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Jumpin' posts [Updated]

Things have been hopping lately a bit in the apologetics blogosphere. Richard Carrier, a skeptic who has reinvented himself recently as an expert on probability, had an interview with Luke Muehlhauser (with whom I had an interview not long ago) in which he airily dismissed Tim's and my work on the resurrection and Bayesian probability as (please excuse the term) "crappy." Evidently there has been a certain amount of rejoicing over this among skeptics who had begun to be a bit worried by our work and by my interview with Luke. I discuss the particular criticism and respond to it at my personal blog here.

Probably most W4 readers interested in Philosophy of Religion already knew about all of this, but in case you didn't, I thought I'd direct you to the action. Links provided in that post. Enjoy.

Update: Both Richard Carrier and Luke Muehlhauser have apologized in the comments to the post at my personal blog. See our interaction there.

Comments (27)

‘what we do argue is, if we are successful, of great epistemic significance concerning the resurrection, because it means that this evidence is so good that it can overcome even an incredibly low prior probability.’

Lydia, it’s not clear to me what it is that may have ‘an incredibly low prior probability’; it would be helpful if you could sketch your argument for the resurrection by fleshing out a bit your imaginary example: What’s the proposition H that is less probable than its negation, and what’s 'this evidence'?

Can I get a map of the apologetics blogosphere? I mean this, sincerely, because I thought I had a fair sense of its scope and I am finding the I have been sitting on an island. I haven't been able to find any good apologetics blog that use math/science without bias or with adequate knowledge, but, apparently, there are a ton I haven't heard of. Any Google hints that would uncover this vast landmass?

The Chicken

Overseas, H in my post is simply a stand-in--some H, any H. I'm walking the reader through the odds form of Bayes's Theorem using imaginary probabilistic examples (H has prior odds of 10 to 1 against it but a Bayes factor for some evidence of 10 to one for it, or 1000 to 1 for it). I'm showing the readers what the odds form of Bayes's Theorem is, how it works, and why it is possible to do significant work without arguing for an actual, specific prior probability. This was the point at issue with the atheist in this case. Obviously, in our article, the hypothesis in question is the resurrection of Jesus. You're welcome to read the article, which discusses the evidence we're using. I link it from the post.

Chicken, I don't actually have a map, but I understand that Vic Reppert's blog, linked from my post at Extra Thoughts, is a medium-sized mammal in the phil. of religion blogosphere and gets aggregated a fair bit. Tim is doing some fun stuff with Bayes's Theorem and Carrier's various blunders in a "handout" on the subject in the post I link.

I read your post, but I didn't read your critic or any of the other links. I know some probability, but the whole idea of using probabilistic inference in this kind of reasoning is very foreign to me. Here are a few comments.

1. You're obviously right that if you can show such a high likelihood ratio that's of epistemic significance.

2. My guess is that the best and most common criticism, and probably the one I'd offer myself if I were familiar with this stuff, is Garbage In, Garbage Out. In other words, I would expect most critics to disagree with the actual numbers you assign to each of the conditional probabilities you use in your calculation. That would be a disagreement on ancient history more than philosophy. Of course there's nothing formally wrong with Bayesian inference, but I don't think it would be convincing if there's disagreement on the "inputs".

3. I'm skeptical of the whole use of probabilistic inference in this kind of reasoning. Presumably you know the work of people like Amos Tversky who've shown that when people actually reason about degrees of belief, they don't follow the axioms of probability measures. Of course that might just mean that we're all irrational and that Bayesian or other probabilistic inference is right and we're wrong. But that seems like it would be an awfully hard thing to prove. I understand that my comment must be extremely naive - I'm sure there's an extensive literature on this question, but I don't know anything about this area. Like I said, it just feels very foreign and...odd.

Sorry if I'm being presumptuous, commenting like this on an article I haven't even read in a field I know nothing about. Take it as an indication of where a naive reader would be coming from.

Of course that might just mean that we're all irrational and that Bayesian or other probabilistic inference is right and we're wrong. But that seems like it would be an awfully hard thing to prove.

Well, the axioms are what they are. I mean, they're mathematical theorems. So in that sense, of course they're "right." But you're right that people will be better and worse at making estimates and applying them to concrete situations.

I don't mind your type of criticism, Aaron, nearly as much as this huge song and dance about the priors and the extremely ignorant implication that if one does not estimate an actual prior one cannot be doing anything of epistemic significance with the probability.

I was also, as you can imagine, rather annoyed (which is putting it mildly) at the implication that we were "misleading" or "slippery," when in fact we were painfully clear. The long quotation from my interview shows exactly how painfully clear I was, yet Carrier actually induced Luke M. to say, in their interview, that there seems to be "something slippery going on" and to throw in a "yeah" or two as Carrier was going on about how confusing we were being to people because of our not estimating an actual prior. This is frustrating, because this implication just arises from a poor understanding of--or a pretense of a poor understanding of--the odds form and Bayes factors.

So I thought it important to clear this up for the record, so that the atheists do not have the excuse of saying (which, I'm sorry, is really just a joke) that Carrier had somehow shown that Tim and/or I had misused probability or had misled people about it.

Certainly, there is no question that the place (or one place) where the action is in all of this is the estimates of the Bayes factors. Another place of action, which we only touched on in the article and on which we, and particularly Tim, have much more data that we didn't give there, concerns the truth of the actual facts themselves, which is in turn related to textual issues and the authenticity of the texts as true records of what the disciples actually taught and died for.

I would point out, though, that it is the multiplication of the Bayes factors that really ratchets things up so hugely. None of the individual factors we estimated was more than 1000/1, if I recall correctly. But the fact that there were 13 witnesses and that we argue that it is legitimate to multiply these naturally yields a very large product.

Is the "prior probablility" Carrier refers to mean the resurrection itself or the existence of God?

I can't tell if Carrier is just being careless or if he's deliberately being a weasel. I could sort of understand if he didn't read the Blackwell chapter carefully enough to really follow the argument. I had to read it two or three times myself to finally feel like I had the structure of the argument pretty well down. Once you get it it's relatively simple (but not simplistic), but until you get it I think it's easy to miss the forest for the trees. But that's not a good excuse for Carrier, since he's trying to publish against it (and since he tries to pass himself off as reasonably competent in probability theory). My guess is he just feels a need to appease his fan base (ie. internet hacks), but doesn't actually have anything substantive to say. Now the hacks out there can say, "yeah, well Carrier said the McGrews' argument is crap." Won't that be great?

It is interesting to me that when I've used this argument against skeptics, the most common response that I've gotten is that using Bayes's Theorem for history isn't legitimate. But Carrier seems to argue quite stridently that it is, and that he would just as soon see Bayes's Theorem become the basic tool for historiography, replacing all of the other criteria which have been used in the field in the past. I can't really see that happening. Carrier has given us an example of what Bayes's Theorem in the hands of a historian might produce!

Bill, the prior is the prior probability of the resurrection, but the prior probability of the existence of God is relevant to that. That is, if you have good reason even independent of the evidence for the resurrection to think that God exists, you're going to think it much more plausible that a man could rise from the dead (because he could be raised by God) than if you think the prior probability that God exists is terrible. So they're related, but the prior Carrier is talking about is for R--the resurrection.

John, I hate to see people say that. Carrier isn't a terribly good historian, either. Witness the fact that he's apparently trying to dress up the Christ myth theory as historically respectable. I think a good historian can be a good historian without Bayesian probability, that's for sure, but I wouldn't like Carrier's use to become Example A of "Bayesian probability messes up historians."

Thanks, Lydia, and congratulations to you and your husband for carrying out an impressive interdisciplinary exercise; I bet there's a lot of hard work and commitment behind this feat, which few people must be able to appreciate fully though this hasn't exactly shielded you from criticism! If this is the article in question, then I agree with you that the particular criticism is unfair. Though I can only skim-read hybrid texts on a Friday afternoon, the assumptions are laid out clearly, and I did manage to reach p12, where you set out what you take to be a couple of ‘unproblematic background facts’: That Jesus died and was buried in a tomb. I think you know I have reservations about singular events, but then I don't expect anyone to 'argue' for any particular prior probability distribution; and these 'background facts' sound innocuous enough, unless one’s a Muslim: I understand that the Quran says that Jesus’ death was an appearance or illusion and that he was assumed in heaven. So compared to a Muslim, an atheist is in principle more likely to be willing to grant the ‘background facts’; which makes me wonder why you keep putting so much emphasis on arguments for the existence of God.


I readily concede the point!

John, here's something I'd love to have time to say more about, but I'll just throw it out there for what it's worth:

Probability theory is meta-level, so it will never and can never be a substitute for good object-level sense and knowledge. Moreover, someone can be excellent at object-level reasoning in some field without any awareness at all of probability theoretic modeling.

But Bayesian modeling can be useful in helping us avoid certain characteristic errors that people who argue about controversial subjects fall into. Here are two; there are probably more, but these come to mind off the top of my head:

It's too easy to think that, when one is defending a controversial proposition, one must compare the probability of that evidence on the controversial hypothesis to whatever the best hypothesis or set of hypotheses is that one's opponent can come up with that will give some non-dismal probability to the evidence--that we're always and only comparing "possible explanations" of the evidence. This is not true. To argue for one's hypothesis, one is ultimately going to want to compare the probability of the evidence on the hypothesis and on its negation. This comparison may reveal that the evidence is _far_ more probable on H than on its negation and that the rivals brought forward from the other side are ad hoc and have a very low probability on ~H; hence, they do little to benefit ~H in the likelihood comparison. This is especially true if several of them (plausibly independent of each other) have to be strung together to account for the evidence piecemeal.

Second characteristic error (especially where the miraculous is concerned): Thinking that something is too improbable prior to the evidence to be justified by evidence. I needn't belabor that one.

Actually, I was suggesting that mathematical probability itself might be inherently invalid for this kind of inference, even if people could specify all the probabilities without error. Obviously probabilistic inference is formally valid: the conclusions follow logically from the axioms. The question is whether it's epistemologically valid.

The point made by Tversky is that the axioms of probability, additivity in particular, don't match the way people actually reason about degrees of belief. This has nothing to do with estimating probabilities correctly or incorrectly. Tversky showed that degrees of belief in real-life reasoning are superadditive. They're neither additive as are probability measures nor subadditive as are Dempster-Shafer degrees of belief (if anybody remembers those). I think that Tversky concluded that this is just more evidence that people are irrational even when we think we're being rational. But I think you could easily turn that around: the way we actually reason is "rational", and applying Bayesian inference (to this kind of problem at least) is "irrational".

Hi Aaron!

Do you think priors should be determined in any particular way? In a set-up involving a singular event the way people in fact acquire religious beliefs certainly exacerbates the problem of the priors becoming ‘the whole story’: By the time they’re exposed to the argument, people will have already been brought up into some religious tradition or other. The result is that those who already believe in Christian revelation will be confirmed in their belief while those who don’t will remain unmoved, which is not exactly surprising by any Bayesian metric; that’s what I took you to mean by ‘Garbage In, Garbage Out’. But irrespective of the tradition they’ve been brought up in, people can still get to agree that one scientific theory should be dropped in favour of another, unless perhaps they feel that their religious beliefs are threatened e.g. by the theory of evolution.

Well, Aaron, I'd be very hesitant simply to use the "way people reason" as giving the meaning of rationality. I have a more normative view than that of rationality. I haven't reviewed the Tversky material lately, but my impression from what I do remember was that in some cases risk aversion came into the question, so that part of what was happening may have been a combination of probability theory with decision theory. (Plenty of Bayesian philosophers also combine these in some ways, and deliberately so, so it's difficult to blame the ordinary guy for doing so, but I'm a purist and disagree with it.) To make up an example that doesn't come from K. & T., if Joe is very risk-averse he may subconsciously exaggerate the actual probability that some bad event will happen because he wants to be sure to take all possible precautions to avoid it. In my epistemically ideal world, Joe knows the actual probability that it will happen and admits that his extra precautions arise from his very strong desire to avoid that event, not from a particularly high probability that it will happen.

Lydia, Carrier's criticisms seem off the mark, but I have to wonder what your thoughts are on the standard view of critical scholars that Jesus was a failed prophet, pointing to the Olivet Discourse ("this generation") and the expectation of a 1st century Parousia by the apostles. Allison and Sanders are good representatives of this view and certainly more respectable than Carrier. One could argue that this evidence outweighs that for the Resurrection. Personally, I don't feel comfortable assigning probabilities to the veridicality of the Easter appearances and am somewhat reluctant to do so for the empty tomb (one author I read in my graduate seminar suggested that multiple attestation could be evidence for the invention of the story, in the absence of other criteria).

Let me add that there is a general failure of the majority of apologetic material to deal with this dominant, mainstream view of historical Jesus scholarship. Too much time is spent on the Carriers and Jesus Seminar types, which most skeptical academics disdain.

John H., I would take C.S. Lewis's view (not just taking it from him as some sort of authority, merely referring to him as someone who has put the point particularly well) that it "won't do" as an historical matter to hold Jesus to have been merely a prophet or teacher. He did claim divine prerogatives and hence was not merely presenting himself as a prophet. This argues some form of insanity if he was not who he said he was, unless he thought he could get something out of making the claim fraudulently. People who say that they are God used to get locked up, and now they at least require, shall we say, assistance to live their lives normally.

I think it is not implausible that the discourses of Jesus's that combine references to the end times with (plausibly) references to the fall of Jerusalem may be compilations of things that, at the time they were uttered, had clearer references, either because they were actually said at different times or because of gestures and other indications. In other words, the appearance of continuity in the discourses which makes them seem such a mish-mash may itself be an illusion. The "this generation shall not pass" verse seems to me most plausibly to refer to the destruction of Jerusalem in 70. The fact that the disciples expected Jesus' soon return may not be a result of any strong unclarity on Jesus' own part but merely an assumption on theirs. (John points out how quick they were to make such assumptions when he corrects the common saying regarding his own death at the end of the Gospel of John.) It is also possible that Jesus left it "up in the air," so to speak, as to when he was returning, not deceiving them but merely letting them form the conclusions they would form, because telling them clearly that it would be a couple of thousand years and more might have influenced them to do things in terms of organizing the church, etc., that he did not want them to do.

I suspect that your seminar prof. was thinking of collusion in the argument regarding multiple attestation, though that's a conjecture. Collusion can, of course, result in multiple attestation. But there are problems with that hypothesis in this case--for example, the extreme external stress and risk on the witnesses would be likely to break a conspiracy to collude in the story. We do discuss the independence issue in the paper and argue that if anything, the disciples' ability to encourage each other to stand firm (if they did; we're not asserting that they did) would argue that they must have shared some top-notch evidence to which they could allude. In a collusion situation, there would in this case be no good argument to make to the other person as to why he should keep on telling his story.

We need to distinguish between assuming that the resurrection accounts are veridical (which would _entail_ that Jesus rose from the dead, and hence cannot be a premise of the argument) and taking them to be actual accounts of what the disciples claimed. The latter is, I think, the forward position that apologists should hold (though they don't always do so) and is closely related to the generally early and intended-to-be-historical nature of the Gospels themselves as texts--a rich and fascinating topic on which my husband has a great wealth of material.

That the apologies were offered is rather a refreshing moment, don't you think? And, of course, that the Christian lady accepted.

Overseas: That's not what I meant by Garbage In, Garbage Out. I was talking about conditional probabilities, not priors. For instance, what is the probability that a witness is consciously lying given a particular report? That's a question of ancient history more than philosophy. My guess is that Christian apologists and atheists will come up with different numbers for probabilities like that.

Lydia: Tversky also gave examples unrelated to decision or risk. For instance, he did an experiment where people were asked to estimate probabilities of tomorrow's high temperature being in the following ranges: 40-50, 50-60, and 40-60. (This is from memory; the point is that one range is a disjoint union of the other two.) The sums of the probabilities given for the first two temperature ranges were significantly higher than the probabilities for the third range. This superadditivity was repeated in a variety of examples. To emphasize: it's not a problem with estimating probabilities, it's that people do not reason about probabilities consistently with the axioms of probability.

I'm not saying we should take real-life reasoning as normative, and Tversky himself seemed even more strongly against that. I'm saying that it takes some justification to select Bayesian inference, or probabilistic inference in general, as normative despite the fact that it doesn't match real-life reasoning. A case could be made against it on that basis, but anyway the burden of proof is on the Bayesians and probabilists in general. It would have to be justified not only against real-life reasoning, but also against other non-real-life modes of reasoning.

But if those weren't really the probabilities that the temperatures would be in those ranges... :-)

Yes, Bill, I was surprised. I think the post was a necessary condition of the apologies.


I agree that the Christians are likely to trust witness reports contained in a book they take to be a revelation from God. Atheists may not be the best contrast class though: There are Muslim theists who trust reports in another book they take to be divine revelation, which says that Jesus did not die but was assumed in heaven by God instead. So Muslims are unlikely to grant what the argument assumes as an ‘unproblematic’ background fact. For Muslims the ‘fact’ is problematic because it clashes with revelation and they’re off the train before we even get started.

Atheists on the other hand are much more likely to get on board if they’re prepared to accept Jesus as a historical person and don’t particularly mind eliminating the hypothesis that Jesus did not really die. I don’t know how Hindus might react; if people who believe in re-incarnation would be particularly impressed by a resurrection anyway. Jesus was not the first person to be resurrected after all; he himself raised another person from the dead and he wasn’t pronounced God on the spot. It may be different if a resurrection is seen as fulfilment of a divine promise, but then all questions are begged.

‘Real-life’ reasoning has been studied more by psychologists rather than philosophers. I suspect people may be more responsive to Dutch Book arguments, where they stand to lose money; but perhaps some people don’t care about money! I agree it’s a fascinating subject, and here’s a little test for you if you care to indulge: You have four cards laid out on a table, marked ‘A’, ‘P’, ‘4’ and ‘7’. The rule is ‘If there’s a vowel on one side of the card then there’s an even number on the other side’. Which cards do you need to turn to find out if the rule holds or not?

It is also possible that Jesus left it "up in the air," so to speak, as to when he was returning, not deceiving them but merely letting them form the conclusions they would form, because telling them clearly that it would be a couple of thousand years and more might have influenced them to do things in terms of organizing the church, etc., that he did not want them to do.

I consider that to be a type of deception. If Jesus could have easily clarified something but didn't because he wanted them operating under a false assumption, he is not "the way and the truth and the life".

The safest bet in my view is just to say that the Resurrection was a mystery and one that can never be solved. Of course, one can make a leap of faith and have certainty in a belief system based on the Resurrection and all the promises entangled with that, but that leap of faith cannot be replaced with mathematical deduction.

Which cards do you need to turn to find out if the rule holds or not?
All but the second card.

Step 2

Hi! I found the whole para from which you quote a bit pathetic; perhaps when we're favourably disposed to a hypothesis we’ll want to restrict ourselves to considering its nebulous negation, while when it comes to a hypothesis we don’t like we’ll want to map things out and even indulge in ad hoc assumptions. I have no objection to what you say about the resurrection; but why would you turn the card marked ‘4’, or else not the card marked 'P' also?

I was looking at from the perspective of "If X, then Y". Which I thought meant you have to show X first, before you confirm Y. But it doesn't affect the validity of the rule if the card marked 4 has a consonant on the other side. So you don't need to turn that card over after all. Likewise it doesn't matter if the card marked P has an even or odd number on its reverse. Thanks.

Congratulations Step2! You’re a member of a tiny minority who get this test ‘right’; it was devised by Peter Wason, a psychologist at UCL, who pioneered studies in the psychology of reasoning and confirmation bias.

Tversky's experiments do not prove superadditivity of probability, at least the temperature experiment mentioned. The human perceptual mechanism for numbers and sound ranges have a known tendancy to overestimate on the high side. We don't know why. This is not confirmation bias, but a systematic logarithmic error in perception in the neural net. If Tversky had used a logarithmic measure set where the higher equal number ranges are scaled logarithmically, the effect of superadditivity would vanish. In other words, his experimental set-up is biased from the start.

How this affects more complex, non-merological reasoning I can't say.

The Chicken

"this huge song and dance about the priors and the extremely ignorant implication that if one does not estimate an actual prior one cannot be doing anything of epistemic significance with the probability"

Tim has a lovely paper on the Rational Reconstruction of Design Inferences; it deals with some of these issues and has a very helpful overview of explanationist approaches to inferential reasoning. (I was a bit disappointed that Licona didn't take some of these points on board in "The Resurrection of Jesus: a New Historiographical Approach".)

I also found the post on your personal blog (which you link to above) very helpful Lydia.


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