Wednesday, July 27, 2011

Tim and Lydia McGrew on the Resurrection, part 2

[Continued from Part 1].

Recall the odds form of Bayes' theorem:



where R denotes the Resurrection and F the set of salient facts. The left hand side of this equation is the posterior odds on R, and the RHS is the Bayes factor (or likelihood ratio) multiplied by the prior odds on R. Recall from Part 1 that the Bayes factor is the primary topic of discussion in the McGrews' article (CCRJ).

The first question to ask is whether or not this is a sufficiently general formula, and I do not think that it is. I think that there is room for reasonable doubt of F as characterized by the McGrews. Further, I think that this reasonable doubt, if established, should limit the cumulative Bayes factor. To see why, treat F as an element of a more general set of characterizations of the texts which we all agree exist; call it `History' (H). Then trivially, F and its complement in H partition H.

Before going further, I should defend what I am about to do - you may see it coming already if you are a practiced Bayesian - since the McGrews provide plausible reasons for the approach that they take. They recognize the divisions amongst serious historians (p.4), asking in response to those divisions concerning source hypotheses the following question: "Faced with such a Babel of conflicting voices, what should the interested layman do?" (p.9). Their response is to sidestep inessential(?) questions about sources and propose a well-evidenced set F from which to argue. On the contrary, I think that layman should accept a consequence of a lack of expertise where expertise exists: namely, to be more general, and to account for that in argument. As Bertrand Russell proposes in his Sceptical Essays, a layman should regard no matter as certain where experts are divided. I think that the tools of Bayesianism allow us to formally obey this principle so as to allow experts to continue the argument.

How important is this? It allows us to be more cumulative, though if we pursue the details we are forced to do more work. But as it is, a truly comprehensive case is beyond the capabilities of any single person, as far as I am aware. I do not think we should shy away, and I think I can argue for the all-importance of this work as follows: consider the following generalization:



where this equation is the analogue of the previous. The crucial difference is that we are conditioning on H, not F:

.

Since {F} is a subset of H - simply F for clarity - it and its complement partition H. So



By finite additivity, this becomes



Here we can notice something important before attempting to calculate the details. Suppose that given the salient facts, the Resurrection is certain, i.e. p(F|R)=1 and p(F|~R)=0. Then this becomes



As we will see, the McGrews argue from F for a Bayes factor of 1044. Let's see what this requires of the more general formula in order to be even nearly as large as that even on the generous assumptions made so far: given that

,

we have that

.

So for the Bayes factor found by the McGrews to be realistic, must be very, very tiny indeed.

The significance of this should be obvious, but allow me to elaborate further: you can accept that the salient facts of the McGrews are plausible - the most probable characterization of the record, even - and still reject their conclusions as outlandishly overstated without calculating an alternative Bayes factor yielded by conditioning on those facts. The reality of proposing cumulative Bayes factors as extreme as 1044 is this: you gotta be damn certain of your assumptions, not merely reasonable in accepting them.

For this reason, I do not have to step into unfamiliar territory and survey the conflicts in scholarship in order to reject such a large Bayes factor. I leave the details to my betters, and I propose a way of pursuing them now.

The partition of H as I have presented it may be refined to make explicit the various hypotheses of interest. Write it out as follows:



Reserve the first blocks for the most accepted hypotheses, which are probably too general to be wholly satisfactory for analysis, and reserve later blocks for outstanding refinements of those hypotheses. To ensure that you are being comprehensive, reserve the nth block for a non-descript `other' category. Next, you can apply probability kinematics to this partition:



where q is your posterior probability and p is your prior probability. Note that in this case the debate over the record is taking place in q, and the debate over how that record influences probability(R) is taking place in p. Alternatively, one can approach it as I did above using normal conditionalization as follows:



I did warn you that it would be complicated. To keep it simplish, one can focus on the most important hypotheses and keep the `other' block stocked with the rest.

If we accept that a layman may reasonably ascribe odds to characterizations in light of which the evidence for the Resurrection is at all distant from one, we are stuck with far humbler conclusions about acceptable cumulative Bayes factors. I'm not going to suggest a number at the moment, but I think it safe to say this: whatever you think of Hume's argument, the Bayes factor for the Resurrection derived from the textual record is insufficient to overcome a reasonable ascription of prior odds, barring some natural theology which substantially increases the odds from those suggested by treating the Resurrection as a potential exception to a regularity. Up to this specific miracle, I contradict the conclusions of the McGrews otherwise. (If you are feeling ambitious, you can suggest a number and attempt to make this statement general by limiting the evidential strength of historical investigation. I have ideas in this direction, but I leave that for a discussion of Hume's argument.)

I still think it's worth going into the details. After all, such a natural theology might be found. So in the next post, I will begin to analyze the Bayes factor as analyzed in light of the textual assumptions made by the McGrews. I also add that the McGrews mention their dependence on F (p.39) and are not blind to the possibility of saying more about the issue. But I am not sure that they anticipated the extent to which that assumption prevents the strong conclusions which they make.

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