Correction of worked example of dependence factor in undesigned coincidence article

Part of being a conscientious scholar is admitting one's earlier mistakes. While it can be tempting to toss earlier mistakes into a memory hole and hope nobody notices them, it's something I try to avoid. Another temptation can be retconning what one said so as to imply that one meant something that one can still maintain. I don't suffer from the latter temptation, but I do sometimes experience the former temptation. I try hard to resist both.

Let it be said here at the outset that the correction here does not mean that "undesigned coincidences are not a thing," "undesigned coincidences have been refuted," "professional publication on undesigned coincidences is fraudulent and needs to be retracted" or anything of the kind. Skeptics, I'm lookin' at you; if you try to use this post for that purpose, you're making a mistake. Also, even if you don't slog through this whole post, please read the last paragraph.

In 2020 an article (by me) on the probabilistic analysis of undesigned coincidences was published in the high-level journal Erkenntnis. The accepted manuscript version of that article can be found here, and it now has a dated cover sheet with a short update note linking back, in turn to this blog post for further details.

Recently I have been reviewing that article while preparing for a talk on the mathematics of undesigned coincidences. I've realized that in the appendix to that article, where I give some illustrative worked examples, Example 3 has a couple of errors. Here are the errors and the corrected work:

The first (and greater) error is that, when discussing dependence under H, I treat the fact that, when we are considering the conditional probability given H, we are taking H to be true, as if it confers additional reason to take the witness to be reliable. This is incorrect, since this is a mere conditional probability and does not actually involve receiving further independent evidence that H is in fact true.

The second error is a sheer matter of incorrect calculation using the model. While I intended (and even emphasized) that the subhypothesis of H that I dubbed H', and its negation, should screen off testimony 1 from testimony 2, under the assumption of H, I did not in fact build that screening off into the model correctly. This error was related to my not properly taking account of the fact that, in this example, I am creating dependence under H by stipulating that a portion of the probability space under H actually gives each of the individual items of testimony higher probability than the rest of the space under H. This differs from the nature of the dependence under ~H modeled in the previous two examples; the relevant subhypotheses there directly hypothesize collusion or copying and make no difference to the probability of an individual item.

Here is a corrected worked example, modeling a dovetailing undesigned coincidence:

H is the umbrella hypothesis that some event took place. (In the text of the article the event is a picnic.)

Let H' be the subhypothesis that the event took place and also that some further fact about the event is true (in the paper, this was that a picnic took place and also that a Joyce scholar was present at the picnic).

Let P(H'|H) = .05

Suppose that what each of the witnesses says confirms not just H but also H'.

Contrary to what I did in the published paper, the credibility of the witnesses 1 and 2 should be kept at a Bayes factor of 10/1 (as it already was) for their information that confirms H', even though this is a detail of their testimony that H, since no independent information (aside from their testimonies) confirming H is being modeled.

For the sake of not complicating the model yet further, suppose that each of their testimonies confirms H' to the same extent.

Both testimonies are screened off from one another given H' and given its negation, modulo H, but they are not screened off from one another by H itself, because they both confirm H'. The probability of the conjunction of the testimonies is higher given H' than given H and than given (H & ~H'), because the probability of the individual items is higher given H' than given (H & ~H'). This produces the fruitful dependence that actually creates a kind of "bonus" probabilistic boost for H from the conjunction of these testimonies.

Let P(T2|H & ~H') = .02

And the same for T1.

Let P(T2|H & H') = .2

And the same for T1.

Therefore, the overall probability of T1 individually and T2 individually given H is

(.95)(.02) + (.05)(.2) = .019 + .01 = .029

(It might be possible to set all the probabilities just right so that this comes out as .03 as it previously did in example 3 in the published paper, but I did not keep trying all possibilities, and .029 was the closest I was able to get in the numbers I attempted, while keeping everything else as it should be—for example, the BF between H' and ~H' as 10/1. Since this is similar to the other worked examples where the individual probability of each item of testimony given H is .03, and since this is merely an illustrative example, I decided to go with this modeling. Nothing rides on having the total individual probability given H be .03 rather than .029, as long as everything else is done accurately within the assumptions stated.)

Let the individual, independent Bayes factors between H and ~H be 10/1 as in the other appendix examples, so that P(T1|H)/P(T1|~H) = .029/.0029 and the same for T2, so that treated as independent the joint BF would be 100/1, as in the other examples.

Confirmation of H' within H by a 10/1 internal Bayes factor:

Since P(T1|H & H') = .2 and P(T1|H & ~H') = .02, under H, T1 confirms H' by a factor of 10/1, and so does T2.

By the odds form

P(H'|H)/P(~H'|H) x P(T1|H' & H)/P(T1|~H' & H) = P(H'|T1 & H)/P(~H'|T1 & H)

.05/.95 x 10/1 = .5/.95

To turn these odds (under H, giving us the probability of H' given either of the testimonies and H) into a probability,

.5/(.95 + .5) ≈ .3448

So given H and either one of the testimonies (say, T1), the probability of subhypothesis H' is approximately .3448 and the probability of its negation is approximately .6552.

We have stipulated that, under H, both H' and ~H' screen off T1 from T2 and that they each have the same probabilities (.2 and .02 respectively) given (H & H') and (H & ~H'). Given such screening, the conditional probabilities of .2 and .02 remain the same.

 Therefore, the probability of T2 given (H & T1) =

 (.3448)(.2) + (.6552)(.02) = .06896 + .013104 = .082064

 To calculate the fruitful dependence factor between T1 and T2 under H:

 P(T1|H) x P(T2|H & T1)/P(T1|H) x P(T2|H) = (by cross canceling P(T1|H) on the top and bottom)

 P(T2|H & T1)/P(T2|H) =

 .082064/.029 ≈ 2.82979/1

 This, then, is the correction factor by which T1 confirms T2 given H—which will be helpful to the confirmation given by the conjunction (T1 & T2) to H.

This is almost exactly half of the size of the dependence factor of 5.617 given in the published paper in the worked Example 3, modeling dovetailing details.

In the text of the paper other than the appendix, the closest that I come to stating the erroneous extra boost of the witness's credibility is this sentence on p. 21:

Given that P (asserted by Source 1) is true, there is some reason to think that Source 1 is truthfully reporting about that day, including the extra detail about the Joyce scholar's presence.

 This should be reworded along these lines:

 Since Source 1 has, as stipulated, some credibility for what he attests, this credibility can be applied to the extra detail about the Joyce scholar's presence.

Concerning the correction factor for dependence under ~H, I argued in the appendix section on Example 3 that dependence on ~H (which weakens the case for H) should not be any higher than in example 2 (varied but not dovetailing details), where it is 2.33 to 1 against H, and plausibly should be lower.

I stand by that. In that calculation, the only dependence arose from modeling two subhypotheses under ~H of PC (partial/possible copying) and Cr (crafty copying) as having a relative advantage in predicting the conjunction (T1 & T2) over complete independence modulo ~H. If that relative advantage is kept proportionally identical in Example 3 (the one I am recalculating), then of course that dependence factor will be exactly the same.

If the dependence given ~H is kept the same as in example 2, on the new calculation the approximately 2.82979 dependence given H more than offsets the approximately 2.33 dependence given ~H, though obviously not as much as the dependence factor of twice that much in the worked example in the paper.

I then pointed out that arguably in the dovetailing case hyper-crafty collusion/copying would be necessary so as to produce the dovetailing between the details with the appearance of casualness/undesignedness. In other words, partial/possible copying would do an even worse job at uniting the testimonies, as would crafty copying. (Full copying of course is just out of the picture here, as it is in example 2.) I suggested that we should even reasonably consider eliminating a dependence factor under ~H altogether in the case of dovetailing details, since in order to have T1 and T2 as described here we have to have not just variation but extra craftiness to think of varying in a way that is both unobvious and dovetailing, which might simply be overlooked by an audience and which colluders are unlikely to think of. And it's even less likely than non-dovetailing variation to come about by partially copying the content while also varying in some unthinking fashion.

As stated in the paper, if dependence given ~H is eliminated due to these considerations, then the dependence given H is all gain for the confirmation of H from the conjunction of the testimonies, over and above the 100/1 Bayes factor treating them as independent.

It is also worth emphasizing that these numbers are merely illustrative. If the two witnesses have even higher credibility in the first place, that will affect this type of example particularly strongly, since the witness's existing credibility should apply to the subhypothesis H'.

Entanglement concerns don't rule out use of all granted propositions

 It was recently suggested to me that, if my concerns about epistemic entanglement in the minimalist arguments for the resurrection were legitimate, they should preclude ever stating that any opponent acknowledges or grants anything that one is going to use in an argument with a conclusion that the opponent doesn't grant. This was supposed to be a reductio of my concerns on the assumption that it's obviously sometimes okay to state that an opponent grants something that one wants to use.

 To recap, I object to the use in a resurrection argument of the statement that the vast majority of scholars grant that the disciples had experiences which they believed were appearances of Jesus risen. I also object to the use of particular quotations mentioning appearances by specific mainstream scholars (like Gerd Ludemann, Norman Perrin, and Dale Allison) in the same way. My reason for the objection is that it equivocates on the term "appearances." What is needed for the argument is that the disciples at least claimed that they had experiences of a particular phenomenological kind. It seemed to them that Jesus was with them on earth in 3D, in groups, able to eat and be touched, etc. But these scholars don't grant that they had experiences of that kind. Please note this isn't just an objection on the grounds that some of these scholars don't really think Jesus was risen. I'm objecting on the grounds that the phenomenological nature of the experiences they think occurred was very different from what is actually needed for the argument. E.g. Perrin seems to think the disciples merely had an experience of theological enlightenment, which he calls a "vision," but doesn't seem to think even had a visual component. All three of these scholars think that the group bodily "seemings"—Jesus eating, having long conversations—were not something the disciples experienced. The experiences rather were fleeting at most, and the physical manifestations to groups were made up later.

 When one says that someone grants appearance experiences and then uses this "fact" later in the argument in a way that the person does not grant, this is at best a fallacy of equivocation. Moreover, if the person literally means something by talking of the appearance experiences that would be evidence against one's conclusion, one is using something that is evidence against one's conclusion as if it is evidence for it.

 The example given as an alleged reductio of my concerns was "Richard Dawkins concedes that there are gaps in the fossil record" in the course of an argument for intelligent design in biology—a conclusion that of course Dawkins does not agree with. 

But it is not true that any use of an opponent's concession runs into an entanglement problem. I may do a video on this some time, but for now: Entanglement is a specific type of issue in which the very meaning of an opponent's admission (and/or the reasoning he uses to reach it) is such that, if that meaning were clearly understood, the proposition in question would (if true) either be quite worthless to one's conclusion or even tell against the conclusion for which someone wants to use it.

 I've given multiple examples: E.g. Turning "A is from Georgia" into "A is from the United States" in the service of an argument for "A is from Massachusetts." Turning "The number in the box is 20" into "The number in the box is less than 21" in the service of an argument for "The number in the box is 5." Here is a video I did on entanglement. This can also be thought of as using the popular fallacy of equivocation, if that is an easier way to think of it.

In the case of the minimalist argument, this involves turning what would be accurately stated as "The disciples had only very brief group appearance experiences of Jesus in which he definitely did not eat and at most spoke only briefly" into "The disciples had appearance experiences of Jesus" in the service of an argument for "Jesus rose bodily from the dead." In some cases the person in question (e.g., Bart Ehrman) doesn't grant group appearances at all.

 In the case of Dawkins, as far as I can tell, there is no necessary entanglement issue with using "Dawkins acknowledges gaps in the fossil record" in the service of an intelligent design conclusion. It would be a good idea, for the sake of clarity, to state immediately in the context that Dawkins thinks that such "gaps" occur millions of years ago and over periods of millions of years. Hence a term like "suddenness" if Dawkins uses it cannot be used to support a young earth. But there are intelligent design arguments (about, say, the Cambrian explosion) that are not arguments for a young earth. It would also be good to point out that Dawkins makes much of what he considers the similarly abrupt appearance of flatworms as a sort of reductio of the use of gaps in the fossil record to argue for design. This, however, would be solely for the purpose of avoiding all appearance of taking Dawkins out of context, not because Dawkins is literally acknowledging gaps in a sense of that term that is incompatible with or tells against a design conclusion. And whether or not it is useful for supporting a design conclusion is something that can be debated explicity with followers of Dawkins without any equivocation or "fuzzifying" of the meaning of Dawkins's statement about gaps in the record.

It is possible to envisage entangled uses of Dawkins' admission that could be made. For example, if a YEC attempted to use Dawkins's admission in a way that implied (or had to be taken to mean) that Dawkins acknowledges that fossil gaps appear over a much shorter period of time.

Or, to change the actual situation, suppose that Dawkins had acknowledged only very small gaps, where small is understood in terms of phenotypical appearance. This is not actually the case. It seems that he acknowledges what appear to be "large" gaps in terms of appearance, but he thinks these do not have the significance that creationists attribute to them. If Dawkins meant only small gaps and one used the statement that "Dawkins acknowledges gaps in the fossil record" in an argument that requires large phenotypical gaps, while giving the impression that Dawkins acknowledges what one needs for the argument, this would run into an entanglement problem. But this doesn't appear to be what is going on.

Another example of an unentangled granting is Bart Ehrman's defense of the existence of Jesus against Jesus mythers. It's quite easy to use a definite description of "Jesus" that is reasonable and understandable by both Christians and non-Christians. E.g., "A Jewish teacher who lived during the reign of Tiberius Caesar, was known as 'Jesus of Nazareth', and was crucified by order of Pontius Pilate." One can state correctly, without any sliding or equivocation, that Bart Ehrman acknowledges that Jesus existed. Even Ehrman's arguments for the existence of Jesus are such that Christians can also use them—e.g., that Paul said in a letter that he met Jesus' brother, and that Paul appears in his letters to think of Jesus as a real, historical person. 

Ehrman doesn't even seem to think of this as something he "grants" or "admits" in the sense that he does so reluctantly. He's quite overt and even emphatic in rejecting Jesus mythicism. But insofar as one thinks of Ehrman as a "hostile" source (to Christianity), one might wish to use some sentence like, "Even Bart Ehrman thinks that Jesus existed." There doesn't seem to be any entanglement here. Entanglement would arise if one implied that Bart Ehrman thinks that Jesus existed in such a sense that "Jesus" is defined as including his actually working miracles. Entanglement would also arise if one seemed to be implying that Bart Ehrman understands "Jesus" to mean, among other things, "A person who claimed to be God," while in fact, Ehrman strongly denies that Jesus claimed to be God. But I don't know of anyone who does that.

 When checking for entanglement, two things to check for are term meaning and the actual arguments that are used by the person admitting the proposition in question to support that proposition. On this basis, as well, Dawkins appears to be acknowledging apparent gaps in the fossil record for the same reason (namely, simply that in some cases we don't have a smooth set of fossils showing a lot of transitional forms) that a proponent of ID would use.

Another hypothetical example of entanglement (which perhaps someone out there gets snared in) would be if a person arguing for a young universe used the premise, "Such-and-such a scientist acknowledges that the universe had a beginning" when the scientist in question believes that the universe had a beginning only as the conclusion of arguments supporting a Big Bang occurring many millions of years ago. In that case the entanglement occurs at the level of the reasons for which the scientist grants that the universe had a beginning. If the scientist didn't believe that the universe is very old, he would not be able to use these scientific reasons to conclude that it had a beginning, but the hypothetical reasoner here wants to use his "acknowledgement" in the service of an argument that the universe is very young.

 In the minimalist case it is often assumed that, if some skeptic or liberal scholar "admits" that the disciples had appearance experiences, he must be doing so on the basis of strong arguments that are the same as arguments that the Christian would use. It is sometimes left vague what exactly this shared argument is supposed to be; usually we just find some sentence like "The underlying data are so strong that the vast majority of skeptics acknowledge these facts."

 But perhaps the shared argument is supposed to be "multiple, independent attestation to group appearances." Gary Habermas has stated, incorrectly, that the Jesus Seminar admits multiple, independent attestation to group appearances. This is not true; as I've argued elsewhere, the pages that Habermas cites from the Jesus Seminar are merely lists of ancient documents that mention group appearances. In context it's very clear that the Jesus Seminar is not treating these as independent of one another. Bart Ehrman has been explicit that he doesn't grant that there were group appearances (though he says that if there were, they were like "seeing at a distance"). Yet Habermas has boasted in a video that he can argue strongly for the resurrection using only "Bart Ehrman's facts."

Entanglement is a tricky epistemological subject. But no, avoiding entanglement does not make it impossible ever to use something that an opponent admits in the service of one's own argument. One just has to be very careful when doing so.