enough to overcome even an extraordinarily small prior probability on R and make belief in R reasonable.

Of course, they aren't independent. This is recognized and responded to on pages 40-46. While dependence could lead to overestimating the factor, it could go the other way too. While it's possible that killing one martyr could encourage more, the more likely effect is that it scares off other people, who now realize that their life is in danger. So while the McGrews recognize that these aren't independent, the claim is that factoring in the dependence makes the case stronger.

Before beginning my response, I want to mention an alternative approach that I'm not taking: I could list contrary evidences C and argue that the Bayes factor of P & D & W & C is fairly small. Sure, it works. Stronger evidence to the contrary is a valid reason to not be persuaded. But here, the topic is the strength of the evidence P & D & W, and bringing up C's do not help answer this.

I will be responding to the claim from which the majority of the Bayes factor comes. I will be objecting to the factor of 10^39 for D by arguing that the dependence among D1-D13 means the Bayes factor has been overestimated horribly. (D1-D13 are similarly dependent on W. I'm leaving out W from here on because it complicates the notation without really adding anything. On the other hand, P could not be added.)

Without a doubt, I can imagine some likely circumstances (call them A), where once fixed, the D's are negatively correlated or correlated little enough that the odds against D & ~R are at least as bad as the independence assumption provides. But on the other hand, the disciples spent a lot of time together in the time leading up to their belief in the resurrection. So I can also imagine some plausible circumstances (call them B), where the D's are highly positively correlated and thus P(D1-D13 | ~R & B) is much, much larger than the independence assumption estimates. Actually, B need not even be plausible for my first point. It's enough for B to be highly implausible.

A & B are not two hypotheses that we should choose between purely based on which one has a higher prior probability. D changes the odds of A and B, and it will turn out that B is the hypothesis that matters, even if its prior probability is extremely remote. To show this I will suppose the odds in favor of A over B are a billion to one. Now condition on D1-D4. The odds against these four testimonies happening is a trillion to one under ~R & A, while vastly less under ~R & B. It is true that ~R & A started out only one billion times more likely than R & B. But once we conditioned on D1-D4, suddenly B is probably more likely than A. ("Probably" depends on the specifics of "highly positively correlated.") While A & B are not a dichotomy, setting up a more comprehensive list of possibilities won't change the idea – every Di shifts the odds enormously in favor of a stronger and stronger positive correlation.

Speaking of ~R without specifying which of A or B is true, we can see that D1-D13 are not even close to independent. Even if B is extremely unlikely, my argument still goes through. The event ~R & D1-D4 consists mostly of ~R & D1-D4 & B, and thus P(D5-D13 | ~R & D1-D4) is not even remotely close to the value reached by the independence assumption.

This make sense anecdotally as well. Suppose 13 soldiers/civilians/terrorists/we don't know what have been captured and are being interrogated, and these 13 are each capable of giving the desired answer. "Tell us where the whatever thing is, or we start killing you." A gun is pointed at the first person – there's a good chance he gives in. The first person is shot, and the gun is pointed at the second person. With the second person, it's not clear which factor is stronger – what we have learned of the group from the first death, or the intimidation factor. But if you start going down the line, and the first four die rather than sharing their secret, the interrogator's expectation that anyone will speak dwindles to a mere hope. The fact of the first four deaths is reason to think that these are not 13 random people, but a group of Navy Seals, thoroughly dedicated ideologues, or something else that makes this group so tough that they can stand up in the face of certain death, and this something else will cause the other 9 to be willing to die too. This "something else" includes both R and ~R & B.

So to compute how large of a Bayes factor D1-D13 produce, we can find an upper bound by computing P(B | ~R) x P(D1-D13 | ~R & B). The independence assumption gave a factor of 10^39, which is of no use whatsoever in computing this probability.

I would like to also hold a stronger position than simply that the factor is much less than 10^39. What could count as B, and more importantly, how probable is it?

***Edit 4/19/10*** The rest of this post is a terrible argument - I still stand by the first half.

First of all, I would like to defend a dismission of ancient historical arguments in defense of the supernatural that scarcely depends on even looking at the specifics. Even under the assumptions about the reliability of Acts, D1-D13 are not mathematical certainties – they are historical probabilities. When the highly probable is plugged into a mathematical equation that takes input in the form of certainties, the results are not necessarily reliable. What is the probability that believing in Peter's martyrdom is the best conclusion to reach based on the evidence available to us, and yet he didn't? One in a hundred? One in a million? If the odds of Peter's death given that the evidence supports it are a million to one in favor, Peter's death cannot produce a Bayes factor greater than a million – while this is actually larger than the thousand to one factor placed on a single Di, it combines differently with the rest of D. When D1-D13 are not taken to mean the events that the disciples actually died for their testimony, but rather to be that the historical record has come to support these conclusions, their positive correlation is exceptionally strong. The circumstances that would lead to made-up stories about Peter being believed, written down, repeated in the historical record, and mistakenly believed could easily lead to the same thing happening with the other 12, so the additional Bayes factor from D2-D13 is relatively small, giving a total factor of “a small factor” times one million – this is far, far out of the range of 10^39. The Bayes factor coming from historical evidence cannot exceed the odds against the evidence itself being false.

Returning to the position that the 13 disciples really did testify about their experiences, the first thing that needs to be brought up is the effect beliefs in an afterlife can have on one's behavior. For a Christian to die for their faith is gain – or at least, it is perceived as gain, which is all that matters here. It doesn't require any special level of devotion. It requires actually believing. To learn that God wants one to die soon is not merely a sacrifice that is well worth it. It's a non-sacrifice – it would be a good thing for the person who dies. This is even true in the relative paradise of middle-class America. How much more persuasive would this reasoning be in first century Palestine? So what would it take for people who believe in heaven and hell to give their lives for their beliefs? Only the same level of dedication that countless groups all manage to reach every single generation.

The intuition of Pascal's Wager is clear even without the math behind it. “But whoever denies Me before men, I will also deny him before My Father who is in heaven.” Circumstance forced the disciples to deny Jesus and risk hell or believe he was still alive. If the disciples thought there was any chance that Jesus rose from the dead and that living for it would influence the afterlife, they could convince themselves to give their lives for this chance out of faith that it was true. So what remains to explain is what could cause the disciples to consider the possibility of the Resurrection and think there was, say, a 5% chance it was true. That's not to say their beliefs only reached a 5% level – but that is to say the 5% level would be more than sufficient to not only explain their actions, but also to explain the rationality of their actions, given their level of belief. This greatly increases P(B | ~R), for what each of the disciples believed about the influence of being a faithful witness on eternal destiny is definitely not independent.

Once beliefs in hell and heaven have taken hold of your mind, it is nearly impossible to escape. When trying to hold onto such a belief that is looking more and more false, and while struggling to escape the mind control of hell, your head plays tricks on you. A pencil isn't where I left it – did God move it as a sign? If yes, then that is a way to escape the torment of the cognitive dissonance, and without risking hell. If God didn't do it, and I say that God did it, what do I lose? After the disciples left their lives behind to follow a Messiah who died, they had nothing left to lose. I've been there – I know how it works. When you want to believe something badly enough, God does nothing at all and this is interpreted as a surprisingly detailed conversation. Upon retelling, a semi-metaphorical “God told me” can become literal. I don't believe these stories when it's a friend telling me in person, and I definitely don't believe these stories when they are written down and aged for nearly two thousand years. While Christians then and now usually have nothing to gain by directly lying, they have everything to gain by deceiving themselves. Pascal proves this – or at least Christians tend to accept his conclusion, which is all that matters here. Under ~R, the disciples' beliefs that they saw something and their deaths for these beliefs was little more than the disciples' failure to escape the mental prison that is the doctrine of hell. While the strength of these factors could have varied from disciple to disciple, how strong it is with each is decidedly not independent, and they all depend on what Jesus actually said, as well as content of the discussions among the disciples about what Jesus meant.

For the sake of contrast, consider something that requires much, much more sincerity than martyrdom: de-conversion. You can't really escape from the logic of Pascal's Wager as a defense of the rationality of trying to believe. You can only find yourself as unable to believe in your invisible friend are you are unable to believe that 2 + 2 = 5. It is far more difficult for a Christian to admit to themselves that it's all in their head than it is to give their life for Christ. My sincere disbelief proves as little as the sincere belief of martyrs, but it still means I can look at Christian martyrs or a passionate testimony and not be that impressed. For a Christian to give their life while believing God wants them to is a very little thing. To push through the “am I going to hell?” stage without relapsing into belief: now that's hard. And yet appeals to a supernatural experience are not needed to understand how it could happen.

With this is mind, a specific explanation of what happened to the disciples is not needed to be unpersuaded by their deaths, for they are not even the kind of outliers who make you scratch your head and wonder "How did they do it?" Or at least nothing beyond "they believed in heaven and hell."


  1. Hi - I'd really like to see your updated post on this.

    Any chance of doing it as an idiots version for non-mathematicians ?

    There's no need to do it as one massive post either.


  2. I think that with this version, I only succeeded in communicating with people who are mathematicians or close. This was not necessary - the fundamental ideas going on are simple enough that I should be able to do better.

    I should be done with a completely re-done version sometime this week. My rough draft is already substantially clearer.