## Wednesday, July 27, 2011

### Tim and Lydia McGrew on the Resurrection, part 3

[Continued from Part 2.]

I think that Part 2 alone accomplishes the goals which I set out in Part 1, most importantly, that the Bayes factor proposed by the McGrews, 1044, is a gross overestimate of the strength of the textual record. I also provided Bayesian formulas which are sufficiently comprehensive to be acceptable to all concerned and that provide a framework in which layman and experts can further pursue the various details and arguments. Albeit in passing, I further concluded that a layman, so long as he treats the disagreement in the expert community as even minimally legitimate, may be quite confident in the following statement: a cumulative Bayes factor produced by the textual record should not confirm the Resurrection against any reasonable prior odds on the same, assuming that a well-formed natural theology which confers a quite substantial prior is not in play.

So why pursue the topic further? As I mentioned, it is possible that natural theology can do the prerequisite work; and it may one day be the case that any legitimate disagreement whatever about the salient facts vanishes, unlikely as the latter prospect seems. There are other reasons which I also think are important.

Careful readers might have noticed that I have regularly referred to a set F of salient facts discussed by the McGrews without ever mentioning what these are. This was deliberate: I wanted to avoid thinking that detail overly important in the previous discussion, as I have explained. Another reason is that I do not wholly accept the McGrews' F as being relevant. This is obviously important when analyzing their argument, but it is especially important since the method rigorously implemented by the McGrews takes a fairly common starting point by arguing from salient facts'. And of these, the trouble I mention is quite common, and I think it can only be clearly explained in the Bayesian framework employed by the McGrews.

The McGrews include the following as elements of F: the eyewitness testimony of 13 disciples (The original ones minus Judas, plus Mathias and James the Just) (D), the testimony of the women (W), and the conversion of Paul (P). The McGrews posit the conditional independence of these facts modulo R and ~R. (They do not do this casually. I'll explain what this means and discuss it later.) So their final Bayes factor is calculated as follows:

$\beta=\beta_P\times\beta_W\times\beta_D$

where

$\beta_P=\frac{p(P|R)}{p(P|\sim R)};\ \beta_W=\frac{p(W|R)}{p(W|\sim R)};\ \beta_D=\frac{p(D|R)}{p(D|\sim R)}$.

As I have mentioned, their final Bayes factor is 1044. The contributions of each term are, respectively, 103, 102, and 1039. So obviously, the really big factors are the independence assumptions - these are internal to D as well, each disciple multiplicatively contributes 103 to the total - and the strength of D. Minor variations in that specific estimate will swamp the other contributing factors. That said, I still want to argue the following: the conversion of Paul contributes nothing.

This will be our first voyage on the sea of calibrating a conditional probability, so some things need to be said before we undertake this adventure. In fact, there is so much to say in advance that Paul will have to wait for my next post.

Apologists often claim - and I think need claim to be successful - that the circumstances surrounding the Resurrection are in many important senses unique. And I agree. What this means in practice is that the most important means of calibrating a probability, frequency data, are in many decisive ways unavailable by virtue of there being no suitable reference class. In even more jargony terms, a frequentist, empirical probability p(A) is the number of occurrences of A over the number of trials, N. The set that N counts is called a reference class, and the indeterminacy or disputability of such classes is a powerful objection to objective interpretations of probability like frequency. Even if in a very limited sense some data are available, I do not think they capture realistic' probabilities, as cases are so limited.

So how can we sensibly calibrate such conditional probabilities? I think that there are boundaries on the reasonable values which we might maintain; if you argue for a Bayes factor as small as 2, you are at one or several points uninviting Christians from the conversation, and your pretension to argument is merely an exercise in self-gratification. Yes, I think that cumulatively, the textual record should be recognized as providing an increased likelihood for the Resurrection in any sensible discussion. But do not worry about it too much - the question is the strength of that likelihood with respect to priors. Still, we can at least stop saying that there is no evidence for the Resurrection or Christianity.1 However, if we are faced with numbers like 1044, we should be immediately suspicious: how can you assert such a bold number with any confidence when we lack frequency data and are discussing unfamiliar events?

I think that the McGrews attempt this in a sensible, plausible way, though it only constitutes a rough beginning. To put it loosely: they look at the relevant conditional probabilities concerning a fact. They claim, usually plausibly, that the numerator p(fact|R) is high. Then they attempt to show that outstanding alternative explanations affecting the denominator p(fact|~R) are not plausible. But they are not pretending that this is comprehensive. They clearly illustrate and discuss why many of the important alternatives to the Resurrection are not at all explanatory of the fact in question. I will quote from this explanation, since it helped me to correct a common error in probabilistic thinking, namely the over-estimation of the importance of unlikely alternative hypotheses:
...we must be on guard against a plausible error. It might seem that our analysis of cumulative case arguments in terms of Bayes factors puts the emphasis on likelihoods in such a way that finding any sub-hypothesis under ~R that gives a high probability to some piece of evidence always represents a significant gain for the proponent of ~R. But when an auxiliary hypothesis Ha is very improbability given ~R, its contribution to the explanation of a fact F is negligible even when it has high likelihood [...] It is easy, also, to slip into a different false assumption -- that in making a probabilistic argument of the sort in question for R, we are obliged to restrict ourselves to those sub-hypotheses under ~R which make some attempt to explain the facts in question. (CCRJ, p.27)
I omit a formal illustration in the ellipses, but I think this sufficiently important that I have talked about it elsewhere already. The moral of the story here is that by focusing on hugely improbable theses like mass hallucination', skeptics are needlessly driving themselves to distraction and discredit.2 A very important question is as follows: what sub-hypotheses actually dominate ~R in the sense that p(sub-hypothesis|~R) is significant? The McGrews answer correctly as follows: "The answer is that most of it is going to the generic hypothesis that Jesus died and that all went on as usual thereafter," but they continue with a more difficult assertion: "which provides no explanation, not even an attempted explanation, of the evidential facts in question" (p.28).

As I will elaborate when discussing the particular facts, the important question is as follows: what exactly does normal mean when we've accepted the background assumptions of the McGrews? Would we expect as normal' in the circumstances that the disciples would abandon the teachings of Jesus altogether and fail to evangelize?

As we lack suitable frequency data or other accepted calibrating methods, we have to rely heavily on intuitions - intuitions about expectations of the psychology, interests, and behavior of 1st-century followers under duress in exceptional circumstances. One may think of the method as a psychological analogue to propensity: one attempts to empathize with the people in question and attempt to imagine what they might do. A very limited knowledge of their circumstances, cultural differences, and a layman's knowledge of the topic means that we have to be very cautious. It also means we should be suspicious of Bayes factors which are supposed to be large enough to convince us that a well-evidenced, putative law of nature has been violated.

That all said, I can start with Paul.

1. It is very often said, by e.g. PZ Myers and Massimo Pigliucci, that one cannot evidence Christianity or gods because they are not coherent hypotheses. More needs to be said about this, but I would at least suggest the following: if the evidence for the Resurrection really is extremely convincing to reasonable people on the assumption of coherency, we should take an attitude similar to that which I think we take to science: some conceptual fuzziness is to be tolerated, barring flat contradiction, where overwhelming evidence for an aspect of a theory is available. Were I to find the evidence for the Resurrection convincing, I know I would be working very diligently to craft a coherent Christian philosophy to accommodate it. So to me, the coherency difficulty is in many ways secondary, unless that difficulty is so severe that one cannot even begin to discuss relevant evidence. I think we usually manage to do so. Wouldn't you agree?

2. You'll notice that this is a common theme of mine. I agree with the McGrews (p.11, elsewhere) that critics of Christianity have often been unreasonably prejudiced against reasonable claims which seem to credit Christianity - and apparently for that reason. I think that this prejudice can be remedied by having skeptics understand that they do not need to invest heavily in this-or-that dubious alternative to the Resurrection by understanding the significance of likelihoods, properly understood. In failing to understand, they send the (perhaps accurate) impression that their views are prejudicial, and where observers are led to believe that skeptics have to rely on an absurd confidence in a mass hallucination to argue against the Resurrection, they will understandably feel that the argument favors the Christians, or at least feel safe in equivocating between skeptics and Christians on the matter. And they'll tend to make familiar noises about the true ulterior motive of atheists, that though they deep down believe in God, they hate Him and want to live in sodomy, adultery, abortiony, or other sinnery. If you ask me what exactly I think happened in 1st century Palestine, I would say very little to satisfy you. What probability teaches us is that we must learn to be comfortable with generalized alternatives instead of specific, highly detailed explanations. Those are by nature hard to come by with any confidence on any issue, much less issues of distant historical inference. Proving a particular negative is very difficult, but a generalized negative like ~R is often manageable. As we will see, it requires work and some cleverness, but it can be done.