The resurrection, Independence, and Dependence

Introduction

In Timothy McGrew's and my co-written  lengthy paper on the resurrection of Jesus, published in 2009 in the Blackwell Companion to Natural Theology, we argued for an extremely powerful Bayes factor representing the force of the testimony of the male and female disciples for Jesus' resurrection.

Over the years questions have arisen repeatedly about the fact that we treated those testimonies as independent of one another, which gave rise to the enormous force of the Bayes factor. In the case of the male disciples, we counted thirteen persons who ostensibly encountered the risen Jesus on earth (the eleven, plus Matthias, who replaced Judas Iscariot, plus James the brother of Jesus), not merely the number of resurrection accounts in the Gospels. 

We did address the issue of independence in the original article, a point that is sometimes ignored, and we emphasized especially the relevance of persecution to independence, which this post will emphasize as well. But there is a lot more that can be said. Over the years since then I've published a large number of technical articles on independence and testimony (see here, here, here, here, here, and here) and three more blog posts specifically relating these issues to the resurrection. Two of these posts (here and here) were concretely about the resurrection. (When possible in this post, I am linking free versions of my own articles from this post, so that you can read them. These are published articles. If you want to get the citation to the published version, see my curriculum vitae here.) One post, just published, is purely technical and is merely a ground-clearing project responding to some incorrect modeling and criticism by skeptical statistician and physicist Brian Blais.

This post is meant to be wide-ranging. It will cover both technical and substantive issues, and I plan to include a link to it on a cover sheet at the beginning of the free online version of Tim's and my resurrection paper. There is no getting around the need to include technical discussion here, but I'll try to make the technical stuff as accessible as possible, and there will be specific connections to the resurrection even in the more technical sections.

1.1 Independence and the trilemma for the disciples

Worries about whether or not we should count the different disciples as independent witnesses of Jesus' resurrection arise from a tacit analogy to something like a magic trick--a case where you have a bunch of people all saying the same thing, and their doing so can all be explained away by one hypothesis about how "it looked like that, but that wasn't really what was happening." When skeptics wave their hands and talk about dependence and witness testimony, I can nearly guarantee that, among other things, they're accessing the image of a situation where everybody makes the same mistake for the same reason. The assumption that the disciples' testimonies would be all dependent in something like this sense lies behind the argument made by philosopher Arif Ahmed, here. Ahmed argues that even if witnesses are not influencing each other, and even if their perception of what has happened on some occasion is the only influence upon their individual testimonies, we can still have undermining dependence among testimonies because the testimonies may all be influenced by the same cause of false positives, namely, some kind of trick.

This type of dependence comes up in actual magic shows. Just interviewing more and more members of the audience who say it looked like the magician sawed the lady in half has diminishing returns as far as confirming that the thing actually happened. And the returns diminish rapidly. It doesn't take long before you realize that you're going to get more or less the same story from a new audience member, even given that it was a trick. That's because the trick was designed to make it look that way to the entire audience, there are plenty of resources for making things appear that way to a large number of people at once, and no matter where you were sitting in the audience, it would probably look that way to you.

Before the technical material, here is a useful, more or less non-technical way to look at this subject. Let's go back to the simplified trilemma form of the maximal data argument for the resurrection, which I often use (see here): Either the disciples were lying about the resurrection, they were mistaken, or they were telling the truth. The content of their testimony shows that they were not mistaken. The context of their testimony shows that they were not lying. Therefore, very probably, they were telling the truth.

Complaints about our taking the disciples' (and women's) testimony to be composed of many separate and independent lines of support for the resurrection are based on a tacit assumption that all of that testimony can be accounted for well by some more specific version of non-resurrection, ~R. If that were the case, there would be a dependence problem, since the conjoined testimony would all serve to confirm that version of ~R. (In what follows I'll call that a subhypothesis of ~R.) 

Now, getting slightly more technical: If H is the hypothesis confirmed by individual items of evidence, and if all of the items confirm some one more specific version of ~H, then, unless all those items are even more dependent given H (see below), the impact of that set of evidence is overestimated if we treat them as all independent evidence for H.

But what if different parts of the evidence push against different specific versions of ~H? As the trilemma illustrates in a simplified way, some of the evidence that we have, about the repeated warnings and persecution of the Twelve in the very early days of the church, is evidence against one specific version of ~R--namely, lying, including conspiracy to lie. That evidence favors the resurrection. It also, just as far as it goes, favors the "mistaken but sincere" version of ~R. But now the advocate of ~R has another problem: There are aspects of the evidence that are evidence against the disciples being sincere but mistaken. Namely, the detailed content of their testimony, including claims of groups to have had certain detailed experiences, is not well-explained by any known type of mistake or hallucination, including bereavement hallucinations. That might seem to push the advocate of ~R back in the direction of lying. But we already saw that lying is also a bad explanation. And some things, like the sheer number of people involved in detailed experience reports, can't be accommodated well by either lying or mistake.

In other words, unlike the magic trick, this is not a case where some more specific version of ~R can explain all the evidence naturally. Some is evidence against one more specific version of ~R (lying). Some is evidence against another version (hallucinations of any kind). And some is evidence against both.

1.2 A primer on dependence and confirmation: Relative conditional dependence

When lines of support coming from two or more pieces of evidence (call them E1 and E2), both individually relevant to H, are conditionally probabilistically independent of each other, we say that both H and its negation ~H "screen off" E1 and E2 from one another. Formally, this can be expressed by 

P(E1|H & E2) = P(E1|H) and P(E1|~H & E2) = P(E1|~H),

read as, "The probability of E1 given H and E2 equals the probability of E1 given H" and "The probability of E1 given not H and E2 equals the probability of E1 given not H."

Informally, what this means is that, if we were given that H is true or if we were given that H was false, in either case knowing E1 wouldn't tell us anything more about E2, or vice versa, beyond what knowing H told us, or what knowing not-H told us. A way to think of this is that in such an independent case, any probabilistic relevance that E1 has on E2 is exhausted by its impact on H. E1 changes the probability of E2 only "through" H.

A simple example of this type is the somewhat artificial "witness separated" case where you have two witnesses who never have any contact or communication with each other of any kind and who both testify to similar contents (e.g., "Smith shot Jones before my eyes on July 1"), though presumably in different words. The fact that witness 1 says that he saw Smith shoot Jones does raise the probability that witness 2 will say something similar, but it does so only insofar as it raises the probability of H ("Smith shot Jones"). If we take H to be certain, this of course gives us some reason to think that either witness will testify to H. But if we were already certain of H, E1 doesn't do anything more to raise the probability of E2. All its work for raising the probability of E2 is done by raising the probability of H. And the same, here, for ~H, since we are stipulating that the items are independent given ~H (an issue that will loom large in what follows). If we knew for sure that Smith didn't shoot Jones, then having witness 1's witness-separated testimony in hand would give us no reason to expect witness 2's testimony.

(In this post I'll try consistently to make H the hypothesis that some E is evidence for, just to keep things as clear as possible. But it doesn't really matter what you call H vs. ~H, as long as you're clear and consistent about it.)

When you have this conditional independence of E1 and E2 given H and given ~H, to calculate the force of their conjunction you just multiply their Bayes factors:

P(E1 &E2|H)/P(E1&E2|~H) = P(E1|H)/P(E1|~H) × P(E2|H)/P(E2|~H)

Given conditional independence for both H and ~H, the multiplication of the individual factors doesn't distort the force of the cumulative case composed of both items. As Tim and I point out in our article and as John Earman points out in the book Hume's Abject Failure, such multiplication can yield a formidably powerful case made up of many individual items of evidence, each of which is positively relevant to H. 

Contrary to what is sometimes assumed (and what Brian Blais assumes in the post linked above), dependence between two items of evidence, even two witness testimonies, isn't per se a bad thing for the cumulative force of a case. What matters is relative conditional independence, and what downgrades the force of the case, all else being equal, is dependence given ~H.

This is actually pretty easy to see even informally. What do we really have in mind when we talk about some kind of bad (for H) dependence between the testimonies of witness 1 and witness 2? It's a situation in which, if H is false, one of these testimonies will give us some reason to expect the other testimony. For example, if W1 and W2 are husband and wife, we may be concerned that one of them wants to pin the crime on Smith falsely and has enough influence to induce the other to lie as well. This would be causal dependence given the negation, and if we have reason to suspect such causal dependence we also have probabilistic positive dependence given the negation--that's the bad kind, in the sense that a proper representation of the force of the two testimonies would yield a cumulative factor less than that represented by multiplying them. 

P(E1 &E2|H)/P(E1&E2|~H) < P(E1|H)/P(E1|~H) × P(E2|H)/P(E2|~H), because

P(E1|~H & E2) > P(E1|~H)

We can say that in such a case E1 and E2 are positively relevant to each other given ~H. If you knew that Smith did not shoot Jones, and you heard the husband's testimony, you'd still have more reason to expect the wife's testimony than if you merely knew that Smith was innocent. Why? Because you think the husband may influence his wife to say that she saw Smith shoot Jones, even if Smith is innocent.

Another way to put this is that this kind of dependence is good for ~H. That is, the concern about influence softens the impact of the force of the two testimonies against ~H (for H). In general, positive relevance between E1 and E2 given H or ~H is, all else being equal, helpful to that hypothesis.

But that's not the only kind of situation we can have. Ultimately, what matters is whether H or ~H unifies the evidence more than the other. If they're completely conditionally independent, that's not an issue. But when they aren't, you can get complex, interesting, and messy cases. This is because H can unify as well (see below for examples), and also because two items of evidence can be negatively relevant to each other given H and/or ~H.

To take one relatively simple illustration, suppose that W1 and W2 absolutely hate each other. Suppose that if one of them says something, the other one often tries to argue with it or deny it. They have no positive causal impact upon each other. If anything, when they know about each other's statements, they are likely to be negatively relevant. 

Now suppose that despite this, they both say that they saw Smith shoot Jones, even though they know about each other's statements. At this point what we have to ask is this: Since their statements are often negatively relevant to each other, are they more negatively relevant to each other given H or given ~H? Obviously, the answer here depends a lot on our background evidence, but here's one plausible way to argue: W1 would rather disagree with W2 if possible (and vice versa). But in a case of murder, if Smith really did shoot Jones and W1 really witnessed it, he may feel bound by the seriousness of the issue, legal worries about perjury, etc., to admit that he witnessed it, even though he knows that W2 is saying the same thing. He may feel, as it were, forced to agree with W2. In contrast, if Smith is innocent, if W1 and W2 were both present on the alleged occasion, and W1 hears W2 say that he saw Smith shoot Jones, he has every reason to disagree, both to refute his hated rival and to promote the cause of truth and justice and prevent his hated rival from maligning an innocent man. Hence, arguably, dependence in this case favors H, since negative dependence is stronger given ~H than given H.

There are also cases of positive dependence given H. Undesigned coincidences are good examples here. Suppose W1 says not only that he saw Smith shoot Jones but also that Smith tripped as he ran away. W2 says that he saw Smith shoot Jones and that he noticed that Smith had an untied shoelace at the time. This is a simple undesigned coincidence. If Smith really did shoot Jones, and if he also really had an untied shoelace, this increases the probability that he would trip. (This isn't saying that it would necessarily lead us to expect it. Merely that it would increase the probability.) W2's testimony increases the probability that Smith had an untied shoelace, which increases the probability that he tripped. W1's testimony increases the probability that Smith tripped, which increases the probability that he had an untied shoelace. The creates positive relevance "under" H (more on "under" in the next section)--namely, that Smith shot Jones--since the tripping and shoelace details are portions of the two witnesses' shooting affirmations. 

That brings us to (drumroll), subhypotheses and dependence.

1.3 A primer on dependence and confirmation: Subhypotheses

One of the best ways to think about dependence given a hypothesis is to think about subhypotheses. Simply put, a subhypothesis of H (or ~H) is a proposition that entails H (or ~H) but is not entailed by it. So "Smith shot Jones and tripped as he ran away" is a subhypothesis of "Smith shot Jones." "Smith did not shoot Jones and W1 is influencing W2 to lie about it" is a subhypothesis of ~H.

With some exceptions that I won't get into here, evidence is dependent given a hypothesis due to confirmation of a subhypothesis. This is why I sometimes speak of dependence "under" a hypothesis, which means the same thing as dependence given that hypothesis. I sometimes refer to H or ~H as an "umbrella" hypothesis, in contrast to its own subhypotheses.

It's also helpful to distinguish what I'll call Type 1 dependence-creating subhypotheses from Type 2 dependence-creating subhypotheses. A Type 1 dependence-creating subhypothesis produces dependence between/among items of evidence, under the umbrella hypothesis (to the extent that it is probable given that umbrella hypothesis), due to the fact that the Type 1 subhypothesis gives higher probability to the individual items of evidence than the average probability of those items, given the umbrella hypothesis.

1.4 A primer on dependence and confirmation: Type 1 dependence-creating subhypotheses

Here are a couple of examples: Let the umbrella hypothesis be, "I am currently in sensory contact with a mind-independent external world." Let the subhypothesis be, "I am currently in sensory contact with a mind-independent apple." Note that the subhypothesis is more specific than the umbrella hypothesis. Let the items of evidence E1, E2, etc., be things like, "I have a visual experience that looks like what I seem to remember calling an apple," "When I try to reach out and touch what I seem to see, I have a tactile experience that feels like what I associate with an apple, and I seem to be able to pick up the apple and bring it toward myself," "When I try to bite, I have a taste sensation that I associate with an apple." You get the picture.

These propositions about sensation have individual relevance to the umbrella hypothesis. My seeming to see an apple is some evidence (given past memories) that I'm in contact with the external world. But there is a bonus, beyond that. The non-hallucination hypothesis also unifies the sensory data like this: If we were given the umbrella hypothesis at probability one (that I am currently in sensory contact with a mind-independent external world), and I received the visual apple-like sensation, this by itself would raise the probability that I'm currently in contact with an apple, specifically. This would have an indirect effect "through" that subhypothesis of raising the probability that I would, upon trying to reach out, have a particular touch sensation. If I then have the touch sensation as well, this raises the probability of the subhypothesis that there's an apple in front of me yet further, which raises the probability that I'll be able to do something that seems like putting it in my mouth and tasting it. And the same for smell. Hence, the individual sensory experiences taken together have more force for the proposition that I'm currently in contact with a mind-independent world (I'm not just hallucinating or dreaming), than the product of their individual forces. For sensory propositions E1 & E2 through En,

 P(E1 &E2 &...& En|H)/P(E1&E2&...&En|~H) > P(E1|H)/P(E1|~H) × P(E2|H)/P(E2|~H) × ... ×  P(En|H)/P(En|~H) , because

P(E1|H & E2) > P(E1|H) and P(E2|H & E1 & E3) > P(E2|H), and so forth.

A way to put this is to say that H unifies the sensory propositions more than ~H does. If I'm not in touch with a mind-independent external world right now, and I have a visual sensation of an apple, I may very well feel nothing when I try to touch it, or I may feel something totally inappropriate to an apple (like a fur sensation), or it may disappear when I try to bring the apple closer, or the taste may be totally unlike an apple. Most of us have had such experiences in dreams. Speaking for myself, when I try to drink anything in a dream, feeling very thirsty, it never refreshes and always tastes terrible. When I try to bring what seems like a book closer to myself in a dream, the words always fuzz and become illegible. And so forth. 

This of course is the basis for the "dagger of the mind" scene in Macbeth--" Come, let me clutch thee. I have thee not, and yet I see thee still. Art thou not, fatal vision, sensible to feeling as to sight? Or art thou but a dagger of the mind, a false creation, proceeding from the heat-oppressed brain?"

Polysensory evidence provides a particularly good example of unification under an umbrella hypothesis. In fact, though my husband Tim first drew attention to relative unification in our resurrection article, one of the reasons I took the idea and ran with it was my years'-long fascination with the problem of the external world and a Cartesian Deceiver scenario. You can see here how I eventually used relative conditional dependence in connection with that question.

You can perhaps anticipate one way that dependence can actually be helpful to the resurrection hypothesis, via this type of unification. If the male and female disciples really claimed such extensive polysensory experiences as are recorded in the Gospel accounts, and if we are (for the moment) setting aside the possibility that they were merely lying (don't worry, we'll come back to it!), the hypothesis that Jesus rose bodily R unifies the sensory aspects of these reports better than its negation ~R. A relevant subhypothesis under R would be, e.g., "Jesus rose bodily and appeared bodily to eleven or more people on Easter Sunday evening."

It's important that I give one more example of a Type 1 subhypothesis, because one of the functions of extensive and varied evidence is to (virtually) eliminate Type 1 subhypotheses under the negation. Going back to the murder example, let's change the testimony of the two witnesses. Now, instead of saying that they saw Smith shoot Jones, they testify that the gun that has been found on the scene  belongs to Smith. Ballistics confirms that this gun was the murder weapon. The witnesses may say that it looks just like a gun that they saw Smith use on different occasions. One of them may say that Smith discussed the weapon with him. Etc. Or one "witness" could just be state government records that the gun is registered in Smith's name.

Here there are parallel subhypotheses under both H (Smith killed Jones) and ~H (Smith is innocent). Under H the obvious subhypothesis is "Smith killed Jones using his own gun." Under ~H the subhypothesis is "Smith is innocent, and someone else used his gun to kill Jones." It doesn't necessarily follow that both H and ~H unify the testimonies about the gun equally. That will depend on other factors, like the absolute probability of the individual testimonies given each umbrella hypothesis and, crucially, the probability of each subhypothesis given the umbrella hypothesis. 

Considering just ~H, we can see how the unification works: E1, the first testimony that this looks like Smith's gun, comes in. If we were given Smith's innocence at probability 1, and we received this testimony, this would all by itself raise the probability of the subhypothesis that someone else killed Jones using Smith's weapon. That would, in turn, greatly raise the probability that the state registry would show the gun as belonging to Smith. So, given ~H, the two testimonies together have higher probability than the product of their individual conditional probabilities.

1.5 A primer on dependence and confirmation: Type 2 dependence-creating subhypotheses

A dependence-creating subhypothesis is Type 2 because its probability is directly raised only when we actually have two or more items of evidence. I say "directly raised" because here I'm not talking about its prior probability but about the way that it interacts with what we can think of as the directly relevant evidence--the evidence that we're trying to model the impact of, concern an event or an hypothesis. In the case of testimonial evidence, this would be the testimonies themselves. The probability of a Type 2 subhypothesis isn't raised by one of the items of testimonies all by itself. 

A way to think of it is that a Type 2 subhypothesis directly asserts relevant causal dependence between two or more items of evidence. Therefore, the reports or testimonies raise its probability only when we notice a similarity between them, which requires us to have more than one in hand.

An obvious example of a Type 2 subhypothesis is collusion/conspiracy, which I'll have a lot to say about in connection with the resurrection. Despite the fact that modern skeptics tend to pooh-pooh conspiracy hypotheses, they also speak very confidently about some sort of undermining dependence between the disciples' testimonies. There is something of a tension here. Collusion in falsehood is one of the most obvious types of hypotheses to reach for under the negation of an event. And as I've noted elsewhere, modern skeptics and/or liberal scholars often are really talking about conspiracy anyway. They just don't like the word. Dale Allison, for example, suggests that maybe a large number of the eleven just "went along" with the claim that they had a group experience as "the twelve," even though they knew quite well that at the time in question they themselves saw nothing. 

In the murder example the obvious Type 2 hypothesis for unification under the negation is that Witness 1 and Witness 2 are directly influencing one another (or one is influencing the other) to testify falsely that they saw Smith shoot Jones. Background evidence (such as the degree of influence that one witness has over others) can increase the prior of that subhypothesis. Direct evidence from the testimonies arises only when we have more than one. There's no reason to hypothesize that Witness 1 is testifying falsely and influencing Witness 2 to do so when we don't even have Witness 2's testimony. After all, he might directly contradict Witness 1 and say that he was there and Smith didn't shoot Jones. So only a conjunction of testimonies can directly confirm this Type 2 subhypothesis of ~H. 

1.6 A primer on dependence and confirmation: The probability of a subhypothesis given the umbrella

It's surprising how many people don't know this: When considering evidence for some event, such as the resurrection or the Battle of Gettysburg or the moon landing or anything else, and comparing how well that event explains the evidence and how well its negation explains the evidence, you must not confine yourself to thinking only of alternatives that have some hope of explaining the evidence.

Rather, in order to think rightly about how strong the evidence is for the event, you need to compare the explanatory power of the salient hypothesis H (that is, the hypothesis toward which the individual items prima facie point) to the power of ~H overall. I can't stress this strongly enough. And it's pretty obvious in the case of the Battle of Gettysburg or the moon landing. If man never landed on the moon, the most probable thing that would happen is that we would have no attestations that man did land on the moon! The most probable outcome would be...nothing. No video, no astronaut interviews, no pictures of the American flag, etc. You won't get a good sense of how strong the evidence is for the moon landing by comparing "Man first landed on the moon on July 20, 1969" only to some gerrymandered conspiracy theory that man didn't land on the moon at all and that this, that, and the other pieces of evidence were falsely manufactured. Where M is the moon landing hypothesis, the gerrymandered conspiracy hypothesis is merely a subhypothesis of ~M. Of course that subhypothesis explains at least some of the evidence. That's what it was designed to do. But the evidence is still strong against ~M because the subhypothesis takes up only a very small bit of the probability space given ~M. Since the conspiracy is very improbable given ~M, it doesn't help much to improve the explanatory power of ~M.

You must use a partition--a mutually exclusive and joinly exhaustive set of options. M and ~M are a partition. M and "~M and a conspiracy to make it look like M" do not form a partition. See here for a video in which I point this out. It's astonishing that not only most skeptics but also many Christians don't realize this. If you don't use a partition, you're going to find it virtually impossible to get a true sense of the strength of the evidence.

This point about a partition is relevant to any subhypothesis that raises the probability of even a single item of evidence. If the probability that even Witness 1 or Witness 2, alone, would say that Smith shot Jones is extremely low, given that Smith didn't shoot Jones, then that lying subhypothesis just isn't all that helpful to ~H in explaining that piece of evidence. Most of the probability space under ~H just doesn't contain that witness lying. And the same for mistake. If the only alternatives for mistake under ~H are overwhelmingly improbable, given ~H (maybe due to the specific content of the testimony, how close the witnesses claim to have been, and so forth), then they just aren't very helpful to the likelihood ratio--the probability of the evidence given H over the probability of the evidence given ~H.

This is also relevant to dependence-creating subhypotheses. Take the above example of Witness 1 as a husband influencing Witness 2, his wife, to lie about Smith shooting Jones--a Type 2 dependence-creating subhypothesis. This is where the relevance of background evidence, or of evidence that comes out in the course of the investigation, about the relationship between Witness 1 and Witness 2 comes into play. If the investigator can tell that W1 and W2 are at odds, that one of them seems to have no good interest in lying, if they seem to have no influence or even a negative influence upon each other, then that subhypothesis isn't very helpful. 

1.7 Indications of independence in verbal details

The words in which the testimony is conveyed are also relevant to the explanatory power of the subhypothesis itself. Variations in wording may indicate at least partial or even total independence given ~H and, for that matter, given H. If there are surface contradictions between the two testimonies, which would have been easy for colluders to smooth over in advance, that also reduces the extent to which the Type 2 subhypothesis under ~H explains their joint evidence.

This is obviously connected to common skeptical and liberal complaints about the alleged contradictions in the Gospel resurrection accounts. While we can't clearly assign each of the accounts to one particular person, and while the alleged witnesses are much more numerous than the verbal accounts that we have, the twin skeptical complaints that "these accounts are all dependent on each other" and that "these accounts are hopelessly contradictory" pull in opposite directions--a tension that most skeptics don't even seem to notice. (Recently Paulogia and Brian Blais seem to be saying that even John's resurrection appearance accounts are just dependent on Mark--quite a feat on the part of John, depending on Mark for appearance accounts, since Mark without the longer ending doesn't have any appearance accounts! I think probably the original ending of Mark was lost early, but that definitely means that it is hardly even meaningful to speak of narratives of Jesus appearing to people as dependent on Mark, since we don't know what was in the original ending of Mark.)

1.8 A primer on dependence and confirmation: Evidence can confirm two mutually exclusive hypotheses

Something else that people often don't know about subhypotheses: It's possible for the very same evidence to confirm two mutually exclusive hypotheses. Maybe this seems obviously true; or maybe it seems obviously false. In any event, it's true. Even without talking about subhypotheses we can see it in a simple case. If the winner of some game must be A, B, or C, and we learn with certainty that it isn't A, the chance that it is B rises and so does the chance that it is C. In a more complex case, a given body of evidence can confirm both "Smith is the murderer" and some subhypothesis of "Smith is innocent," such as "Smith is innocent and these two witnesses are colluding in a lie." Even though these are mutually exclusive, they aren't jointly exhaustive (certainly not in the prior probability distribution). Therefore, when the other portion of "Smith is innocent" is eliminated (the part in which nobody says they saw him shoot Jones), that probability space is redistributed to both "Smith is the murderer" and to whatever subhypotheses of "Smith is innocent" are left standing. This does not mean that those end up with equal probability. And it does not mean that the evidence doesn't confirm "Smith is the murderer" overall, disconfirming "Smith is innocent," perhaps even drastically. It just means that some subhypothesis of "Smith is innocent" may also receive a probabilistic boost.

1.9 A primer on dependence and confirmation: The importance of varied evidence

The biggest thing to know (for this post) about the probability of a dependence subhypothesis (either Type 1 or Type 2) is the importance of varied evidence. Philosophers of science have long known that varied evidence that all points in the same direction is especially valuable to confirmation, but there has been a certain amount of controversy over the proper probabilistic rendering of this intuitive point. I made my own contribution to that literature here: Valuable varied evidence for the confirmation of H is evidence that, by its variety, pushes against dependence given ~H. 

This is true in science as in history. It's true everywhere. The reason that a doctor does a variety of kinds of tests before telling a patient definitely that he has cancer is because that provides a crosscheck on false positives. A blood test will have a certain false positive rate. An x-ray showing what appears to be a mass will have a certain false positive rate. Clinical discussion of symptoms may point in the direction of a diagnosis but only imperfectly. Even a biopsy could be subject to some sort of lab error (like accidently switching patients' samples), however rare that is. It's when all those things point in the same direction that the case becomes especially strong. In testimonial cases, testimony from people with different social backgrounds, different perspectives on the event, or concerning different events that all point in the same direction, is especially valuable. 

In criminal cases, a combination of evidence that can't all be explained away in the same way, is especially valuable. Suppose that, in addition to two lines of evidence that the gun belongs to Smith, we also have Smith's fingerprints on the gun, unsmudged, in the right places for a person who held and fired the gun. This would be evidence against the Type 1 subhypothesis under ~H that someone else committed the murder using Smith's gun. Even if a hypothetical different murderer used gloves, we wouldn't expect to find Smith's unsmudged prints in the right places.

Even a somewhat improbable ~H subhypothesis can gain a lot of traction if it explains all of our evidence really well, without being further gerrymandered. Just getting still more evidence that the gun belongs to Smith doesn't keep raising and raising the probability that Smith is the murderer. If the only evidence that we have for Smith's guilt is more and more and more evidence that the gun belongs to Smith, we run into diminishing returns, because the Type 1 subhypothesis of ~H that someone else used (what is now undoubtedly) Smith's gun can account for all of that in one fell swoop. (This is something I've called the bottleneck problem for the minimal facts approach to the resurrection. Minimal facts theorists don't realize  that they can't keep raising the probability of the resurrection just by saying more and more about evidence that the disciples believed the resurrection. One needs to address the issue of mistaken belief in a stronger way. And no, I don't think N.T. Wright's "they were Jews so they wouldn't have believed it if they didn't have really good evidence" does much for that purpose.)

Again, this doesn't mean that the evidence that the gun is Smith's doesn't raise the probability of his guilt. But it can't raise it any higher than the probability of "Smith is guilty" given "the gun belongs to Smith." That's an upper bound. At that point, everything comes down to how hard or easy it would be for someone else to use Smith's gun. And all the evidence that the gun is Smith's can't get past that ceiling; it's all explained by a subhypothesis of "Smith is innocent" in which the gun really does belong to Smith. The evidence is unified under ~H by that subhypothesis. It would behove the detective to go out there and find some evidence about how difficult it would be for someone else to use Smith's gun!

This relates to the magic trick theory of Jesus' appearances, already discussed. Just getting more and more evidence that the gun belongs to Smith is like interviewing more and more witnesses in the audience of the same magic trick. 

But in the case of the resurrection, it isn't like that. We do have varied evidence, including evidence that pulls against particular dependence-creating subhypotheses of ~R.

1.10 A primer on dependence and confirmation: The ultimate gerrymander and independence

Before launching into a detailed discussion of Jesus' resurrection, I want to make one more technical point: Merely listing all of the evidence and tacking it on to a hypothesis to make a gigantic subhypothesis that entails all of the evidence does not in itself create dependence under the umbrella hypothesis. To see that this is true, consider a very long series of coin flips that all come up heads. Suppose that for some reason you want to try to argue that  the coin is fair despite this very long series of heads. So H is "the coin is loaded" and ~H is "the coin is fair." 

You can create a subhypothesis of ~H which simply lists all of the instances of heads, calling them part of the hypothesis by fiat: "The coin is fair and flip 1 came up heads, flip 2 came up heads, ..., flip n came up heads." But that isn't actually a dependence subhypothesis, despite the fact that it entails all the evidence. Here's the reason: Since the umbrella hypothesis is that the coin is fair, the fact that the first flip comes up heads can only confirm part of that ultimately gerrymandered subhypothesis. It confirms only "The coin is fair and flip 1 came up heads." That means that ~H still screens off flip 1 from flips 2 through n. The probability that flip 2 comes up heads, given ~H and flip 1, is still .5. So this ultimately gerrymandered subhypothesis doesn't actually produce dependence under ~H. It is neither a Type 1 nor a Type 2 dependence-creating subhypothesis.The relevance of this point will become clear when I talk about "crazy hallucinations" and also, to a lesser extent, large conspiracy theories concerning Jesus' resurrection.

2.1 The reportage model and Jesus' resurrection

When I say what follows, I expect some skeptics and even some others to check out of this post. That's your choice, but I would urge you not to do so. In the rest of this post, as in our 2009 article, I am assuming the reportage model of the Gospels and Acts. (Though I hadn't yet coined that term then.) As I've said ad nauseum over the past six years or so, the reportage model of the Gospels and Acts is not question-begging concerning the resurrection. It does not entail that Jesus really rose bodily from the dead, and we're taking as premises only that the stories in the Gospels and Acts represent the content and context of what the original alleged witnesses claimed. Note: What they claimed. 

With respect to the persecutions in Acts and the direct reports of what the disciples said, I do take the reportage model to narrate what really happened, but these are in themselves mundane events, not miracles. Concerning the miracle of the resurrection itself, as a premise I only take it that these reports tell us what the people listed as present claimed. We then seek to discover what the best explanation is for their making these claims, with this content, in this context.

I also note, for the record, that the copy on my website of our 2009 paper now contains a cover sheet in which Tim and I disavow all references to the alleged consensus of scholars as shown by Gary Habermas's research. We now know that that research is so shaky as to be almost entirely valueless. But I also stress, as I have elsewhere (see here), that even our case in that article depends on the fact that they made these detailed claims in this context of persecution. As our work since then has defended the robust, un-redefined reliability of the Gospels (the reportage model), we are entirely willing to defend that proposition, regardless of the fact that it isn't (as it turns out) granted by a scholarly consensus. So despite the references in the article to Habermas's work, ours was never a minimal facts approach.

I'm not going to defend the reportage model here, as I've already done so in four books. That is taken as background--the Gospels and Acts are sufficiently reliable that their accounts can be taken to tell us at least who said what when, what the original alleged witnesses, named explicitly in the books, claimed had happened.

Here's the point of all of that: In what follows I will frequently make reference to the content of the Gospel narratives and how hard they are to explain given ~R (that Jesus didn't rise again). This lead-in explains why that isn't question-begging and how Gospel reliability feeds into that.

2.1 Resurrection evidence and subhypotheses of ~R that don't get off the ground

A big part of the value of copious, detailed, and varied evidence for some H is the way that it just destroys the ability of Type 1 subhypotheses under ~H to cope with all of it. In the above case where we have two lines of evidence that the gun belongs to Smith, I pointed out that if we also had unsmudged fingerprints in the right places on the gun, that would not be explained at all by "Smith is innocent and someone else used his gun." In fact, the fingerprints would disconfirm that theory. 

In the case of the resurrection, here are some attempted Type 1 dependence subhypotheses: bereavement hallucinations, wrong tomb, moved or stolen body, pareidolia (like seeing shapes in clouds or weird light effects), vague "mistake of some kind" (such as seeing a real person at a distance and thinking he looks like Jesus), and (not widely proposed) an imposter such as a twin pretending to be Jesus.

With some of these, it's obvious on the face of it that the proposal can, at most, account for some small part of the evidence of what these people claimed in this context. All proposals that people went to the wrong tomb or that the body was moved really do nothing to account for the rest of the evidence. Usually an attempt is made to connect the empty tomb with the appearances by saying that the discovery of the empty tomb caused the disciples to expect resurrection appearances, and that they then had these experiences due to their expectation. But there is no reason why even the truthful report of the women (and Peter and the Beloved Disciple, per John) that they found a tomb empty would cause groups of the male disciples to have, or even to claim that they had, the detailed group experiences recorded in the Gospels and Acts, in which various groups seem to eat with Jesus and so forth. 

There is also no reason why even finding a tomb empty would cause the women themselves to experience jointly, or to report jointly, that they not only saw Jesus but were able to grasp his feet, as recorded in Matthew. The robustly bodily-type appearance reports are related to the wrong tomb/moved body theories as the fingerprint evidence is related to the theory that the other person used Smith's gun. The first theory just doesn't account for that other evidence, which is very different in type. One can see here how the maximal data use of the details of the claims is important.  

So these theories don't create dependence among the many different types of evidence that we have. In fact, they don't even account for other portions of the women's testimony, such as seeing angels and being able to grasp Jesus' feet.

Pareidolia and vague mistake of some kind (such as seeing a real person at a distance and misidentifying) don't even account for part of the evidence! So they don't even get off the ground. If all we had were reports that the disciples said "We saw Jesus" or "Jesus is risen," maybe such theories would have some hope of unifying by explaining the fact that multiple people made such generic claims. But the so-called "creed" in I Corinthians 15 makes no claim to be a narrative of what was experienced; it's clearly intended just to be a list. And all of the Gospel and Acts narratives say things that are not accounted for by these types of theories. Nobody in these stories merely thinks that he sees Jesus at a distance and nothing more. Jesus is always said to be on earth, and his being on earth is always indicated in specific details. The account in Matthew 28 of the appearance in Galilee is probably the least detailed, but even there it explicitly says in vs. 18 that Jesus approached the group. And he speaks highly specific words, which a cloud shape wouldn't do and a person merely seen briefly at a distance couldn't do. And all the rest of the accounts say much more, including the claim that Jesus gave long discourses, that he ate, that he invited touch, and so forth. So these vague theories, though they would be Type 1 dependence theories if they had some hope of explaining the evidence, don't even get off the ground. (I mention this because skeptics do sometimes use pareidolia or seeing at a distance to account for the evidence. Kamil Gregor has suggested a pareidolia account of the appearances, not saying that he believes it but saying that it could explain the evidence. Bart Ehrman has said that he doesn't in fact acknowledge group appearances, but that if he did, they would be at most seeing at a distance. Years ago I watched an interview with Walter Sinnot-Armstrong, which I haven't been able to find again, in which he suggested that perhaps someone thought he saw Jesus at a distance and mentioned it to the others at the same time, and they went along with it for fear of appearing weak in faith.) 

Different subhypotheses which don't probabilify one another have to be strung together in an ad hoc manner (see here my professional paper on ad hocness) to try to explain the evidence that we have. James Fodor's RHBS theory is of this sort. He postulates a planned reburial of Jesus' body, which the disciples didn't know about and didn't find out about, followed by private (individual) hallucinations, followed by group religious experiences, followed by cognitive bias modifying memories, followed by social pressure that supposedly reshaped and then harmonized the accounts, making them more impressive and similar to one another. Fodor's theory is very explicitly aimed at the minimalist (not minimal facts but still minimalist) resurrection argument of William Lane Craig. To be perfectly honest, I don't think Fodor's model, despite all its ad hoc-ery, does a very good job explaining even evidence that Craig should be able to access. For example, why would these cognitive biases and social harmonization not get rid of the surface alleged contradictions that still exist between the Gospel resurrection accounts? (Craig could access this point even though he doesn't use a reportage model; the existence of some appearance of discrepancies among the accounts doesn't require the reportage model to notice.) And why didn't the supposed development of the accounts under social pressure go in the direction of Jesus' exaltation, yielding accounts similar to but even more elaborate than the vision of Stephen in Acts 7, rather than the on-earth appearances that we have?

This point is relevant both for Craig's argument and for a maximalist argument. But the mutual independence of these elements, and the explanatory weakness of others, is even clearer for a maximal data argument that uses a reportage model. The Bayes factor is very high, and the elements are very resistant to dependence in the maximalist argument. "Group religious experiences" that we have good reason to postulate in history just aren't like the reports that we have in the Gospels and Acts, and the kinds of experiences that are recounted there would be virtually entirely independent from merely finding an empty tomb. Nor are "bias" and "socialization and marginalization of doubt" likely to produce a situation in which more than thirteen people, just six weeks after the event, are willing to die for explicit, detailed, bodily, polymodal experience claims.

2.2 Resurrection evidence, independence, and crazy hallucinations

What we might call "mere crazy hallucinations" don't constitute even an apparent dependence hypothesis. That one person suddenly goes nuts and thinks he sees a pink elephant (or a resurrected teacher) in the corner of the room doesn't raise the probability that someone else who happens to be standing near him will suddenly go nuts in the same way. So mere crazy hallucinations on the part of the various disciples remain independent of each other. 

An important point against crazy hallucinations, even for one person, and the evidence we have concerns the ascension claim. We allude to this point in our resurrection paper. Suppose that a single disciple has such severe mental illness that he really believes that he has experienced a whole series of detailed, polymodal, bodily meetings with Jesus, when these were really just hallucinations. He even believes that other people were there too and can confirm his accounts. This degree of mental illness is highly unlikely to end abruptly and for no medically apparent reason with his recounting a further hallucination in which Jesus ascended to heaven, followed by his having no more such hallucinations. To postulate that "for some reason" a disciple or disciples suddenly stopped having these earthly, bodily hallucination experiences, is just a further ad hoc (and hence independent) epicycle added to a hallucination theory in order to explain the cessation of the on-earth-like experiences. Below I will discuss the fact that the ascension and cessation are also not well explained by bereavement hallucinations.

Here I will very briefly say something about the idea of "contagious" hallucinations, which was (in some form) held by Gerd Ludemann. Even Ludemann, however, didn't apparently think this could explain the robust bodily manifestations in the Gospel stories. He just said that they were added later; in other words, they weren't what the original people experienced or claimed. The original seeing, he said, was a "seeing in the spirit" (What Really Happened to Jesus, p. 69), whatever "seeing in the spirit" might be.

In point of fact, there is as far as I've been able to tell no documented instance where lying is strongly unlikely (see below on conspiracy), in which about twelve people all claim to have eaten meals and met on multiple occasions with a dead person, in groups, whom they were able (in group settings) to touch, and where this appears to be due to a "contagious hallucination." References to, e.g., the Angel of Mons just show the double standard of the critics. Some take that to be a data point showing that there really are contagious, group hallucinations, but in point of fact it isn't at all clear that there was a group of people who were actually present making a clear, detailed claim to have seen the Angel of Mons, much less doing so under conditions of persecution. The story appears to have gotten started from a newspaper piece that was intended to be fiction, which was misunderstood as fact despite the author's later attempts to make it clear that it was fiction. In general, the weirder and more startling the claim to "group hallucinations" as a known fact, the more we seem to find a) "group" is being applied to individuals at different times, not to a group together at one time, b) it is radically unclear that any original set of people really did claim to have had the specific group experience in question (so skeptics are using a double standard by taking such an experience to be established fact, while insisting that there was no such experiential fact in the case of Jesus' resurrection), and/or c) the "lying" option has been neglected, as there was no strong reason not to lie.

And of course no one claimed to have had lunch with the Angel of Mons! Even bereavement hallucinations don't claim to be able to explain the robust group experiences reported in the Gospels, as I'll discuss below. 

It's fairly predictable that at this point a skeptic will say, "Well, miracles don't happen either, so anything is more probable than a miracle." It's not that simple. If there really are such things as "contagious hallucinations" that are phenomenologically robust to this extent, these would be part of the order of nature. We would therefore expect to find them happening with at least a certain amount of frequency.

Miracles, on the contrary, are supposed to be rare, performed by God for special purposes. The whole point is that they are not part of the order of nature. See my article here on the value of miracles as signs and its dependence upon rarity. This article is similar to, though not verbally identical to, my professionally published article in The Routledge Companion to Theism.

2.3 Resurrection evidence and an imposter theory

Perhaps surprisingly, the twin theory (or other imposter theory involving a physical double), proposed at one time by Greg Cavin, actually unifies more of the evidence than any of these. This illustrates an important epistemological point: When evidence is varied and copious, often the more far-fetched and silly a theory is, the more of the evidence it's able to "explain." A strong cumulative case thus pushes the alternative in more and more radical and implausible directions. At least on a twin imposter theory, we have a real, bodily person actually talking to people, going around, trying to make people think he's Jesus risen, and he looks like Jesus. Such a twin could, of course, really eat, be touched, and the like. 

But of course there's a reason why even Robert Greg Cavin no longer pushes the twin theory. Not only is such a theory blatantly ad hoc, because there's zero independent evidence of Jesus having a twin brother. Moreover, such an action by such a twin is also completely unmotivated and, indeed, counter-indicated. This hypothetical twin gains nothing by his actions and could even be at risk from Jesus' enemies if they heard about him. He must at some point disappear and run away to live out his life in obscurity somewhere else. What would be the point? 

Such a theory has its own problems explaining the evidence. For instance, while the wrong tomb theories do nothing to explain the appearance accounts as we have them, an imposter theory does nothing to explain why the women said they found the tomb empty and talked with men in white. One would have to introduce more "moving parts" to the theory, uniting it with a stolen body or wrong tomb theory, and perhaps even add some accomplices of the twin who are willing to play the men in white! And what do these accomplices get out of all of this? This has no value for dependence, because (as in the case of simply adding more and more heads tosses to a fair coin theory) these additional elements are independent features of the theory.

According to Luke's accounts, Jesus gave long and impressive discourses on OT prophecy and how it was fulfilled in himself. This is of course a characteristic of Jesus' teaching prior to his death--that he taught with authority (Matt. 7:29) and that people were very impressed by his teaching (John 7:46). If Jesus had a double or a twin, why would we expect him to be able to imitate this characteristic of Jesus himself? Yet this was apparently at least claimed by Jesus' disciples about his post-resurrection appearances.

While we're at it, while an advantage of the twin imposter theory is that he can really eat, presumably he can't make himself go up in the air. So the twin imposter theory doesn't explain the claim that Jesus' disciples personally saw him ascend into heaven.

Notice too that even though sometimes (though not in all the accounts) Jesus is not at first recognized, he is in, even those cases, eventually recognized as the scene continues (John 20:16, 21:7, Luke 24:31-32). An imposter would have to worry about making missteps that would give away the fact that he wasn't really Jesus. And to some extent, given an imposter theory, we would expect the order to go the other way--initial apparent recognition of a person who really looks like Jesus, followed by later doubts that arose because the person couldn't keep up the imposture. An imposter would be well advised to keep his interactions with Jesus' closest friends short rather than getting overly ambitious and taking greater risks. The accounts we have point to someone who had no such worries. 

Again, why am I taking so much time on this crazy theory? Because this post is about dependence. Surprisingly enough, it's one of the better attempts to create dependence among the evidence items and to explain the agreement of so many of Jesus' disciples that he appeared to them raised in bodily form, even though that was false.

2.4 Resurrection evidence and bereavement hallucinations

Supposedly things are somewhat different, and somewhat better for the skeptic of the resurrection, when it comes to bereavement hallucinations. 

Bereavement hallucinations have for a long time been regarded as one of the strongest candidates for a ~R theory. While skeptics take them to be non-veridical, other scholars, especially Dale Allison, are more inclined to regard at least some bereavement hallucinations as true paranormal apparitions. In either case, the claim is that this is a relatively normal thing, that this kind of thing is known to happen after a loved one has died, and that it can explain the evidence well as an alternative to bodily resurrection and bodily appearances--the traditional Christian theory that Jesus really came out of the tomb and appeared bodily to his disciples. Allison is an "objective vision" theorist and thinks that Jesus wasn't really present bodily to the disciples. (Even William Lane Craig, who has recently been rather strangely defensive of Allison, admits at least that Allison takes an objective vision view of the appearances. This despite the fact that in that very podcast Craig is still defensive about Allison in other odd ways, making a pointlessly big deal of the fact that Allison doesn't strictly identify the resurrection and the as Ascension.) Full-fledged skeptics of course think that all this sort of thing is just some strange and naturalistic phenomenon without any veridical content. 

In both cases, whether the claims are naturalistic or paranormal, the idea is that bereavement apparitions (or bereavement hallucinations--here I'm using the terms interchangeably) can even happen to groups and that they are a better explanation of the disciples' appearance claims than literal bodily interactions with the risen Jesus.

It might seem, then, that bereavement apparitions are the ideal candidate for a Type 1 dependence subhypothesis under ~R, a subhypothesis that raises the probability of most or all of the appearance evidence in one fell swoop. Bereavement apparitions are (supposedly) a known phenomenon, they (allegedly) involve some kind of experience in which it seems like the dead person is present, they (allegedly) often involve visual perception in which one seems to see the dead person, and (suppoesdly) they even happen to groups. But they aren't indications of a bodily resurrection. Allison even claims that such apparitions seem tangible at times. For anyone who denies the bodily resurrection, what's not to love?

I should say up front that I think Allison himself is far too uncritical of the apparition literature. Oddly enough, he seems to take any report, even one distant in time, that someone saw a ghost or an apparition to reflect truly at least what someone claimed about his own experience, but he doesn't do this for the Gospels and Acts. There, he constantly throws out parts of the evidence that don't comport with the apparition theory. This is a pretty blatant double standard. He also never takes seriously the possibility that those who claimed such experiences in the apparition literature were lying, even when he also doesn't record negative consequences (to themselves) due to their claims. So even if Allison or someone else did turn up some old case in which a group of people claimed that they had a meal with an apparition, the "lying" option would still remain to be dealt with. 

But there's more: There are very large disanalogies between the Gospel and Acts accounts, just taken as claims from the people who were supposedly present. I've recently done an entire video series on these disanalogies. As this post is already going to be very long, I mustn't try to reproduce all of that content here. But in short: With the reportage model as background, here are things about the reports that the bereavement apparition theory can't account for. Again, take all of these to be a description of what, given the reportage model, the relevant followers of Jesus claimed. I'm just not putting "alleged witnesses claimed that" at the beginning of every item, for convenience. All page citations here are to Allison's book from 2021, The Resurrection of Jesus: Apologetics, Criticism, History.

--Jesus ate with them, in groups. Allison admits that apparitions rarely eat, and as far as I could find, he doesn't relate any case in which allegedly an apparition ate with a group. p. 221

--Jesus had long conversations with them and gave long discourses. Allison emphasizes that this is not representative of apparitions. p. 219, see especially footnote 40

--Jesus was not merely tangible to individuals, like Mary Magdalene, but also to groups, whom he expressly invited to touch him and who did touch him (in the case of the women in Matthew). He invited Thomas to touch him in the presence of other members of a group. Group tangibility doesn't appear to be a characteristic of grief apparition experiences. At least, Allison doesn't say that it is, and he writes off cases like the Doubting Thomas case and Jesus' invitation to the disciples to touch him as apologetic additions. pp. 64, 152, 217, 227, 229

--Jesus' interactions in his appearances allegedly on earth are consistently polymodal, rather than sometimes being merely auditory, other times merely visual, and the like. But as Allison himself lays out series of bereavement apparitions of a given person, they take various forms, ranging from (allegedly) individuals feeling that the person is tangible, through single-sensory (just visual, just auditory), to mere "sense of presence." pp. 215-216

--Jesus' on-earth appearances end abruptly with an account of a group of people who witness him leaving our space-time continuum (the ascension). Grief apparitions, though they do tend to come to an end not long after the person dies, do not seem to end in this abrupt fashion with a group who believes that the person was on earth and that they personally witnessed him leaving. Rather, they just sort of stop. p. 332

What all of this means is not only that bereavement apparitions don't even unify all the evidence, and not only that the accounts we have are evidence against the theory that these were bereavement apparitions. Those things are both true. But there's more: What all of this means is that the theory that Jesus rose bodily unifies the evidence of these claims better. Think of my example above about the apple. It's just like that. And it operates even in individual encounters, which even in Allison's account are not consistently tangible and polymodal. But of course if Jesus is really physically present, then he is of course accessible to multiple senses. For Mary Magdalene herself, when she seems to see someone there, whom she eventually recognizes as Jesus, this raises the probability of the subhypothesis of R that includes, "And he's here right now in the garden." She then grasps him and finds him to be apparently tangible. And the fact that she both sees and hears him is also well explained by his really being there. This is one reason why even the Bayes factor for an individual encounter report should be highly favorable to R over ~R. 

For the disciples in groups, the fact that one of them has visual and auditory experiences of an apparently physically present Jesus raises the probability of the subhypothesis of R that includes "And he's here in the upper room now." That in turn makes it nearly inevitable that someone else who is present will also be able to see him and interact with him in polymodal ways. If he's present bodily at a particular place and time, of course multiple people can see him, hear him, and touch him as well. 

One person's experiencing Jesus' apparent bodily presence in a particular place and time also raises the probability that someone there will be able to hand him things, while the others can see this handing occurring. And, of course, if Jesus is really present bodily at that place and time, he will be able to eat before them and with them. And of course if Jesus is really present bodily on a particular occasion, there is no reason why he can't give a long lecture on Old Testament prophecy or carry out long conversations. 

It's noteworthy thatAllison speculates (p. 63) in an entirely evidence-free fashion that maybe some members of the eleven in the upper room weren't able to see him at all and simply went along with the others. Why even bring that up? Obviously, because it would fit better with his own visionary view! In fact, Allison says right there in a footnote (note 124) that there are accounts of apparitions who are not visible to everyone present. This is quite unsurprising, of course. To put it mildly, people who are visible and audible merely in a vision, rather than bodily, are not automatically expected to be visible to everyone present. But that isn't what the evidence says about Jesus. As I'll emphasize below when I discuss conspiracy, we have ample evidence from Acts that all of the twelve asserted stoutly, under persecution, that they really were witnesses of Jesus' resurrection. It doesn't look like they saw and heard nothing but just "went along."

2.5 Resurrection evidence and the Peter Hypnotist hypothesis

I'm going to start here with a super-duper-silly attempted Type 2 subhypothesis and then move on to one that is merely super-silly. Why do I do this? Because, again, while skeptics are big on how foolish it is to think that the disciples independently testified to Jesus' resurrection, they typically don't realize just how hard it would be even to try to come up with dependence subhypotheses of ~R. Remember again that a Type 2 dependence subhypothesis directly postulates causal dependence between reports. It can therefore be confirmed, if at all, only when we have at least two reports in hand.

A good example of skeptics' (and others') suggesting direct causal dependence concerns mutual influence. There is typically a sense that merely by saying that the disciples probably influenced each other, we've said something profound that helps to explain their joint testimony. Here's Dale Allison doing this:

Whatever the answers, the twelve were gathered before Jesus appeared to them. This means that, despite the crucifixion, they were still together; and if Peter was among their number, his claim that Jesus had appeared to him, like Mary Magdalene’s similar claim, cannot have been without effect. They could not, furthermore, have been united in their conviction that “he appeared to the twelve,” if united they were, until they had spoken with one another about their experiences; and to imagine that none of them, in the process, influenced the recall or interpretation of others would be naïve in the extreme. pp. 63-64 

Here Allison just waves his hand in the direction of the idea that somehow Peter and Mary Magdalene influenced the other disciples and that they compared their stories. Even setting aside the fact that Luke explicitly says, to the contrary, that the male disciples weren't impressed by the women's accounts (Luke 24:11), this vague reference to influence and mutual comparison does virtually nothing to explain the stories we have, if these were indeed what the disciples claimed. 

You don't just generically influence someone else in the following way: In reality (this would be the more minimal, vision-like experience that Allison believes occurred), a figure of a very close friend, who is dead, appeared nearby, indoors, pretty briefly, but he didn't ask for food and eat it before you, he didn't invite touch, he didn't show scars, and he didn't talk at length. Nonetheless, by your entirely ordinary influence you induce others who were there at the time to have false memories that he  actually seemed to do all of these things in your presence. Vague influence doesn't account for about a dozen people who are all influenced in this way. Of course, it's quite convenient for Allison that he simply ditches all of that as not something that the disciples experienced at all or even claimed to experience (pp. 64, 152, 217, 219 footnote 40, 229, 258; see also an explicit statement that these are later developments in an interview with Alex O'Connor, here). But if, per the reportage model, those things really were part of the joint testimony, then ordinary interpersonal influence and conversation doesn't tell us how they got there.

"Mutual influence" is a gesture in the direction of a Type 2 dependence hypothesis under ~R. But when we have robust and detailed testimony, it is a non-starter.

So I'm going to up the ante on mutual influence and make it a more unifying, but by that very token much sillier, Type 2 subhypothesis. Let's call this the Peter Hypnotist theory. As far as I know, nobody has proposed this, though mentions of Peter's influence do come up. I'm bringing it up because it's the kind of Type 2 subhypothesis that does a little bit of work explaining the evidence--though still not much. The Apostle Peter was a master hypnotist and could use his Jedi mind power to implant lengthy, detailed, empirical, but completely false memories in other people. Like the twin theory, this is enormously far-fetched, but it illustrates the same thing: To try even to make a stab at a theory that unifies some portion of a large body of diverse evidence, you are forced to go very far afield.

There is little to commend this theory even before we get to the resurrection accounts. All the Gospel pictures of Peter are of a blunt, rather blundering personality without subtlety whose influence (if he had influence) with others lay in the force of his strong feelings, his love for Jesus, and his willingness to rush in where angels feared to tread. Peter denies Jesus, not smoothly but with bluster and cursing, because he's scared and he can't think of anything else to do, which is exactly the opposite of what we'd expect from someone with psi powers. Nor is there any other candidate among the disciples of Jesus for such a master hypnotist.

Then of course, as with any theory in which some of the disciples just lie, there is the question of why Peter would engage in such an elaborate scheme. For Peter himself, the Peter Hypnotist theory is one of lying. (So it's a hybrid of the "lying" and "mistaken" options.) This question is all the more urgent since, according to Acts, Peter hung around long enough and was defiant enough to get himself arrested multiple times and beaten at least once--no picnic. 

But even the unification of the evidence is dubious, precisely because we have such wide and varied evidence. Women: Why would Peter bother hypnotizing women, to produce such detailed false accounts, given that plenty of men wouldn't believe them? Then too, as I pointed out in an older post apropos of conspiracy theories, at least some of the women had other men in their lives who exercised influence over them. There would always be a chance that, e.g., Chuza, Joanna's husband, would smell a cult and pull Joanna away from its overwhelming, nefarious influence.

What about Thomas? Thomas is the Grumpy Cat among the disciples, stoically saying that they might as well return to Judea and die with Jesus (John 11:16). One must just raise yet further the power of Peter's hypnotic powers, in an obviously (even more) ad hoc way, to claim that Peter was able to carry out an elaborate scheme of hypnotizing both Thomas and all the others into thinking that Thomas wasn't present at Jesus' first appearance (which didn't really happen), refused to believe, and then was present at a later event (which also didn't happen), when he saw Jesus himself and his doubts were overcome. Thomas is a real killjoy for any hypnosis or influence theory.

James the brother of Jesus is another especially bad candidate for hypnosis or even some lesser variety of influence. This point is automatically relevant to conspiracy theories as well. Both Josephus's and Hegesippus's accounts of James's death indicate that at least by the time he died he was well-thought-of in certainly Jewish circles. This is why, according to Josephus, there was a Roman inquiry into his lynching; those who thought well of him drew attention to the fact that the high priest exceeded his authority by ordering his death. Since he wasn't apparently a follower of Jesus prior to Jesus' death (John 7:5), there is reason to think he wouldn't have been influenced by Jesus' followers at all in the weeks following Jesus' death. James was apparently firmly rooted in his own circles. Why should he be swayed by the influence--ordinary or extraordinary--of Peter or the other disciples to believe that something happened to him that didn't really happen? And why (anticipating the conspiracy section) would he have joined with them in a dangerous conspiracy if he knew better?

As with many deceiver theories, including Cartesian theories, the issue of power creates concerns for this deceiver theory (and even more for the conspiracy theory discussed below) that don't apply to the realist theory (or the bodily resurrection). If Jesus really rose bodily there is no question of his not having the power to be seen, touched, heard, etc., by individuals, by multiple people at one time, and by multiple people on multiple occasions and in different places. All Jesus has to do is show up! Once we have apparent events in which Jesus appears to be showing himself bodily in a given region, and appears to want to do so multiple times, there's no reason to think he can't do so yet again as long as he's on earth. He's the bodily risen Jesus. Power is no problem. 

In contrast, if we're really trying to spin out this Peter Hypnotist theory, we can't just assume that the hypnotist has so much power that he can carry out such powerful mind tricks on these people in this set of circumstances and also on these other people in this different set of circumstances, and also these other people, etc. Just as we can't assume that a Cartesian Deceiver has infinite power to deceive in an indefinite number of different ways, we can't assume that for a powerful hypnotist among the disciples.

Again, why am I spending so much time on this? See what I said above about the twin theory. 

2.6 Resurrection evidence and conspiracy

On to lying and conspiracy (without hypnosis). Here's another attempt from Dale Allison at a vaguely unifying Type 2 theory--this time, ordinary influence by social pressure inducing some people to lie:

If, let us say, two or three of the disciples said that they had seen Jesus, maybe those who did not see him but thought they felt his presence would have gone along and been happy to be included in “he appeared to the twelve.” p. 63

It is easy to envision some among the five hundred or the twelve, on hearing others declare that they were seeing Jesus, decide that they were too, even if their perceptions were indistinct or confused. They might readily have succumbed to the social pressure to go along with the crowd, or have not wanted others to judge them to be of little faith. p. 299 

Whether one wants to call it this or not, this is a kind of conspiracy theory. In this picture, some of the disciples, together in a group, saw Jesus in a vision while others who were present at the time didn't, and knew that they didn't. But they lied and went along with something that they knew they didn't really experience, due to mutual social pressure within a religious group. Needless to say the Gospel accounts do not say anything of the kind and in fact imply the contrary. For example, in John 20:20 the disciples are glad when they see Jesus (and they also don't appear to have any trouble recognizing him) and they tell Thomas what they have seen. Only Thomas dissents, disbelieving the apparently unified testimony of the others (20:25), until he personally sees Jesus in a later appearance.

Multiple people colluding in a lie is a Type 2 subhypothesis of ~R, because by its very definition it postulates dependence between or among some of the testimonies. You can't have collusion if there's only one person involved. This theory is also radically deficient as an explanation, due to the variety and scope of the evidence we have, both about how many people apparently claimed to have seen Jesus after his resurrection and, especially, the persecution they risked and endured.

It's surprisingly typical for skeptics and liberal scholars to engage in a kind of two-step: On the one hand, they sniff at the fact that Christians talk about the disciples' willingness at least to risk martyrdom (and in some cases to be martyred) for their belief in Jesus' resurrection, as an argument that they were sincere. The usual form this takes is to say that this point is already acknowledged, at least in modern times and sometimes even in older skeptical authors, and doesn't get us very far. The implication is that Christians are attacking a straw man. Allison writes,

Likewise, and in connection with the resurrection of Jesus, the argument from sincere belief only negates the long-discarded theory of Reimarus, who envisaged Jesus’ inner circle clandestinely stealing his body and inventing a religion for their own gain. p. 310

But at the same time, as in the quote above from the very same book, the author will put forward the idea that at least some of those involved, even among the original twelve, did not have evidence for Jesus' resurrection and knew that they didn't, but said that they had seen Jesus anyway. What is this if not insincere proclamation? And of course the extremely widespread reference to "apologetical embellishment" in the Gospels, and the casual assumption that early Christians in general were not above making such embellishments to convince the unconverted, raises the question, "If this is your evaluation of the character of early Christians, why not attribute it to the original alleged witnesses?"

I've documented (see here) that C. H. Dodd seems to think that even the tiny snippets in the Gospels that he thinks represent the earliest "kerygma" already contain apologetic embellishments to convince people that the disciples had good reasons for what they claimed. This is an extremely striking claim when one stops to think about it in light of the controversy (which I've inaugurated and been much involved in) between maximal data apologists for the resurrection and minimalists of various stripes. A central point in this controversy is that the minimalists don't leave themselves with enough material to work with to argue that the disciples had good reasons for believing that Jesus was risen bodily. Dodd, who is often cited by Gary Habermas as part of his minimal facts argument, is implying that they did not have good reasons and that even the earliest form-critical layer of the church's proclamation seeks to rectify this problem by making things up!

In an interesting dialogue with Philip Goff (whose position is nearly identical to Dale Allison's), I was forced repeatedly to point out that if a Gospel author knowingly included a made-up appearance or made-up aspect of an appearance of Jesus for apologetic purposes, this would be neither more nor less than a lie. A story doesn't work to, say, counter Docetism or counter the idea that Jesus was a ghost unless the audience believes it. But there is a surprising amount of squeamishness in contemporary liberals and even skeptics in just saying that the disciples or the evangelists (whether or not they were disciples) lied. Apparently that is regarded as gauche; euphemisms are preferred.

I say all of this to point out that, in the case of conspiracy to lie, the 19th-century-style apologetic arguments on this point are by no means misdirected, since some form of conspiracy theory for the disciples is still alive and well, though sometimes labeled differently.

And our evidence of what the disciples proclaimed and under what circumstances they proclaimed it is deadly to such theories. 

First, consider the fact that even a 2-man conspiracy theory has some degree of independence built in due to human free will and motivation. Contrast this with a case of dependence in which we find two sets of fingerprints in two different places and find them to be identical. Fingerprints don't have free will. They can't choose not to agree with one another. If they're the same, they're the same. (Here I'm leaving aside the possibility of fraudulent placement of fingerprints and focusing on the causes of similarity.) When you have even two people choosing together to lie about something, there is (paradoxically) a degree of independence built into the theory even though it is a Type-2 dependence theory (in the sense that it directly postulates dependence). Each one has to have sufficient reasons for entering the conspiracy.

When there is great risk in lying this is an even greater issue. Each person has to decide that it's in his best interests or furthers some cause that is important to him to such an extent that it's worth lying under conditions of risk, and different people typically weigh these factors differently. Also, some people have strong scruples of conscience about lying while others don't, or scruples of conscience may develop as time goes on. 

This is especially pertinent to the resurrection, since even at the time of the first proclamation on Pentecost in Acts 2 the disciples could expect persecution rather than personal benefit from their story. Pentecost was less than two months after Jesus' crucifixion. The very same religious leaders were in power. The resurrection proclamation was bound to anger them, since it involved accusing them of having the Messiah crucified. And it involved the claim that God was on the side of Jesus' movement rather than on the side of those religious leaders, and that this was demonstrated by the resurrection of Jesus. Peter makes such hard-hitting points explicitly in Acts 2:23-36. Those religious leaders had just recently been successful in inducing Pontius Pilate to crucify Jesus on a charge of sedition. Jesus' core disciples were almost certainly not Roman citizens, so, like Jesus, they could be summarily condemned by a Roman governor to a horrific death. And the more followers they garnered for their movement, the more plausible it could seem to Pilate that they were dangerous revolutionaries. What did any of them, even two of them, hope to gain? 

A skeptical response is to point to the early Christian habit of selling property and bringing the proceeds to the apostles for distribution (Acts 4). Perhaps they just wanted money and were hoping to exploit their followers. But this would be a foolish hope given the above considerations. It was at least as likely that they would be killed before they had an opportunity to enjoy ill-gotten gains. (Nor, as a historical matter, do we have reason to think that they waxed rich in any event. Even the Apostle Paul, in his tensions with Peter, never implies anything of the kind. His perhaps slightly wry comment, found in I Cor. 9:4, is that some apostles traveled with their wives. And his harshest condemnation of Peter, found in Galatians 2:11-13, is that for a while he refused to eat with Gentiles.)

These issues concerning independence and risk multiply greatly as more and more people are brought into the conspiracy. The more people involved, the greater the probability that one of them will back out. Knowing this (and it's just common sense), it would be unwise even for highly motivated conspirators to keep involving more and more people and inducing them to tell more and more stories. A conspiracy of three may be manageable (though even then a context of persecution makes it a dubious enterprise). A conspiracy of twelve, fifteen, twenty is reckless, dumb, and likely to fail. 

On this issue, Chuck Colson's comments on conspiracy and the resurrection are on-point:

[F]ast forward nearly 2,000 years, to an event I happen to know a lot about: Watergate. You see, before all the facts about Watergate were known to the public–in March 1973–it was becoming clear to Nixon’s closest aides that someone had tried to cover up the Watergate break-in.

There were no more than a dozen of us. Could we maintain a cover-up–to save the president? Consider that we were political zealots. We enjoyed enormous political power and prestige. With all that at stake, you’d expect us to be capable of maintaining a lie to protect the president. 

But we couldn’t do it. The first to crack was John Dean. First, he told the president everything, and then just two weeks later he went to the prosecutors and offered to testify against the President. His reason, as he candidly admits in his memoirs, was to “save his own skin.” After that, everyone started scrambling to protect himself. What we know today as the great Watergate cover-up lasted only three weeks. Some of the most powerful politicians in the world–and we couldn’t keep a lie for more than three weeks.

So conspiracy is disconfirmed by the sheer number of people involved in this case. In the immortal, facetious words of Antonin Scalia, "So everything from the Easter morning to the Ascension had to be made up by the groveling enthusiasts as part of their plan to get themselves martyred." (Note too that Scalia's and Colson's comments do not require actual martyrdom for their relevance but could be satisfied even by a great risk of martyrdom.)

As in the case of the Peter Hypnotist hypothesis, in a conspiracy hypothesis we aren't allowed just to wave a hand in the direction of collusion and then assume that this is ipso facto a dependence-creating theory for any number of conspirators and any amount of evidence. Just as we can't assume an indefinite amount of power on the part of a hypnotist to implant elaborate fake memories, we can't assume an indefinite amount of influence by a group of leaders to keep in line an ever-larger number of followers. At a certain point one is just stipulating, "And all of these other people went along too," which is much like stipulating that a bunch of additional coin tosses fell heads despite the fact that the coin was fair, or that a bunch of additional people just happened to go crazy at the same time and had similar hallucinations. 

Here's a technical way to look at this: Suppose the initial collusion subhypothesis is "Peter and John colluded to say that Jesus rose again and that they had seen him." In addition to the fact that Peter and John must both decide that this is a good idea in the circumstances, which is improbable, this subhypothesis arguably doesn't raise the probability of, e.g., "Philip lied and said that he saw Jesus risen bodily" any higher than its average probability given ~R. Philip's acquiescence--and not just acquiescence but even active participation--can't be assumed based on Peter and John's reckless decision to perpetrate a hoax. And as for Thomas...! Good luck getting him to go along with this and even add a phony set of stories in which he's first skeptical and then converted.

Similarly, the question of why the leaders should involve a group of women at all in such a fraud is highly pertinent. The women's negligible influence over others would seem scarcely to outweigh the possibility that one or more of them, under other pressures, would refuse to cooperate. Yet the Gospels record a detailed story of a meeting between Mary Magdalene and Jesus, a fairly detailed story of several other women (Luke names three and adds that there were "others" as well) finding the tomb empty, and a story (in Matthew 28) of more than one woman being able to touch Jesus while they were together. Again, the reportage model undergirds the idea that at least this was what these people claimed. Their involvement in a conspiracy is most unlikely. I've written more about this point here).

More: Given the variety of incidents recounted in the Gospels, with partially overlapping and partially non-overlapping groups of people involved, it becomes increasingly difficult to account for these by way of lying. Again, the reportage model supports particular views on who said what when. The conclusion in this case is that, e.g., Thomas and the others agreed in saying that he was initially suspicious but then had his suspicions overcome on a later occasions. Also, that a group of seven people, five of whom are named, claimed to have encountered Jesus and shared a meal with him by the Sea of Galilee (John 21). Also, that the eleven claimed to have encountered Jesus both in Jerusalem and later in Galilee. Again, the reportage model leads to rejecting the idea that the first encounter in Jerusalem (Luke 24, John 20) and the encounter in Galilee (Matthew 28) are partially-invented "versions" of some quite different experience. Instead, they represent at least what the relevant people claimed about locations. And indeed there is no particular reason why Jesus, really risen, could not have met the male disciples in both places at different times. It is only when one starts with the assumption that the Gospel accounts are heavily embellished that one thinks it plausible that the Galilee and Jerusalem meetings are versions of the same event. 

This means that a problem with the conspiracy hypothesis is just how unwieldy this set of lies became and how foolish it would be to try to produce such a creaking edifice of fraud and falsehood. As with the problem of including an unnecessary number of people, so here: A degree of independence arises from the need to stipulate that the conspirators kept on trying too hard and producing far more stories than necessary. (I've made a similar point about Cartesian Deceiver scenarios. See here.)

We must remember that the followers of Jesus could have just kept their mouths shut, and the whole thing would have died down. They could just have gone back to practicing Judaism without antagonizing the religious leadership. They brought the persecution described in Acts upon themselves by getting up and loudly proclaiming that the Jewish leaders were complicit in killing the Messiah and that he was vindicated by God through his resurrection. Feeble comparisons to suicide bombers, kamikaze pilots, and people who drink kool-aid will not serve, since these earliest alleged witnesses were inviting and suffering persecution not for a political or social ideology or an unempirical religion (which is so much easier to believe sincerely but falsely) but for empirical claims about what they had supposedly witnessed for themselves.

It was at just this point in our 2009 article that Tim came up with the inequality that has served as the catalyst for so much of my own later professional work--the claim that dependence considerations serve H or ~H just to the extent that the dependence of the items of evidence is greater, all things considered, given one of them. We then considered the probability that there might even be negative dependence given ~H, due to the fact that some disciples would know that other disciples had indeed suffered (or even for that matter been seriously threatened) for their testimony. I'll emphasize this even more below when I discuss multiple instances of persecution in the early chapters of Acts. As in the case of two witnesses who dislike each other, it is plausible to think that truth will be somewhat helpful in stiffening spines in the face of persecution. This constraint by truth can operate both at an individual and at a group level. If the various disciples know that what they are saying is true, this can create a fruitful ability to influence one another, given truth, by reminding each other of what they know. But a previous determination to lie (for highly unclear benefits, even to begin with) will be likely to be shaken and thus a conspiracy will be likely to fall apart under duress. 

The richness of the evidence, both in terms of the number of people involved and the number and detail of complementary alleged appearances, is evidence for R over ~R (and vastly so) and also, for that matter, evidence for R over the subhypothesis, "~R and there was a conspiracy to make it look like R." For the latter is not indefinitely expandable without simply re-introducing independence under ~H.

We should therefore not be surprised to find that skeptics and liberal scholars sometimes suggest that actually there wasn't such a large number of people testifying to these things at all. This denial is at the heart of the "minimal witnesses" theory of Youtube skeptic Paulogia, and Dale Allison implies something similar. But interestingly, Paulogia overclaims: He (of course) rejects the historical reliability of the book of Acts, so he doesn't take at face value the claims about multiple arrests and the like. But he also says that even if he did take Acts at face value, the disciples other than Peter and John are involved in proclaiming the resurrection only until the first time they are told to stop, which is flatly false. 

Dale Allison (pp. 309-310) is a little more accurate about the contents of Acts than Paulogia, since he acknowledges at least that the twelve are still leading the church through Acts 6, though he apparently overlooks the reference to them in Acts 8:1. But he still conjectures that disciples other than Peter, James the son of Zebedee, and James the brother of Jesus might have gone silent, died abjectly rather than bravely, or recanted and avoided persecution, an occurrence which he thinks a Christian document like Acts would have suppressed.

I have documented what Acts claims and its relevance in detail elsewhere (see the playlist here). To recap briefly: The scene of the election of Matthias (in which, I note, no miracles are alleged) in Acts 1:13-26 is of great apologetic importance. I've realized this forcefully only in recent years. In that chapter, the author of Acts is careful to list, by name, all of the eleven. He then tells how Peter gathers the earliest Christians together before Pentecost and initiates an election for a replacement of Judas Iscariot, so there will still be twelve core leaders of the new movement. He is quite explicit that candidates must have been witnesses of Jesus' whole ministry, must be prepared to be witnesses to Jesus' resurrection, and must have witnessed his ascension. The group nominates Matthias and Joseph Barsabbas, known as Justus. Matthias is elected by drawing lots. In the next chapter (Acts 2:14), less than two weeks later, the author stresses that the eleven stood up with Peter and thus prominently associated themselves with what he was preaching, which of course included God's vindication of Jesus by his resurrection. In Acts 2:37 audience members are said to have asked not just Peter but also the other apostles what they must do. Given the very large number of people baptized that day, and given the absence of public address systems, this is all the more plausible. It seems undeniable that more people than just Peter were answering questions and baptizing. 

Through at least Acts 6 and the story of the appointment of deacons (6:1-6), the word "apostles" seems to be used in its narrow sense to refer to the Twelve, not in the broader sense of referring to any messenger. As one correspondent pointed out to me, in Acts 4:36-37, even Barnabas is distinguished from the apostles, as the term is used here, though later in Acts 14:14 the term is applied more broadly and includes Barnabas. In Acts 4, Peter and John are arrested, threatened, and warned not to preach about this anymore. They categorically refuse. In Acts 5:17ff, a group of apostles are again arrested, and vs. 29 makes it clear that this group includes more than just Peter and John. They are released, the account says, by an angel, but however it came about, they are defiantly preaching yet again on the next day. 

This time the Jewish council considers killing them. It's unclear what the method would have been. Stoning would endanger their position with the Roman government, though that doesn't prevent the lynching (by stoning) of Stephen in Acts 7. Perhaps they were considering trying again to take them to Pilate, as they did with Jesus. Gamaliel talks them out of killing the apostles, or trying to, and instead they are beaten, which would be no mild punishment in itself. Not only is this not the first time they are told to stop preaching (there have been at least two earlier prohibitions), but even after being beaten they express to the other Christians their intention to go on preaching and do so publicly (verses 41-42). In chapter 6 they appoint deacons so that the Twelve can devote themselves to preaching. In chapter 7 Stephen, who evidently combines preaching and debating with his deacon work, is stoned to death. After that a persecution by Saul of Tarsus (later the apostle Paul) breaks out, but per 8:1, the apostles remain in Jerusalem nonetheless.

An interesting side note here is that it's in Acts 6:7 that we're told that many priests became believers in the new movement. This provides a reasonable point at which the Christians could have learned the inside information that the Sanhedrin had considered killing the apostles and had been dissuaded by Gamaliel, as reported in Acts 5. (A later point would of course be when Paul converted, since he had been a pupil of Gamaliel at one time according to Acts 22:3. But 6:7 is also plausible.)

This litany of early warnings and persecutions, and the very clear indication that twelve named people who claimed to be witnesses of Jesus' resurrection remained steadfast and bold throughout, prominently leading the new movement, is highly significant and makes theories of lying by at least any of these twelve simply untenable. 

Not only is twelve an unwieldy number for an enduring conspiracy. These specific leaders were known as leaders by the hostile religious rulers in Jerusalem. They had made no secret of their leadership role, from the day of Pentecost onward. Apostolic leaders were directly threatened on multiple occasions and even beaten. These multiple incidents provided multiple opportunities for apostles to back out. Yet there they are, still explicitly leading the church in Acts 6:2. Each time something new happened, anyone who knew that he was attesting to something he knew to be false, even just "going along" (to use Allison's term), had another opportunity to decide that this wasn't worth it. When they come with the whip to beat you for teaching something that you know isn't even true, it's quite understandable that you will say, "It's getting real now! I'm outta here!" But Acts makes it clear that they didn't. 

There are thus at least three considerations producing a significant degree of independence, given ~R, that weigh against lying. There's the at-least-partially independent inclusion of an unnecessary number of different false stories, making the Big Lie overly complex. For the testimony of, at least, the twelve, there is a dimension of independence among the different individuals, created by the initial evaluation of risk vs. benefits (falling heavily on the side of risk with very little benefit). The initial conspirators have to decide to include such a dangerously large number of individuals. And different individuals all have to decide that it's worth it. This becomes more improbable the more individuals are involved. Third, as we layer in the (even non-miraculous) successive accounts in Acts of repeated threats, beatings, and more, we get independence considerations across time for these various individuals--that is to say, separate occasions on which various members members of the twelve could have backed out of a lie but, per Acts, didn't do so.

2.7 "Something else" won't do the trick

Here is a moral of the story for all of these attempted ~R subhypotheses. Write this in big letters over the resurrection debate: It's not good enough just to say, "The resurrection didn't happen and something else happened instead that made it look like it did happen." "Something else" is not an explanatory hypothesis! And still less is it a dependence-creating subhypothesis under ~H.

There are times and places where a somewhat vague reference to a ~H subhypothesis makes sense. If you already, independently know you're at a magic show, then you don't have to know how the illusion works. By definition, that's what a magic show is--producing impressions that appear to be something that they aren't. It's legitimate to say, "It's some kind of trick."

When we consider instead a complex empirical case for a series of events (as in the case of Jesus' postresurrection appearances), saying, "It's some kind of trick" is just non-explanatory. There is no kind of trick that we know of that produces this whole set of effects.

What this means is that one can't appeal to dependence under ~R by assuming that there must exist some ~R subhypothesis that actually unifies all the evidence we have. (Don't forget the discussion above about gerrymandered hypotheses that don't really make independence go away.) 

That's why I've taken so much time considering potential dependence-creating ~R subhypotheses in detail: To show that I've covered the bases and that nothing does the trick.

2.8 Named individuals, independence, and other Gospel miracles

While I've been considering writing an update post on independence for a long time, I had a conversation in mid-August of this year that made me all the more determined to do so: A good friend asked me why it was legitimate for Tim and me to count the number of people involved in the Gospel resurrection stories (at least the male disciples and James the brother of Jesus) and make a cumulative Bayes factor representing all thirteen of those people, but why (in conversation with him) I was balking somewhat at treating different people who were present for miracles during Jesus' ministry in the same way. Why not say that we have at least thirteen attestations to, say, the raising of Lazarus--Jesus' twelve disciples and Jesus himself? (Lazarus is my example, not his. But it's the kind of thing we were discussing.)

I don't want to be misunderstood: I think we have good evidence for the raising of Lazarus. If the story even represents what people who claimed to be present claimed to have witnessed, it would be very difficult for them to be merely mistaken. And a bizarre conspiracy involving Mary, Martha, and Lazarus to pretend he was dead is really beyond the pale. But I would hesitate to create a cumulative Bayes factor involving Jesus and all twelve of his disciples as separate witnesses. Why? It was a good question, and one that I think I should answer in case it occurs to anybody else.

Four considerations seem to me to support making a distinction in probabilistic modeling between miracles in Jesus' ministry and the resurrection. First, in the election of Matthias passage Peter is absolutely explicit about being witnesses to the resurrection, specifically (Acts 1:22). It's true that in the same passage Peter broadly mentions Jesus' ministry (vs. 21), and in the sermon on Pentecost Peter, flanked by the remaining members of the twelve, mentions "mighty works and wonders and signs that God did through him in your midst" (2:22). But no specific wonders and signs are mentioned. In contrast, Peter says in so many words, "This Jesus God raised up, and of that we all are witnesses" (2:22). 

So, although I definitely think that the story of the raising of Lazarus came from at least one person who was there, and though I defend traditional authorship on the fourth Gospel, so the Beloved Disciple is one such person, we don't actually have evidence about how many different people specifically attested to the raising of Lazarus, even among the twelve. In contrast, the book of Acts is extremely emphatic that twelve named individuals were willing to stand up (2:14) and be counted as witnesses of the resurrection.

Second, the Gospel accounts themselves are surprisingly explicit about who was and wasn't present at various resurrection appearances. Although there are occasional listings of who was present at a given ministry miracle (e.g. Peter, James, and John at the raising of Jairus' daughter in Mark 5:37), usually we don't find this. Usually we're left guessing about how many people were present for a given miracle and in a position to tell about it. These can be educated guesses, based in part on the implications of the stories and in part on differences in wording and salient details among different Gospel accounts of the same event. But for the most part that's what they remain.

In contrast, we have things like Luke's list of women, including three named individuals--Mary Magdalene, Joanna, and Mary the mother of James--attesting to the empty tomb (Luke 24:11), Mark's list of women who went to the empty tomb--a list that doesn't entirely overlap with Luke's (Mark 16:1 leaves out Joanna and includes Salome), Matthew's slightly different disambiguator for "the other Mary" in the tomb story and his failure to list either Joanna or Salome (Matt. 28:1). Richard Bauckham himself has taken note of this point about the named women who went to the tomb (Jesus and the Eyewitnesses, pp. 48-50).

In Luke, we find Cleopas named in Luke 24:18 in the Emmaus story, but no name for his companion. If Luke were simply making things up, he might as well have made up a second name! Luke specifies that Jesus' appearance that evening to a larger group of male disciples included the eleven and others with them (24:33). If one is not convinced that "the eleven" here is a group designation only, one may even think that Luke was slightly mistaken here, since John carefully says that Thomas wasn't present on what appears to be that occasion on Easter evening (John 20:24) but was present eight days later. But we can wonder about such a mistake only because both John and Luke try to tell us who was present. Nothing like this happens, that I can recall, in the case of any of the ministry miracles. John lists five out of the seven people allegedly present at the appearance at the Sea of Galilee--Peter, the two sons of Zebedee, Nathanael of Cana (apparently not a member of the twelve), and Thomas (John 21:2), and he says that two others were with them but doesn't specify names for them. Again, why not invent more names if one is inventing as opposed to remembering and (perhaps) unsure after the passage of years who the other two were?

Third, when the Twelve attest to Jesus' resurrection, they are attesting not just to a single experience but to multiple experiences that all support R. The Gospel accounts instance one occasion in Jerusalem and one in Galilee where all of the eleven were allegedly present, and one additional occasion in Jerusalem where ten out of eleven were present. For Peter and the sons of Zebedee, the occasion in John 21 is yet another story of their encountering Jesus alive. In Acts 1:2-3 the author emphasizes that Jesus appeared to the apostles (remember that at this point in Acts the term appears to be used narrowly) over a period of forty days, providing "many proofs" that he was alive. This is different from any particular ministry miracle, since each of those was a single event. Thus the disciples had more opportunities to confirm Jesus' resurrection, and that should be represented by our modeling. In part it increases the strength of a single person's affirmation of the event, since that person had more opportunities to verify the fact in question (in this case, that Jesus was risen bodily). But as argued above it's also relevant to the issue of independence given ~R, since it is negatively relevant to theories like conspiracy, given the elaborateness of the set of stories that would have to be concocted. 

Fourth, the question raised by my friend specifically concerned whether Jesus' knowledge of what occurred at the time of his own miracles should be modeled separately from the knowledge of the other disciples. I had pointed out that, without assuming that Jesus had supernatural knowledge of such things as whether the person was really ill and was really healed (which in the context of the argument would be circular), we usually can't assume that Jesus had any special way of knowing about the situation that went beyond what the disciples knew. So, for example, without assuming supernatural knowledge, we can't assume that Jesus had any greater knowledge that the crippled man in Mark 2 really was crippled (though I have no doubt that he was; I'm just playing devil's advocate here) than anyone else who was standing as close to him as Jesus. And the same for whether the woman with the issue of blood (Luke 8) was really healed and didn't relapse later. 

A major question at issue, from a skeptical perspective, is whether these things happened at all. The reportage model of the Gospels and Acts doesn't tell us that Jesus himself attested to these things later on. It does, I think, permit us to say that these accounts came from people close to the facts, but the disciples fulfill that description, and as pointed out above, we don't know how many of them talked about one specific ministry miracle. In fact, it's pretty improbable that an account given by Jesus is the source from which a given ministry healing miracle made it into the Gospels. There was no reason for Jesus to tell disciples who were present what had happened. Jesus wouldn't be interviewed by Luke or Mark. And none of the Gospels were allegedly written by Jesus. It's much more likely that the ministry miracle accounts came from one or more of the disciples, telling them for the benefit of people who weren't there. And as already noted, it's difficult to know how many separate disciples testified, except by noting variations between the accounts which could come from separate witnesses present. (Digression: Jesus is a not-implausible source for the stories of some events in the Gospels for which he would have been present and presumably his disciples would not have been--the temptation in the wilderness being a good example. Further digression: I don't think Jesus had to be alone during his conversation with Pontius Pilate. Guards, at least, could have also been present, and perhaps one or more friends were permitted as well. Mark 15:39 makes it not wholly implausible that one or more Roman soldiers involved in Jesus' death may have joined the Christian movement. The accounts of Jesus' conversation with Pilate don't say that he and Pilate were alone, though skeptics typically say that they were alone based on an overreading of John 18:33. But in any event these are not healing miracles.) 

In contrast, the reportage model of Acts indicates that the twelve disciples did witness to Jesus' resurrection. 

2.9 Simplification and potential underestimation: The testimony of the women 

The question that lies behind this entire post is whether, in treating (especially) the testimony of thirteen disciples (the twelve plus James the brother of Jesus) and the testimony of the women as independent in our estimated Bayes factor, Tim and I overestimated the strength of the total evidence.

In a couple of ways, I would argue that we actually underestimated the evidence. One of these where I think this can be argued most strongly is the testimony of the women, to which we gave a Bayes factor of 100/1. I now think that this doesn't give enough force to the following considerations, especially when one emphasizes that, on the reportage model, these things were at least what the women claimed: 

1) There were multiple, named women involved. Four are named in the accounts--Mary Magdalene, the other Mary, Salome, and Joanna. 

2) They claimed to have had encounters with one or two men in white and to have heard specific words from these men/angels. 

3) At least two or more of them claimed to have been able to touch Jesus. 

4) Mary Magdalene apparently gave a very detailed account of a one-on-one, polymodal meeting with Jesus which, despite claims of detailed grief hallucinations, is still better explained by his actually being there physically. (I've discussed above why this is so.) 

5) The testimony of the women incorporates the evidence that they found the tomb empty. I think we were right not to break out "the tomb definitely was empty" as a separate, undeniable fact, as is more commonly done, especially in minimalist cases. But it's still worth noting that W (the testimony of the women) requires an explanation of the fact that the women claimed to have found the tomb open and empty. 

6) The variations among the accounts of the women's movements, much harped upon by skeptics, argue for independence between the accounts, and we can even connect at least two of the accounts of finding the empty tomb with plausible specific sources--e.g., Mary Magdalene for John's, Joanna for Luke's. (Luke mentions Joanna in more than one place in his Gospel and is the only one to do so.) Possibly "the other Mary" for Matthew and Salome for Mark, though this is more conjectural than the connections for Luke and John.

A consideration on the other side suggesting caution is that we don't know for sure that those specific women were persecuted. This is in contrast to the direct targeting of the twelve apostles narrated in Acts. On the other hand yet again, we do know that Paul's persecution targeted women generally as well as men (Acts 8:3), and the fact that most of these women were known by name to be witnesses of Jesus' resurrection would have made them at least somewhat more likely to be targets.

All things considered, and bearing in mind that there isn't some way to churn these things out mechanically, I think that it would perhaps still be somewhat conservative to assign a Bayes Factor of 10⁴ or 10,000/1 to the cumulative testimony of the women. You can think of this as giving each named woman a Bayes factor of 10/1 (less than that given to the named male disciples in the paper, due to the male disciples' more specific persecution) and then multiplying, recognizing that this is still a simplification. It probably still underestimates the force of the very detailed testimony from Mary Magdalene, individually. It takes into account the failure of subhypotheses under ~R to account for the shared polymodal contents of the claims when the women were in groups (e.g., that they collectively heard the man/men in white speak, that several of them collectively were able to touch Jesus). It is modest about individual Bayes factors given that we don't know if these specific women were persecuted, though if they weren't stupid they must have realized that they were at risk of persecution. It also takes into account the independence indicated by variation among the Gospel accounts concerning the women.

2.10 Simplification and potential underestimation: Other named individuals

Another way in which I think we were conservative in the original paper concerns several other people who are named whom we did nothing with: Nathanael (John 21:2), Cleopas (Luke 24, who apparently made claims to a two-man experience with an unnamed companion), Joseph Barsabbas, who evidently claimed to be, and was believed to be by the earliest Christians, a witness of Jesus' resurrection making him a candidate for replacing Judas (Acts 1:23). There is variation among these, since, on the one hand, Nathanael and Cleopas both were allegedly involved in specific, detailed, polymodal appearances, while we don't know exactly when Joseph Barsabbas was allegedly present. (I'd probably guess at the first appearance to a larger group of the male disciples as told in John 20 and Luke 24, especially since Luke says that there were others there as well as the eleven. But that's a conjecture.) For what it's worth, Joseph Barsabbas was so prominent in the community that he was nominated to be a member of the Twelve.

The same consideration applies here that applies to the women--namely, that we don't know if these men were persecuted as the Twelve were. But, again, the persecution from Saul of Tarsus definitely didn't confine itself to the Twelve, not by a long shot. Even if we assigned a Bayes factor of merely 3/1 to the evidence about each of these men who is named, that would have significance.

Digression: We did absolutely nothing with the "appearance to the 500," and I'm inclined to go on doing nothing with it, due to the fact that it's mentioned only in a list with no details. This is another way in which we were, and are, trying not to overestimate, though it's probably not strictly valueless. The most evidential aspect of Paul's mention of the 500 in I Cor. 15:5 is that he says some of them are still alive at the time of writing. He obviously considers this to have evidential import. But we don't know if those he has in mind are some we're already counting, especially the women and the three additional men just mentioned. It's very likely that in a group of 500 all or nearly all of Jesus' followers would be present. We should be careful not to double-count someone like Nathanael, and we have no way of knowing if we're doing this for the 500 or even those alive at the time Paul is writing, since he doesn't name anyone.

2.11 Simplification and averaging Bayes factors

Beyond doubt, our argument and Bayes factor included some simplification. Sometimes simplification involves a kind of averaging of considerations for members of a group. For example, Peter and John's testimony should arguably be ranked more powerful than the testimony of the rest of the Twelve due to their prominent role as spokesmen and due to the explicit statements attributed to them in Acts, such as Acts 4:20 where they explicitly allude to what they have "heard and seen." In I John 1:1, John even refers to what they have touched or handled. Though he doesn't speak of the resurrection there, it hardly seems coincidental that the fourth Gospel also emphasizes touch in the resurrection accounts.

Due to the Fourth Gospel's explicit listing of James the son of Zebedee as present on an occasion when not all of the Twelve were present, and the claim in Acts 12:2 that James the son of Zebedee was martyred (apparently the earliest one of the Twelve), arguably James the son of Zebedee could have a higher Bayes factor than some others. 

Thomas for sure could be assigned a higher Bayes factor, since we have a complex story in which he resists belief and then capitulates on the basis of evidence. (And John 21 also says that Thomas was present at the Sea of Galilee.) If Thomas himself claimed this, and the other disciples claimed this, that is particularly good evidence. Contra C. H. Dodd, I do not think that Thomas is a mere "type" of the doubter. Dodd asserted that Thomas isn't a believable human figure, but on the contrary, I think his bluntness and Puddleglum-like nature in other passages in John (e.g., "Let us go and die with him" in John 11:16, "No, we don't know what you're talking about," John 14:5) suggests a real person with a specific type of personality.

But given the repeated allusions to the Twelve in the early chapters of Acts, the explicit listing of all of their names in Acts 1, even of Matthias's name, and the explicit association of the Twelve both in preaching and in persecution, it's justifiable not to try to model these differences among the Twelve and instead to choose a Bayes factor (we chose 10³) which we think strikes a reasonable average. Note too that these Bayes factors include the individually polymodal nature of the claims and the apparent claims to having multiple experiences. Every single member of the Twelve was supposedly present for at least two or three group appearances.

James the brother of Jesus is an interesting example of simplification and averaging in action. On the one hand, we have no account that explicitly says that he was present with a group. Paul lists an appearance to James in I Cor. 15:7, but since that is just a list we get no details. As far as it goes, this provides an argument for downgrading James the brother of Jesus to a lower Bayes factor than the other disciples. 

On the other hand, we have an explicit account of James's being killed by the order of Jewish leaders in Josephus in Antiquities 20.9. None of the other disciples have such an assertion from a source that is non-canonical, early, and non-Christian. We also have a somewhat different (and hence arguably independent) account of his death from the Christian author Hegesippus. This specific evidence for martyrdom is an argument for upgrading his Bayes factor. Moreover, despite attempts to obscure this, the canonical documents do indirectly indicate that James, as a brother of Jesus, was first skeptical (John 7:3-5) and then a member of the Christian movement; a natural interpretation of these implications, given the reference in Acts 1:14 to the brothers of Jesus as being together with the Christians prior to Pentecost, is that James was converted by seeing the risen Jesus on earth, or at least that this is what he claimed. (James could also have influenced the other brothers.) Dale Allison (pp. 77-79) pooh-poohs this inference in part because he thinks the entire story of Jesus' ascension after forty days is late and invented, so he deliberately flattens the distinction between pre-ascension and post-ascension appearances and implies that James's experience might just as well have happened after Pentecost. In contrast, per the reportage model, given all of the evidence for the care and veracity of the author of Acts, it should be taken to indicate at least what the disciples themselves claimed. This means that it has evidential value in various directions, most notably in the direction of sharply distinguishing what they claimed about earthly appearances from what they claimed about any later vision (such as that of Stephen or Paul's Damascus Road experience). This is relevant to an evaluation of when James had a resurrection experience. 

Putting all of these considerations together, it doesn't seem unreasonable to give James the brother of Jesus an average Bayes factor equal to that of the other disciples.

2.12 Simplification and diachronic modeling

Another way in which it is a simplification to treat thirteen disciples and the collective testimony of the women as independent lines of argument for R is exemplified by considerations I've laid out in the rest of this post concerning unification under R. We are trying to represent many things here--for example, the simultaneous facts that so many people claimed to have seen Jesus, that so many claimed to have had various robustly bodily group experiences, and that the Twelve (and apparently James Jesus' brother) persisted in these claims despite beating, repeated warnings not to continue, and in some cases actual death. 

My usual proceeding in modeling complex empirical cases is to use an imaginary diachronic model: Suppose that you first found out this, then you found out this, and so forth. That is very difficult to do in this case. One tries almost in vain to think of a cleancut way of "layering in" the fact that such-and-such number of people testified while separating that from the detailed content of their testimony and the detailed circumstances, and then layering the specifics in subsequently. In the case of group claims, how does one separate the claims of different people to have had group experiences? One wants to take into account both the fact that particular individuals apparently claimed to have had multiple group experiences and the fact that multiple people apparently claimed to have been present together for the same group experiences. 

The variety of contexts and groupings of the reports is essential to the strength of the case, but it also makes it difficult to model. Even though we assigned a Bayes factor of 10³ to the testimony of a single disciple, under the conditions of persecution described in Acts, one would not have received the testimony of just one person alone, since according to Acts, Peter and the eleven were standing up together from the first public proclamation. 

But at the cost of being very artificial, let's try to do a little of this diachronic ordering. Try to imagine, if you can, talking with just one disciple who claims that, starting on the evening of the Sunday after Jesus' crucifixion, he experienced at least four different group appearances of Jesus. (This would be the one on Sunday evening, the one eight days later, one in Galilee later as described by Matthew, and the ascension scene as described in Acts 1.) He describes them in detail, as told in the Gospels, including the eating, the inviting to touch, the skepticism of Thomas, the long sermon to the disciples on Old Testament prophecy mentioned in Acts 24, etc., and the eventual ascension after about 40 days. He does this at the time of Pentecost, and he seems aware of the danger of further persecution at that time, given the hatred of the Jewish religious leaders and their recent success in having Jesus crucified. He also calmly declares that the other disciples were there at the time and will back him up, but you haven't had a chance yet to speak to them or to hear Peter preach.

At this point you have some reason to believe his sincerity, you have significantly more reason to think that Jesus rose bodily than you did before (remember that the polymodal nature of his experiences is itself unified and better explained by actual bodily resurrection than by hallucination, even grief hallucination), but you are understandably also wondering if he's severely mentally ill. 

Now things get even more artificial, because I'm going to say "as time passes" and refer to events that would be impossible to know about without also knowing that others confirm this one person's testimony. But bear with me. As time passes, you come to know two additional and very important things: This disciple has a lot more reason to fear persecution even than he had on Pentecost; he's even beaten for his testimony, but he is determined not to recant (Acts 5:41). 

Also, and significantly, he does not have any more of the experiences he originally described to you. They just stop. If asked, he says that Jesus returned to heaven, so he isn't sharing meals on earth with people anymore. This is very important, because if he were really having this whole series of exceptionally vivid, detailed, and polymodal hallucinations due to severe mental illness, they wouldn't be expected to stop abruptly like this, without treatment of any kind. And stopping due to a belief that one has seen the person ascend to heaven is, as already discussed, not typical of bereavement hallucinations either.

This is where a high individual Bayes factor, for just one disciple, for Jesus' bodily resurrection over its negation seems appropriate. The series of events even concerning this one disciple is extremely unexpected given ~R as a whole, and it is even notably better explained by R than by more specific subhypotheses of ~R such as hallucination. Moreover, within the individual stories that he tells, the specifics of his claimed experiences "as of" specific encounters with the risen Jesus are better unified by R than by ~R, as discussed.

But now add in more evidence: You now talk to Mary Magdalene and learn that she, too, claims that she found the tomb empty, talked to two men in white, and then had a detailed, polymodal experience in which she had a dialogue with Jesus (yes, she didn't initially recognize him, but a man was undeniably there, and she was weeping and looking into the tomb when he first showed up), eventually recognized him, and was able to touch him.

You then talk to other women who separated from Mary Magdalene but also claim to have found the tomb empty, talked to two men in white who said he was risen, and later saw and touched Jesus on the road as they were going to tell the disciples. There are differences here so that it's a little difficult to disentangle the exact order of events. Since nobody has a watch, it's difficult to know whether the claim is that Jesus talked to the group of women on the road before or after talking to Mary Magdalene.

You then talk to Peter and John, and they both claim that they, too, found the tomb empty. They also agree with the first disciple's claims about various detailed postmortem group appearances, and they add a story about a meeting at the Sea of Galilee, eating with Jesus there, and a dialogue with him there. Moreover, they are spokesmen for the group and thus are risking more and are arrested more often than the others. They, too, are beaten and are yet unwilling to stop preaching. 

Eventually you learn that the rest of the Twelve make similar claims. Some of the stories have variations in wording and in details of the claimed encounters, just as would be expected. And the Twelve continue to lead the new Christian movement despite being beaten and threatened and despite the persecution of someone named Saul of Tarsus.

Again, none of them claims to have had similar experiences since the event they call Jesus' being "taken up to heaven," which they claim to have witnessed.

The agreements of multiple people with each other, concerning multiple different claimed group appearances, are far better unified by R than by ~R; each specific encounter claim from any one person raises the probability of a subhypothesis of R--that Jesus really did appear bodily on that occasion--which, if true, virtually guarantees that on that occasion others will also be able to interact with Jesus normally. (Remember, R is Jesus' bodily resurrection in a body that can be accessed normally, not just some vague concept that could include exaltation or spiritual resurrection.) 

When, in our older article, we multiplied the disciples' Bayes factors, we did nothing formally with greater unification under R than under ~R, though we alluded to the possibility in the case of persistence under persecution. We simply treated the individual Bayes factors as independent given both R and ~R. 

And I haven't even layered in the story of Paul's conversion. See here for a discussion of a maximalist use of the conversion of Paul, even though he didn't claim to have an on-earth interaction with Jesus like the ones described in the Gospels.

Conclusion

My argument here is a more detailed effort than any thus far to show how many considerations and how many aspects of the evidence for Jesus' resurrection argue strongly against attempts to treat the evidence as dependent in an undermining sense. 

The assigned Bayes factor of 10³ for a single disciple's testimony is arguably legitimate, given the extensiveness of the mental illness that would be required as a ~R explanation and the non-explanation of the ascension claims and cessation of the experiences. (We emphasized both of these points in the original article.) But even if you think it is somewhat too high, or even if you think that, say, averaging still shouldn't put the Bayes factor for James the brother of Jesus as high as the factor for others, remember that the greater unification of subsets of the total evidence under R was not used at all in the argument we presented.

And, as already argued, we probably underestimated the Bayes factor for the testimony of the women, and we did nothing with the presence of other named people alleged at certain events.

Having gone over all of this and thought about it at length and in detail, I am now confident that we did not overestimate the strength of the case, and I am very strongly convinced that we did not significantly overestimate it. This post will hopefully be a resource for others to point to when the question of overestimation via an independence assumption is brought up, whether by Christians, moderate liberals, or skeptics.