Thursday, January 06, 2011

The odds form of Bayes's Theorem [Updated]

It's been brought to my attention that an atheist styling himself some sort of probability expert has been going about implying that Tim and I are deceptive or slippery in our presentation of our argument for the resurrection, that I was misleading in my interview with Luke Muehlhauser on Common Sense Atheism, that our argument for the resurrection in our Blackwell anthology paper is worthless, and heaven knows what else.

The problem seems to stem from the fact that our article focuses on the likelihood ratio (known as the Bayes factor) for the evidence we adduce rather than on the prior probability of the resurrection.

First of all, for the record: No, we were not remotely deceptive or misleading about this in the article. We were painfully explicit about it. (The entire preprint text of the article is available on-line here.) Viz.:

Even as we focus on the resurrection of Jesus, our aim is limited. To show that the probability of R given all evidence relevant to it is high would require us to examine other evidence bearing on the existence of God, since such other evidence – both positive and negative – is indirectly relevant to the occurrence of the resurrection. Examining every piece of data relevant to R more directly – including, for example, the many issues in textual scholarship and archeology which we shall discuss only briefly – would require many volumes. Our intent, rather, is to examine a small set of salient public facts that strongly support R. The historical facts in question are, we believe, those most pertinent to the argument. Our aim is to show that this evidence, taken cumulatively, provides a strong argument of the sort Richard Swinburne calls “C-inductive” – that is, whether or not P(R) is greater than some specified value such as 0.5 or 0.9 given all evidence, this evidence itself heavily favors R over ~R.


But our estimated Bayes factors for these pieces of evidence were, respectively, 10^2, 10^39, and 10^3. Sheer multiplication through gives a Bayes factor of 10^44, a weight of evidence that would be sufficient to overcome a prior probability (or rather improbability) of 10^–40 for R and leave us with a posterior probability in excess of 0.9999.

In my interview with Luke M., I said this (transcribed from the podcast, available here, at approximately 16:10 to 19:30):

In Bayesian terms, what we do in the article is that we try to separate might call...the indirect evidence, which would be relevant to that prior probability, from the direct evidence. So the things that would be relevant to the prior probability would be things like evidence for and against theism, for example, evidence for and against the existence specifically of the God of Israel, the God of the Jews, or other evidence prior to Jesus' purported resurrection regarding who Jesus was, and so forth. That would all be relevant to the prior. And what we focus on in the article instead is what we might call the direct evidence, the evidence that supposedly tells you what happened, what you might call reports...You might call it evidence after the fact. So what we focus on are the testimony of the disciples and of certain women that said that they saw and spoke with Jesus, the evidence of the disciples' willingness to die for that testimony, and the evidence of the conversion of the Apostle Paul. And what we try to do is we use a modeling device known as a Bayes factor. Roughly speaking, a Bayes factor tries to model, number one, which way the evidence is pointing and, number two, how strongly the evidence is pointing that way. And what you're trying to do at that point is you're trying to look at explanatory resources of the hypothesis, in this case, the resurrection, and the negation of the hypothesis. How well does each of these explain the evidence, and is there a big difference between how well each of these explains the evidence? I should clarify that when I say a difference, too, it's actually a's very important that you measure it by the ratio, not by the difference. But you need to look at those two hypotheses and see which one gives you a better expectation of that evidence and how much better is that expectation. So we estimate Bayes factors for these various separate pieces of evidence, then we argue for the legitimacy of multiplying these Bayes factors, because that gives you a lot of kick, and you have to discuss that issue, and we do, of independence, and whether it's legitimate to multiply them in order to combine those Bayes factors, and that ends up with this very high, high combined Bayes factor in our estimate...And so what we estimate is that you could have this overwhelmingly low prior probability (and I don't actually think that the prior probability is this low. I think it's low, but I don't think it's this low) of 10^-40 and still give a probability to the resurrection in excess of .9999. And we don't get to that by saying in fact the evidence gives us a posterior probability in excess of .9999. We just say, well this is the power of the...combined Bayes factor, and a combined power that great could overcome this great of a prior improbability and would give you this high of a posterior probability. So that's the basic method.

This is all exceedingly clear: We were arguing for a certain magnitude of confirmation of the resurrection by the evidence we adduce.

I understand that the current atheist meme on this, which shows a rather striking lack of understanding of probability, is to say that if one does not argue for a particular prior probability for some proposition, one literally can say nothing meaningful about the confirmation provided by evidence beyond the statement that there is some confirmation or other.

This is flatly false, as both the second of the quotations above from the paper and my rather detailed explanation to Luke M. show.

Let me try to lay this out, step by step, for those who are interested:

The odds form of Bayes's Theorem works like multiplying a fraction by a fraction--a fairly simple mathematical operation we all learned to do in grammar school (hopefully).

The first fraction is the ratio of the prior probabilities. So, let's take an example. Suppose that, to begin with (that is, before you get some specific evidence) some proposition H is ten times less probable than its negation. The odds are ten to one against it. Then the ratio of the prior probabilities is


Now, the second fraction we're going to multiply is the ratio of the likelihoods. So, for our simple example, suppose that the evidence is ten times more probable if H is true than if H is false. The evidence favors H by odds of 10/1. Then the ratio of the likelihoods (which is also called a Bayes factor) is


If you multiply

1/10 x 10/1

you get


The odds form of Bayes's Theorem says that the ratio of the posterior probabilities equals the ratio of the priors times the ratio of the likelihoods. What this means is that in this imaginary case, after taking that evidence into account, the probability that the event happened is equal to the probability that it didn't: what we would call colloquially 50/50. (You'll notice that the ratio 50/50 has the same value as the ratio 10/10. In this case, that's no accident.)

Okay, now, suppose, on the other hand, that the second fraction, the ratio of the likelihoods, is

1000/1. That is, the evidence is 1000 times more probable if H is true than if H is false. So the evidence favors H by odds of 1000 to 1.

Then, the ratio of the posteriors is

1/10 x 1000/1 = 1000/10 = 100/1,

which means that after taking that evidence into account (evidence that is a thousand times more probable if H is true than if it is false), we should think of the event itself as a hundred times more probable than its negation.

See how this works?

What this amounts to is that if we can argue for a high Bayes factor (that second fraction), even if we don't say what the prior odds are, we can say something very significant--namely, how low of a prior probability this evidence can overcome. That is exactly what we say in the second quotation from our paper that I gave above. It is exactly what I explain to Luke M. We say that we have argued for "a weight of evidence that would be sufficient to overcome a prior probability (or rather improbability) of 10^–40 for R and leave us with a posterior probability in excess of 0.9999."

In our paper, we concentrate on the Bayes factor. The Bayes factor shows the direction of the evidence and measures its force. We argue that it is staggeringly high in favor of R for the evidence we adduce. Naturally, the skeptics will not be likely to agree with us on that. My point here and now, however, is that neither in the paper nor in my interview was there a mistake about probability, any insignificance or triviality in our intended conclusion, nor any deception. We are clear that we are not specifying a prior probability (to do so and to argue for it in any detail would require us to evaluate all the other evidence for and against the existence of God, since that is highly relevant to the prior probability of the resurrection, which obviously would lie beyond the scope of a single paper). Nonetheless, 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.

I trust that this is now cleared up.

Update: See also this discussion of Bayesian probability and Richard Carrier at Victor Reppert's blog, here.

Update 2: See the comments thread. Luke and Richard have both apologized for their comments in the interview, and I do accept those apologies.


Tim said...

I believe the proper technical term for this post is "uber pwnage."

Carving Ben said...

The Shroud of Turin is direct evidence, I believe. It is also 'evidence' of the sort that modern one-dimensional men, should take sereiously, what with it's three dimensional information, and it having all the apperances of a singularity of the physics sort of kind. Well, that's about it.

Lydia McGrew said...

I published your comment, Carving Ben, but I do consider it off-topic.

Unknown said...

Great post, Lydia.

John Fraser said...

Carrier is such a weasel. I can't believe the guy: "Yeah, I didn't comment on the McGrews' article because it was so crappy. Heh, heh [laughs nervously, hoping nobody notices the obvious bluff]." What a joke. He didn't comment on it because he knew he would get a world-class whuppin'.

I was looking forward to the day when you guys took Carrier to the woodshed. Ever since I heard that he was planning on doing something with Bayes' Theorem I knew this day would come. I'm looking forward to more of the same in the future.

Interesting that Carrier is so hot on applying Bayes' Theorem to history when he is so obviously unqualified for the task.

Lydia McGrew said...

Thanks, Doug.

John, I was just re-reading the transcript of what Carrier said about us to Luke M. It's...astonishing. I can't believe he really believes what he's saying. We "sort of hint" that we don't argue for a specific prior. And our article is "a useless argument. Why would they publish in a companion to natural theology an incomplete argument that doesn’t even argue for the resurrection?"

And then the way he blusters and pontificates about how we didn't even include a "closing paragraph" as we would have to do in a "science journal" where the "peer review process" would force us to say that we hadn't "proved our conclusion."

What he says there would seem to indicate that he is *utterly clueless* about what a Bayes factor is and about all of the things that we carefully explain (and that I explain in this post and explained to Luke M.) concerning what we have argued and why it is significant.

If he is not clueless, he is being exceedingly deceptive himself by pretending that he is clueless and by saying that our article provides "completely useless information."

It leaves me shaking my head. I wasn't going to say anything at first, but once I learned that some atheists were so relieved not to have to think that we know what we're talking about, I decided I needed to lay it out.

Lydia McGrew said...

Oh, hey, John: You probably are already reading it, but in the thread over at Vic Reppert's blog, which I link in the main post, Tim is giving out virtual cookies and peppermints to people who can find Carrier's errors on probability theory in a handout that Carrier has up on the web. Fun for all and free cookies. (But you have to have your browser enabled to accept cookies--haha.)

John Fraser said...


I did see Reppert's blog first, actually. I haven't had a chance to have a go at winning some cookies yet (not yet having read Carrier's attempt at Bayesian probability) but I was enjoying the show.

I should make a small correction to my previous post. Carrier said he "didn't want" to cite your article. Wasn't it nice of him to lower himself to do so anyhow? It's nice that he only has to impress other internet infidels and not real scholars. Great.

Lydia McGrew said...

I can't tell whether he lowered himself in the end to cite us or not. The way he explains it to Luke, he doesn't actually talk about how, exactly, the McGrews are wrong, but after one reads what he's written, one will know why we are wrong. How exciting. I can hardly wait.

John Fraser said...


Sorry to hog the thread, but that was too funny! You're right, I can't really tell what he means, either. But I love this bit of bluster:

"So, if you want to see a reply to that, read that chapter. It’s not like a “McGrew said this and this is wrong because…”

LUKE: Right, right.

RICHARD: But once you’ve read that chapter you’ll understand what is wrong with their chapter."

HAR! Translation: "I can't really tell you specifically what's wrong with it, but if you read my stuff you will get an immediate and intuitive grasp of how crappy the McGrews chapter is. Not that you'll be able to put it into words, you'll just know (almost as if by magic)."

And Luke's response: "Right. I think it is something a little slippery going on."

Something a little slippery is going on, all right!

Victor Reppert said...

Lydia McGrew said...

John, you aren't hogging the thread, because almost no one else is commenting anyway. :-)

That is really an amusing part of the dialogue.

The thing is, if Carrier really thinks what we are doing is useless, then it's hard to see why he doesn't just explain in clear terms why it is useless and be done with it. In this interview, he implies that it is useless because we don't calculate a prior. I have other reason to believe that there was a time when he actually understood something about the Bayes factor analysis we do and that he has changed his critique more recently to make this big deal about the prior. Before, I gather his criticism was that the Bayes factor isn't as large as we say it is, which would at least _address what we are actually saying_. This criticism in the interview with Luke M. is sheer nonsense in terms of the probability theory, and I cannot understand why he has gone to it, as it suggests utter incompetence on his part to anyone who knows the probability theory at all, even to the extent that I explain it in this post.

Lydia McGrew said...

It's been so many years since I saw that, Victor. It does seem...apt.

William Luse said...

I was wondering whether that Luke guy stuck up for your integrity at all. In that snippet John quotes, he sounds like a suck-up.

Lydia McGrew said...

I'm afraid not, Bill. It would have been possible, I would have thought, for Luke just to have said that he would go back and listen again to that section of his interview with me, but he didn't. Here's the relevant portion of the transcript. (I didn't type this; someone else did and sent it to me.) (I have to break it up into two parts for blogger.)


LUKE: Yeah. Now I was speaking a little while ago with Lydia McGrew who is a Christian apologist and wrote an article on the Resurrection and defended the Resurrection by using Bayes’ Theorem. I was speaking with her and asked about inference to the best explanation, or argument to the best explanation, and she seems to agree with you that argument to the best explanation can be sort of an intuitive, semi non-mathematical way to get towards the truth but it’s got to come back to Bayes’ Theorem.


LUKE: And I agree with her even though I do talk a lot about argument to the best explanation but a lot of that is because nobody would read my blog if I just gave Bayes’ Theorem in every post.

RICHARD: Oh sure, yeah.

LUKE: But I do agree that it has to come back to Bayes’ Theorem.

RICHARD: It’s still a useful rule of thumb.

LUKE: Yeah.

RICHARD: It’s still useful to learn it, it’s just, it’s not enough really.

LUKE: Yeah.

RICHARD: It’s better to know that than to know neither, frankly, so…

LUKE: Yeah. But then, so she’s using a Bayesian method at least to try to argue for the Resurrection of Jesus and you very much disagree with that conclusion.


LUKE: So where is the disagreement then if you somewhat agree on the method but you’re coming to very different conclusions?

Lydia McGrew said...

RICHARD: Well, I’m curious to know what she actually claimed on the interview, because the article in the Companion to Natural Theology does not come to any conclusion. One of the conspicuously missing things is the prior probability, one of the key premises of the entire argument. All they talk about are what we call two of the four premises of Bayes’ Theorem and they make an argument from those two premises.

But you can’t reach a conclusion without answering the other two premises and they never do, which I find disturbing because it suggests… and they don’t really explain this very well, I mean they kind of hint at it.

But considering the fact that the kinds of people who are going to be reading this Companion, a lot of them are not going to be sophisticated enough to know that they’ve been hoodwinked like that… in fact, the mere fact that you and other people, I’ve met other people who say, “Yeah, she argues for the Resurrection,” say, that’s not what she even argues at all.

LUKE: Right.

RICHARD: Was she deliberately misleading about that? And that’s kind of the problem I have with it. All they argue is that certain evidence makes the resurrection more probable.

LUKE: Right.

RICHARD: Again, that’s completely useless information. [laughs] We could go from 1% to 10%, that’s 10 times more probable. Yeah, that makes it more probable. It’s still 90% chance it’s false.

LUKE: Yeah.

RICHARD: So, it’s a useless argument. Why would they publish in a companion to natural theology an incomplete argument that doesn’t even argue for the resurrection? What’s the point of that? And not even to explain in a closing paragraph as you would in a science paper, for example. If you did this in a science journal, believe me, the peer review would mandate that you have this closing paragraph explaining that you haven’t actually proved your conclusion you have just done one step of two essentially to do that. So that’s one problem.

Another problem is that her facts are all wrong. Of course, that’s a common problem. That doesn’t indict Bayes’ Theorem. Obviously you put the wrong facts into the theorem, you’re going to get the wrong conclusion. That’s a straight forward problem of all reasoning methods. You don’t attack a theorem, you attack the facts.

LUKE: Yeah.

RICHARD: And there are a lot of things in there where their facts are just plain wrong. In my chapter on the resurrection, it’s called ‘Why the Resurrection is unbelievable’ in the Christian Delusion edited by John Loftus. I wrote that… I didn’t want to bother citing the McGrews’ article. It’s such a crappy article.

LUKE: [laughs]

RICHARD: It’s a bad introduction of Bayes’ Theorem. It really is. Specifically because it doesn’t explain this.

LUKE: Yeah.

RICHARD: And it has all these fancy calculations and stuff that make it look very impressive. It seems to me like it is hoodwinking the public in a way. So anyway, but my chapter does address it. It’s like I specifically crafted the chapter. In fact, I even put Bayesian arguments… I have it all in colloquial English but then in the end notes I have a Bayesian version of what I said.

I talk about all the facts they do, and I hit many of the same points. Unlike them, I talk about the problem of prior probability and get that nailed down.

So, if you want to see a reply to that, read that chapter. It’s not like a “McGrew said this and this is wrong because…”

LUKE: Right, right.

RICHARD: But once you’ve read that chapter you’ll understand what is wrong with their chapter.

LUKE: Right. I think it is something a little slippery going on....

Lydia McGrew said...

By the way, if Carrier is "curious to know what I really claimed in the interview," it's here in this post, and he can go listen to it for himself. I trust it's...helpful.

William Luse said...

That's pathetic. And I don't think Luke understands the points in dispute. For which maybe you can't blame him since Carrier doesn't say what they are, but he could at least have said, "Well, let's clear that up. Exactly what facts does she have wrong?" How hard would that have been?

Lydia McGrew said...

That would have been a very reasonable follow-up question and needn't have seemed hostile, either. That is to say, fellow atheists would doubtless be interested as well in further details on those "wrong facts."

I would have settled for not agreeing that "something slippery is going on," though. More like, "Hmmm, that's interesting" or something non-committal like that.

Mr Veale said...

I agree that it is difficult to believe that Carrier believes what he is saying.
I'm still reeling from his discussion of the criteria of authenticity. He must know that this is not how these criteria are used; and it is not how they were used! They assumed a process of transmitting traditions about Jesus, in a particular social context.


Lydia McGrew said...

One of the commentators at the other blog says that Carrier has now said to him in private e-mail that he concedes that I did not "intentionally" mislead people.

I'm overwhelmed by that generous concession. Next up: Admitting that our article does not provide "useless information" because it doesn't give a prior; in fact, admitting that he was completely wrong in what he told Luke M. on that score.

Breath-holding is not happening.

Luke said...

I've linked to this and the discussion on Reppert's blog from Twitter already, and will link to it from Common Sense Atheism later.

I'm sorry if it sounded like I was "sucking up" to Carrier in my interview; it wasn't my intention. Most of my "right"s and "yeah"s in my interviews are acknowledgments, not deeply-considered agreements - but if that's not clear, then I need to change how I'm doing that!

Carrier apparently has only a few years' experience with Bayes' Theorem, as the historical method section in Sense & Goodness Without God (2005) does not mention Bayes. When asked to guess at the competence in probability theory between two people who have been publishing peer-reviewed philosophy literature on probability theory for at least a decade vs. someone who discovered Bayes' Theorem in the last few years, I'm going to bet on the former in a heartbeat.

I think Carrier is right about the need to do history with Bayes' Theorem, and I suspect you and Tim agree. But I worry about his execution, and would be overjoyed to see him team up with a formal epistemologist to rework "Bayes' Theorem and Historical Method."

I haven't had time to consider Lydia's post above, or the Carrier article linked to at Reppert's site, in much depth. But I'm glad this discussion is happening and I wanted to apologize to you for saying "Yeah, it does seem like something slippery is going on." That was me getting caught up in the "yeahs" instead of taking the time to thoroughly compare Carrier's claims the words in your article! I hope you'll forgive that and harbor no hard feelings!



Lydia McGrew said...

Luke, thank you very much, and no hard feelings. The use of "yeah," I understood, can be a conversational convention that just indicates that you're listening. The other thing about "something slippery," that did bother me, but shake and pax on it!

Please do read the post and consider how it relates to what RC said in the interview, because I think it is a useful and clear answer.

Questions welcome.

Luke said...


I checked the 2009 paper, the Carrier interview transcript, and the transcript of my interview with you, and I think it's fairly clear you're right about both (1) the fact that you've always stated the form and conclusion of your Resurrection argument clearly, and that (2) arguing for a very high likelihood ratio pointing toward the Resurrection is epistemologically significant.

I have a blog post on this scheduled to go up tomorrow morning.

Lydia McGrew said...

Thanks, Luke.

I like to be clear, and I prefer it if people don't just have to take my word for things because all the details are too obscure.

Lydia McGrew said...

A reader asked in another thread (where the question didn't really fit) for some links to info. about Bayesian inference, etc. Whether the following will be helpful or not will depend on the reader, but here are some:

Handouts by Tim McGrew on Bayes & mathematics:

Preprint article by Tim McGrew on evidence:

Tim McGrew's article on miracles in the Standford Encyclopedia of Philosophy:

Richard Carrier said...

Dear Lydia,

I apologize for my remarks in my interview with Luke Muehlhauser. I badly overstated my impressions, and drew the wrong inference from what I took to be the opacity of your paper's explanations. I agree with everything Luke has said in his latest blog on this question.

Your conclusion is useful if it were based on correct facts. Part of my point in the interview was that it was not, hence my conclusion was actually based on that statement (that your declarations regarding the facts were incorrect), not the actual mathematical result you produced which, if it were correctly derived from the facts, would be a strong result that would warrant more serious attention to the prior probability of divine intervention in the universe in general. And whether the facts are correct is a wholly separate issue Luke and I did not delve into in that interview (I do so in chapter eleven of The Christian Delusion, with further support in my book Not the Impossible Faith and chapters in The Empty Tomb: Jesus Beyond the Grave--none of which address you specifically, only the evidence). I should have corrected myself on these points.

And you do explain the absence of prior probability calculations (I said you "hinted" at it, which is inaccurate hyperbole). What I should have said was that this explanation is too opaque to lay readers and most don't understand this caveat. Which is why I keep having people come up to me saying your article gives a Bayesian proof that the resurrection occurred or that Lydia McGrew "proved" the resurrection accounts are true (which even you would agree is not an accurate description of what your article does). I was reacting to those claims, not yours. I shouldn't have assumed this was your design, but only an accidental effect.

Again, I regret my hyperbolic remarks, and I have asked Luke to make public my corrections in these regards. I apologize for misrepresenting you and I hope I can mend fences on this score.

Lydia McGrew said...

Richard, thanks very much; apology is accepted, and I'm glad to have been proved wrong in my comment above re. holding my breath. It's good to be wrong sometimes.

As you say, the factual issues are where the action lies and are not what you got into in the interview with Luke.

I'm glad to have the probabilistic point re. priors and "uselessness" well cleared up.

Doug said...

Lydia, what do you think of some of the comments here on your work -

Lydia McGrew said...

Doug, I want to say loudly and clearly that one thing I _don't_ do is to go running around the blogosphere answering everything that anybody says about me or my work. In particular, I don't get involved in the skeptical blogosphere. I tend to think that the intellectual quality in that context is pretty low, the people are pretty unconvinceable as well as sloppy and inclined vastly to overestimate their own knowledge and grasp of the issues. It's just a time-sink, a black hole.

Playing Bat-Apologist in the skeptical blogosphere is not my vocation, and I don't want, in a sense accidentally, to slip into that role here and now.

In fact, partly for that very reason I was inclined at first to ignore Richard Carrier's interview with Luke. But in the end I decided to respond, partly because the issue is cut-and-dried and what you might call modular: It could be separated neatly from other issues and answered briefly and decisively.

So I'm reluctant to get involved taking a whack at whatever this or that person says about our article at the site you list.

I will mention only two things: The fellow who sneers at our combined Bayes factor on the grounds that we are assuming independence _appears_ to have overlooked the fact that we have an entire section discussing that very issue and offering, as far as I know, a new technical point in the literature concerning the question of whether assuming independence strengthens or weakens a case and relating this to the question of situations of duress.

Second, the earlier commentator who says that the probability is "approximately 1" that there would be made-up resurrection stories (and apparently thinks that this applies to the gospels) ignores various obvious distinctions. For example, the distinction between stories by people who had nothing to gain and everything to lose for making up such stories and people who had nothing to lose and something to gain by doing so. Also, the distinction between people's elaborating stories when they themselves were in a position to know what really happened and people who were not in a position to know what really happened.

We are talking in the paper about what the disciples themselves claimed. They were in a position to know whether what they were claiming was true or false, and they had a great deal to lose and nothing to gain by simply making up such tales.

Naturally, the next refuge of the skeptic is the claim that the disciples never said any such thing, but such skeptics have an interesting time with (among other things) the book of Acts, which is historical in genre and written by someone personally close to the events it reports if anything is and which shows us the disciples making extremely explicit claims about what they saw at the very beginning of Christianity.

There are other points that could be made concerning the Gospels, but I will just add that there is a wealth of material that supports the claim that they were written by or in direct consultation with people who were eyewitnesses of the relevant events--hence, that at a minimum they represent *what the disciples claimed*. The argument from undesigned coincidences is one of the strongest of these.

Paul Baird said...

I've asked John Fraser about this, and I'll now ask yourselves directly - have you conducted this sort of analysis on any comparable non-Christian miraculous event to show that that event could not have happened to the same degree of probability as the Resurrection ?

Lydia McGrew said...

Paul Baird, you might start by listening to my interview with Luke M. That question does come up there. To be clear, I think some of the comments I make there are also applicable to plenty of reported Christian miracles. This doesn't fall per se along Christian/non-Christian lines. As always, the devil (or God) is in the details.

Paul Baird said...

Thank you Lydia - I've posted my response on the Premier Christian Radio's Unbelievable forum. John Fraser is a regular poster there.

Tim said...


Your reply misses the point. Lydia's kitchen reproduction of the Hindu
"milk miracle" raises P(E|~H) to virtually the same level as P(E|H), yielding a Bayes factor of approximately 1. The fact that she did not do this in technical terms in an interview with a non-specialist does not invalidate the Bayesian analysis.

Truth Seeker said...


I could really use some help making sense of a couple of things:

1) I know of some atheists who use Plantinga's Dwindling Probabilities argument to show that we can't be confident that we know what the original NT text said assuming even a .9 reliability of information transmission (one person told another, who wrote it down once, and then again, etc.) I have read George Campbell's article, but I do not know enough math to really understand what sort of legerdemain is going on here. When do these probabilities acquiesce to historical methodologies of textual reconstruction of 99 percent of the original text, or use of criteria to establish the burial, empty tomb, post-mortem appearances, and origin of the disciples belief in a resurrected messiah? Surely, bayesian probaility doesn't overwhelm all of ancient history, but I don't know how to explain this in Bayesian terms.

2) Even more confusing is a footnote in William Lane Craig's book Reasonable Faith that says even if four independent witnesses are only 50 percent reliable, then the probability that all four are wrong is: .5x.5x.5x.5=.0625%. So, instead of a "Dwindling of Probabilities" in the damaging sense that Plantinga meant it, Craig seems to be using a dwindling of probabilites in exactly the opposite manner.

What is the easiest way to make sense of this?

Thank you dearly,

Tim said...


Almost any question about probabilities is going to be at least a little bit technical. In fairness to Plantinga, I should note that his argument, though fatally flawed, is not as simplistic as the one that you are being given.

Judging from your description, I would say that your atheist friends appear to be treating the transmission of texts like a game of "telephone" with a certain probability of an error in transmission at every step. If those probabilities are independent, then to get the probability that there have been no errors, we multiply the probabilities together. So if there is one original document copied ten times, and if the probability of flawless copying at each step is .9, the probability that there has been no mistaken copying throughout the chain is .9 x ... x .9, ten times, or .9^10, which is about .35. Conversely, the probability that there has been at least one copying error in the chain is 1 - .35, or about .65.

The problem, of course, is that this model is nothing like the actual situation we are in for determining the original text. With over 5000 manuscripts and fragments in Greek alone, we have loads of cross-checks. We do find errors, lots of errors -- but they're not the same errors in every copy. By comparing the texts and noting their variations, we can reconstruct virtually all of the original text of the New Testament with a very high degree of confidence. So the whole calculation done by multiplying such probabilities is based on a mistaken model.

I don't have Craig's book in front of me at the moment, but from your description it sounds as though his point is something a bit different -- though it also involves multiplication of numbers. Suppose that four independent witnesses all say the same thing -- for instance, that Jones hit Smith. There might be some doubt regarding each bit of testimony; perhaps the altercation to which they are bearing witness took place late at night under poor lighting conditions. But in order for them to be wrong, every one of those testimonies must be mistaken. The force of their testimony can be gauged by multiplying the likelihood ratios together. If these ratios are equal, we can calculate it as [p/q]^4, where p is the probability of one witness's testimony testimony given that Jones actually did hit Smith and q is the probability of one witness's testimony given that Jones did not actually hit Smith.

The math is useful for those who know how to use it. But if you find it confusing, don't be flummoxed. The key point to keep in mind is that combined evidence can be very powerful.

Truth Seeker said...

Thanks very much for your reply Tim,

1) I would like to be able to understand the math better because as you say, it is useful for those who know how to use it. Could you recommend any number of books for me to read that would help me develop from a beginner to a more intermediate/advanced level?

2) Your closing statement had me thinking: "The key point to keep in mind is that combined evidence can be very powerful."

If a person found four arguments from natural theology each to make God's existence as likely as .60 percent(i.e.):

Kalam= God's existence is .60 likely
Teleological= " " " .60 "
Moral=" " " .60 "
Historical Case for the Resurrection= " " " .60 "

would it make sense to calculate the cumulative probability of God's existence in that case as: .4x.4x.4x.4=.0256 where .0256 represents the likelihood that God does not exist given the cumulative force of the four arguments each making God's existence as likely as 60 percent?

If my numbers are not correct, then how would you mathematically represent the power of a cumulative case built from natural theology?


Anonymous said...


I have an atheist professor who has a book coming out wherein he goes after the historical case for the resurrection of Jesus on a number of different levels. One objection in particular, which I would be very grateful to hear a response goes as follows:

The Lourdes, France example shows us that at best people are reliable only 1/100,000 or 1/1,000,000 or even 1/10,000,000 in the cases where they claim that they have witnessed miracles. Lourdes (and lots of other cases) shows us that people are really, really unreliable at reporting miracles. So, in the case of the miracle reports of the resurrection in the gospels, we can't reasonably accept those reports as veridical given the low frequency of reliability amongst human beings at accurately identifying miracles. This problem is compounded by the small number of independent eyewitnesses to the resurrection as well.

How would you respond to this challenge?

Thank you,

Tim said...


Your professor's attempt to extract a probability from the Lourdes cases is messed up on many levels. But set that aside. Granting that no miracles have ever taken place at Lourdes -- and though I am inclined to agree, it is not a matter that I have looked into in any detail -- here is just one of the things wrong with your professor's reasoning.

Many people make a pilgrimage to Lourdes in the positive hope and expectation of seeing or experiencing a miracle. They suffer no adverse consequences for proclaiming that they have witnessed or experienced such a miracle; on the contrary, they are surrounded by thousands who would be only too happy to have been in proximity to such an event -- for after all, if you are standing next to me and tell me that the pain of your gout has gone, then I can claim to have been present on the occasion of a miracle.

In every respect, this is the opposite of what we find at the origin of the church in the book of Acts. There is abundant evidence that many who professed to be original witnesses of the Resurrection passed their lives in labors, dangers, and sufferings voluntarily undergone in attestation of the accounts they delivered and solely because they believed those accounts, and that, from the same motives, they submitted to new rules of conduct. Their testimony was against their earthly interests; they had every reason to believe that their reports would bring them misery here below, as indeed in many cases they did.

Testimony against interest is a very different matter from testimony that falls in with one's interests. The attempt to generate a probability based on a presumption of the falsehood of all of the cases at Lourdes tells us nothing of interest. Cases where the claim complies like this with the interests and expectations of the individuals form the wrong reference class.

Lydia McGrew said...


I heartily second Tim's discussion. I would add that another difference is that claimed healing miracles are in general much more difficult either to verify or to falsify, for many reasons--for example, it can be difficult to tell if natural processes would have brought about the result in question. It can be more difficult to tell (especially for someone simply witnessing a single event) what the state of the person's health was previously.

The resurrection claimed in the case of Christ was not like this. The idea that they were simply _mistaken_ in thinking that he had actually risen from the dead is absurd. It is not as though they caught a glimpse of someone from a distance and thought that that might be Jesus.

I hold no particular brief for the miracles at Lourdes (I'm a Protestant), but everything about this "reliability" claim made by your professor smells questionable to me. It gives the impression that some sort of actual, objective study was made, but I know of no such study. And if there were one, did it proceed by assuming that no miracles could possibly have taken place (which would be circular) or by checking each case individually without such an absolute presumption? Are the "reliability" numbers cited related to false negatives or false positives? And so forth. This sounds like a made-up number making a pretense to having been established in some objective or "scientific" fashion.

Anonymous said...

Thank you for your reply. I concur with your comments but a question remains for me still. Given Bayes Theorem, and Hume's Abject Failure which only focused on the prior probability of a miracle occurring based on frequency probabilities, is it the case that the specific circumstances surrounding the resurrection that you mention(i.e. testimony against interest) serve to increase the prior probability of the reliability of the testimony we have concerning the resurrection in spite of the frequent unreliable testimony regarding miracles that we have from other miracle reports (i.e. Lourdes, Hundu Milk miracle, etc.) so that the prior probability is not some very small decimal percentage? Or, do the specific considerations of the circumstances surrounding the miracle reports of the Resurrection factor into the specific evidence of Bayes on the right side of the equation in the numerator? I hope your considerations factor into the prior probability of the reliability of miracle reports by human beings, otherwise, we as Christians have to be something like 99.99999...... percent certain that the witnesses to the resurrection are reliable in order to overcome a very small prior probability in which the frequency with which miracle reports are reliable is very very small. In other words, which part of Bayes do the considerations you offered affect? P(H/B), P(E/B&H), etc.

Thank you,

Lydia McGrew said...

Kevin, I think you're rather seriously confusing prior probabilities and likelihoods here, in all sorts of ways.

First, consider the supposed unreliability of witnesses to miracles (There are myriad problems with attempts to calculate this alleged unreliability, some of which I may discuss in a later comment.) Even if we took such claims at face value, they would not have an impact on the prior probability of a miracle. If you had some independent way of checking that a miracle had not taken place in a particular case where a miracle was claimed, this would not drive the _prior_ probability of a miracle down to any greater extent than any other situation where no miracle happened! For example, you woke up this morning and went through your morning and apparently no miracle happened in your vicinity. We may take that to have some impact on the probability that miracles happen. But if you check out some claimed miracle and believe that you have evidence that no miracle happened, the impact on the prior there is just the same: Oh, another "situation" in which there wasn't a miracle. The fact that someone claimed there was one doesn't give it "extra points" driving down the prior probability.

What you should say is that the impact, *if any* (but we've already addressed this above) of claims of "unreliability" in miracle reports should be an impact on the likelihood--on the Bayes factor. *If* the cases were relevantly similar, the past discovery of unreliability of relevantly similar witnesses would mean that the Bayes factor for new testimony of the same sort would be less powerful. This is not a matter of the prior probability at all.

However, we have already argued that the cases are not relevantly similar.

And, yes, to answer your question: The specifics of the situation concerning the disciples' testimony are relevant to the Bayes factor too. They help us to decide how much weight to give the testimony in favor of the event.

Lydia McGrew said...

Some more problems with the "unreliability" estimate (some of this may be repeat from above): This attempt to figure out the unreliability of witnesses from the supposed non-occurrence of the events they testified to runs a very serious risk of being circular. If you decided that the witness was mistaken without some independent evidence that the event did not take place as testified or that no miracle occurred, if you merely decide that probably no miracle took place because of the prior in _that_ case against a miracle, then this should not count _at all_ as a "new instance" of a witness's "unreliability." That's blatant bootstrapping: Your 1000-fold experience of no miracle becomes magically your 1001-fold experience simply by inductively assuming that no miracle has taken place in the 1001st instance, which then acts as, supposedly, _more evidence_ when you run into the 1002nd claim. Epistemologically, that's completely illicit. If you have independent, specific, evidence that the witness was mistaken in *that particular case*, then you can at least in some sense "count" that as a case where you know that the witness was mistaken. Otherwise, you can't just use the _presumption_ that the witness was mistaken as if it constituted new, independent evidence of the unreliability of testimony to miracles generally.

Next, we have an incredibly messy ambiguity concerning the concept of "unreliable witnesses." In a healing claim, this is especially obvious. A witness may _accurately_ testify that his friend was ill before time A, drank the water of Lourdes (or whatever) and then was well afterwards. The witness may therefore be _completely reliable_ concerning the actual facts that he witnessed. Suppose that the witness concludes that this was a miracle, and suppose for the sake of the argument that it was not. This does _not_ make the witness "unreliable," and even if we had some independent evidence that no miracle took place in this case (say, we discovered that the sick person had been treated by a doctor in a way that would probably have brought about the healing naturally, exactly as it took place), it would be sloppy to say that this made the witness "unreliable." Both the witness and we are making a judgement as to the _cause_ of the healing, but the fact of the healing is a datum we are both trying to explain. If the witness has accurately and of his own knowledge attested to the healing, he has actually been reliable, not unreliable. After that, the judgement as to the cause is open for both us and the witness to make.

Interestingly, in the case of a resurrection this distinction between the event witnessed and the conclusion that it was a miracle tends to collapse, because if the witness did accurately report that So-and-so was dead at time T and that he saw that same person alive again at time T + 1, it's difficult to know what else to call this but a miracle. The natural mechanisms for the event are pretty much nil.

Lydia McGrew said...

(cont.) Similarly, a witness may accurately report that he saw a man arrive with a crutch, leaning on it heavily, and leave dancing. The witness may be a dupe; the man may have been only faking his lameness. But the witness may be accurately reporting what he saw.

But it's rather a hard sell to claim that the disciples were duped (by a double, perhaps?) into thinking that they saw Jesus risen when he was not.

All these distinctions must be kept in mind if we are going to talk about what past miracle reports tell us about witness reliability.

If what is happening here is that we have these many, various miracle claims, that we believe no miracles took place, and that we put these all down as instances of "unreliable witnesses," there are multiple problems with this conclusion. It should not be taken to yield any kind of epistemically helpful "reliability number" for witness testimony in later, allegedly miraculous, contexts.

The lesson here is that even if you think no miracle has taken place, the concept of "reliability of witnesses to miracles" is probably too blunt to be very useful.

Anonymous said...

Thank you for your thoughts and patience with me as you help me work through these questions; we are making great progress!

One more question and answer should wrap up the issue for me:

Bayes' theorem applied iteratively yields:

P(H|E¹∩E²)= Λ¹Λ²P(H)/Λ¹Λ²P(H)+P(-H)

I noticed that Bayes has the form of: X/X+Y so that as Y tends toward zero, the value of the ratio tends toward 1, so that the crucial probability would be the intrinsic probability of no miracle occurring multiplied by some naturalistic alternative explanation of an alleged miracle (The Y in X/X+Y). Now, my question is two-fold:

1) Do the considerations of disanalogy between the specific circumstances surrounding the resurrection that you and Tim have offered as compared to other miracle reports get heaped into the numerator as independent pieces of evidence?

2) I want to think that it is legitimate to multiply the probability of the various naturalistic explanations together as a subset of all plausible naturalistic explanations that can of have been offered so that Y (in the X/X+Y) will tend toward zero. For example, if we consider explanation such as hallucinations, mass hysteria, twin Jesus, apparent death, cognitive dissonance, and the like as a pool of naturalistic explanations and find each to have a likelihood of only say 10%, could we multiply .1x.1x.1x.1 etc. while holding the hypothesis of a resurrection fixed in the numerator so that Y would tend toward zero? Is that a legitimate use of Baye's Rule? If so, then wouldn't that be an explanation of how the eyewitness circumstances surrounding the Resurrection could overcome a very low prior probability of my atheist professor’s very blunt and erroneous estimate of the reliability of eyewitness testimony to miraculous events?

Lydia McGrew said...


In answer to your question #1, I think you will find it more helpful for purposes of this discussion to think in terms of the odds form of Bayes's Theorem (per the main post) rather than the form you are using. Yes, they are inter-derivable, but the odds form is going to allow you to separate out that crucial Bayes factor. Now, it is not that the special considerations we are bringing up get put into the numerator of the regular form of Bayes Theorem as pieces of evidence on which we are conditioning. Rather, these considerations influence the ratio of the likelihoods, the Bayes factor, by which the priors are multiplied in the odds form to yield the ratio of the posteriors.

Consider an example where evidence for a miracle appears _weak_. Suppose that a report is made that a person was blind, prayed to a particular saint, and subsequently was able to see. But now suppose that we also learn that in the interim after the prayer the person also had eye surgery with a high rate of success for restoring vision, and suppose that the person simply recovered in the expected way from the surgery and that his sight returned according to a usual sequence for recovery induced by surgery. This means that P (E|~M) is approximately equal to P (E|M). The natural explanation is perfectly adequate. The evidence does not favor the occurrence of a miracle.

Many of the special considerations we are raising are relevant to the probability of specific pieces of evidence (e.g., the disciples' accounts of the post-resurrection appearances of Jesus', their boldness in proclamation, etc.) given no miracle. The most striking effect of these particulars is to make P (E|~M) very, very low and to make the ratio (the Bayes factor) P (E|M)/P (E|~M) top-heavy, because the naturalistic explanations are so feeble.

Concerning your #2, there is multiplication taking place, but I don't think it's the multiplication you are thinking of. Instead, what we argue for in the paper is that it is legitimate to multiply the Bayes factors for various pieces of evidence.

Another way to think of this is to think in terms of multiplying the improbabilities of explanations of _different_ pieces of evidence. (It looks to me like you are considering multiplying the probabilities of different attempted naturalistic explanations for the _same_ piece of evidence.) So, for example, we'd have to have one naturalistic explanation for the disciples' testimony (e.g., hallucination), another for the women's testimony to the empty tomb (e.g., wrong tomb), another for the women's testimony to seeing Jesus (more hallucinations) another for the conversion of Paul (another hallucination that just happens to happen to this persecutor of the church). And in fact we argue in the paper that one should multiply the disciples' testimony as well, because they would all have to happen to have an extremely strange sort of hallucination that they were all talking and interacting with Jesus and with each other simultaneously, and it would have to be such that they never "snapped out of it" later and realized that it was all just a hallucination.

You get the picture. So the multiplication concerns different pieces of evidence.

Lydia McGrew said...

I also wanted to address this final phrase in your question:

"...could overcome a very low prior probability of my atheist professor’s very blunt and erroneous estimate of the reliability of eyewitness testimony to miraculous events..."

I think it would be a mistake to approach it that way. Your professor's estimate is bogus and inapplicable for multiple reasons, as discussed above. One shouldn't take it at face value and then try to "overcome" it, as if one has to first factor it in and then try to come up with such good evidence as to overwhelm this silly number he has made up. On the contrary, one should toss the number out the window to begin with, for all the reasons given, and do the calculation in a completely different fashion.

Anonymous said...

Thanks Lydia,

If you would, I have another question related to Bayes that would be very helpful for me and others that I speak with to get a handle on.

Just as the cumulative force of the likelihood ratios for multiple, independent eyewitness testimony can be mathematically modeled to show the Bayes Factor, or the direction the evidence is pointing, I was hoping to model the cumulative case for Christian theism from the arguments of Natural Theology in a similar manner.

For example, if we have 5 independent arguments for the existence of God, and a person finds each of them to make the existence of God 80% likely, then how would that be modeled along a Bayesian line?

Would it be [p/q]^5 where p=.2 and q=.8? Using these numbers I get 0.0009765625. Does this mean that for the person in the epistemic situation under consideration that the arguments from Natural Theology make the evidence 10,000 more probable if H is true than if H is false? How low of a prior probability could a person in such an epistemic situation rationally claim that the evidence can overcome for them?

Thank you,

Tim said...


This is a fascinating question. A full answer that chased every rabbit down its own separate hole would get very long indeed. So if you don't mind, I'll sketch an answer to a simplified version of the question.

First, consider the case of a single valid deductive argument, one where every premise is required for the inference to go through, for the conclusion "God exists." There may be some doubt about one or more of the premises, but if they are all true, then the conclusion must also be true. The sum of the uncertainties of the premises is the maximum uncertainty of the conclusion. So if the argument has two premises and each premise has a probability of, say, .7, then the conclusion that follows from those premises must have a probability of at least .4 -- perhaps more, though without further information we can't say how much more.

We can get a slightly higher lower bound if we know that the premises are probabilistically independent: then we can multiply, getting .7 x .7 = .49. Once again, this isn't the probability of the conclusion; it's the probability that both premises are true. But unless the conclusion is the logically strongest claim derivable from those premises, that is to say, unless it is logically equivalent to their conjunction, the probability of the conclusion may be higher than this.

Now, suppose that we have two valid deductive arguments like this for that same conclusion -- call them Argument A and Argument B -- each with two probabilistically independent premises, and (for simplicity) let's suppose that each premise of each argument has a probability of .7. Furthermore, let's suppose that across arguments, the premises are also independent: whether premise 1 of Argument B is true is probabilistically independent of whether premise 2 of Argument A is true, etc. How should we assess their combined strength?

[To be continued ...]

Tim said...

The key to this problem is that if both premises are true for either argument, then the conclusion must be true; after all, it follows deductively from the truth of those premises. So in order for the conclusion to be false, at least one premise in each argument must be false. (This is, of course, only a necessary condition for the conclusion to be false, not a sufficient one.) Out of the 16 possible combinations of truth values for the four premises in question, nine meet this criterion:

~A1 & A2 & ~B1 & B2
~A1 & A2 & B1 & ~B2
A1 & ~A2 & ~B1 & B2
A1 & ~A2 & B1 & ~B2

~A1 & ~A2 & ~B1 & B2
~A1 & ~A2 & B1 & ~B2
~A1 & A2 & ~B1 & ~B2
A1 & ~A2 & ~B1 & ~B2

~A1 & ~A2 & ~B1 & ~B2

I have grouped them for computational convenience. Since each premise has, ex hypothesi, a probability of .7, its negation has a probability of .3; since they are, again ex hypothesi, independent, we can calculation the probability of a conjunction of them by multiplying the respective individual probabilities. So in the first group, each line contains two assertions and two negations and therefore has probability .0441 (= .3 x .3 x .7 x .7), and there are four of them, for a total probability of .1764. In the second group, each line has probability .0189 (= .3 x .3 x .3 x .7), and their are once again four of them, for a total probability of .0756. The third group has only one line, with a probability .0081 (= .3 x .3 x .3 x .3). Summing the probabilities of all three groups -- all nine ways for there to be at least some failure in both Argument A and Argument B -- we get .2601. This is the maximum uncertainty for the common conclusion. Hence, the probability of the conclusion is greater than or equal to .7399 (= 1 - .2601). Thus, combining these two arguments, neither of which, by itself, guaranteed a conclusion with a probability greater than 0.49, yields a probability greater than .7399 for the conclusion.

Another way to get to the same result is to consider the possible combinations of truth values that do lead to at least one valid argument. I won't drag you through the calculation, but the partial sums are .0882 + .4116 + .2401 = .7399. This calculation serves as a cross check on our first one.

The key points in this calculation are

(1) The assumption that the arguments were simple deductively valid arguments,

(2) The assumption of a probability for each premise -- for simplicity's sake, we chose .7 for each,

(3) The assumption that within each argument, the premises were probabilistically independent, and

(4) The assumption that across arguments, the premises were probabilistically independent.

Without these assumptions in place, the calculations become more difficult. In theory, it is possible to have so much positive dependence across arguments (failure of 4) that the second argument adds nothing to the first (say, in case we knew that, if either premise of the first argument were true, both premises of the second would have to be true). That is, of course, an unrealistic scenario for many arguments of interest -- a mere limiting case.

I hope that this is the sort of answer you were looking for.

Anonymous said...

Wow, thanks Tim!

Assuming the same values from your post (2 premises each as likely as .7), except this time we add Argument C with the same number of premises and the same probability for each premise, would this be the correct mathematical representation:

~A1 & A2 & ~B1 & B2 & ~C1 & C2
~A1 & A2 & ~B1 & B2 & C1 & ~C2
~A1 & A2 & B1 & ~B2 & ~C1 & C2
~A1 & A2 & B1 &~ B2 & C1 & ~C2

A1 & ~A2 & ~B1 & B2 & ~C1 & C2
A1 & ~A2 & ~B1 & B2 & C1 & ~C2
A1 & ~A2 & B1 & ~B2 & ~C1 & C2
A1 & ~A2 & B1 &~ B2 & C1 & ~C2

8(.3x.7x.3x.7x.3x.7)= .074088

~A1 &~ A2 & ~B1 & B2 & ~C1 & C2 ~A1 & ~A2 & ~B1 & B2 & C1 & ~C2
~A1 & ~A2 & B1 & ~B2 & ~C1 & C2
~A1 & ~A2 & B1 & ~B2 & C1 & ~C2

~A1 & A2 & ~B1 & ~B2 & ~C1 & C2
~A1 & A2 & ~B1 & ~B2 & C1 & ~C2
A1 & ~A2 & ~B1 & ~B2 & ~C1 & C2
A1 & ~A2 & ~B1 & ~B2 & C1 & ~C2

~A1 & A2 & ~B1 & B2 &~ C1 & ~C2
~A1 & A2 & B1 & ~B2 &~ C1 & ~C2
A1 & ~A2 & ~B1 & B2 & ~C1 &~ C2
A1 & ~A2 & B1 & ~B2 &~ C1 & ~C2

12 (.3x.3x.3x.7x.3x.7)= .047628

~A1 &~ A2 & ~B1 & ~B2 & ~C1 & C2
~A1 &~ A2 & ~B1 & ~B2 & C1 & ~C2
~A1 & ~A2 & ~B1 & B2 & ~C1 & ~C2
~A1 &~ A2 & B1 & ~B2 & ~C1 & ~C2
~A1 & A2 & ~B1 & ~B2 & ~C1 & ~C2
A1 &~ A2 & ~B1 &~ B2 & ~C1 & ~C2

6 (.3x.3x.3x.3x.3x.7) = .010206

~A1 &~ A2 & ~B1 & ~B2 &~ C1 &~ C2 1
(.3x.3x.3x.3x.3x.3) = .000729

Maximum uncertainty is (1-.132651)= greater than or equal to .867349 percent.


Tim said...


It's late at night and my eyes are blurring a bit, so I might have missed something, but it looks to me like you've done it right. Remember that this calculation is made simple by the constraints I listed. The independence assumption is critical; without it, the whole problem becomes more difficult (indeed, without further information of some kind as to the direction and degree of dependence, insoluble). So it's not a one-size-fits-all schema. But it's very useful to see how these limiting cases work out.

Anonymous said...

Hello Mr. and Mrs. McGrew,

I have a few questions regarding the maximum uncertainty of two premises, each of which has a .7 likelihood of being true, yielding a maximum uncertainty of .49 given the assumptions Tim listed. Instead of focusing on the cumulative force of n-number of limit case arguments for God I wanted to focus in on the Kalam Cosmological argument for the existence of God, and assume that each premise is .7 percent likely when conditioned on one piece of evidence for each premise.

1)Everything that beings to exist has a cause (Craig offers three lines of support for this premise, but lets just assume one of them for now makes this premise .7 likely)

2)The universe began to exist (Craig offers two philosophical, and at least three scientific lines of evidence for this premise, but lets just assume one of them for now makes this premise .7 likely).

3)Therefore, the universe has a cause.

Now, I think that you would say that the maximum uncertainty for this conclusion would be greater than or equal to .49 given the evidential assumptions listed above. So, my question is how do we augment the probability of the conclusion in the ‘greater than’ direction? Do we condition new, and independent pieces of information diachronically (what does diachronic/synchronic mean?) for each premise using the following version of Bayes:
Using likelihood ratios, we find that:


After each new conditioning of evidence do we retain our updated posterior probability each time by using the following rule:P(-H)=1-P(H)where the value of H in this case would be .49 from our first calculation so that when we condition a second piece of evidence for each of the two premises in the Kalam we get to plug in a value of .51 for –H. Would this be the way that conditioning new evidence augments the probability of the conclusion from .49 in the greater than direction?

Lastly, I guess I am a little concerned that without multiple lines of independent evidence for the premises in an argument we can only reasonably hold to the conclusion according to some maximum uncertainty value which seems like it would be low for any argument that only has one piece of evidence supporting a premise. For example, if the Kalam only had one line of support for each of its premises, should my degree of belief in the conclusion be .49, or could I reasonably maintain a higher degree of belief without any new evidence to condition upon?

Thank you,

Anonymous said...


This is Kevin again... I just wanted to gently and respectfully remind you that I asked a question a while back about diachronic conditioning of new evidence using the Kalam argument as an example and I just wanted to make sure that I haven't been forgotten.

Thank you,

Tim said...


Sorry: we had a busy spell and I did, in fact, forget for a while that you had left another question.

Regarding the Kalam: if the two premises are probabilistically independent and each has a probability of .7, then the probability of the conclusion is greater than or equal to .49.

To increase the probability from there, we need to conditionalize on new pieces of evidence. This can be a bit tricky, and the issues get too complex to discuss in a combox. I’ve been thinking of writing up a technical paper on this topic.

“Synchronic” means “at the same time”; “diachronic” means “across time.” We do not typically get all of our information about any topic of interest in a lump, all at once, so generally our degree of rational confidence in some contingent proposition P is changing across time.

You write:

Using likelihood ratios, we find that:


I understand the set-theoretic notation, and I gather that “Λ¹” stands for P(E|H)/P(E|~H), but I am having difficulty untangling the rest of the formula. Can you rewrite it in a more explicit form?

Your description of conditionalization isn’t very clear. Although P(~H) is always 1 – P(H), conditionalization involves assimilating some shift in one’s evidence by adjusting all of the affected propositions in the probability distribution. If you’d like to explore this more, drop me an email note.

Multiple converging lines of evidence can work wonders – they’re a bit like compound interest. Again, this is something better discussed in email than in a combox.



Anonymous said...

Thanks Tim,

I completely understand busy spells! I would love to see a paper by you on this topic. With that said, I would like to come back to my most recent question a little later in an e-mail as you suggested.
For now, I wanted to ask a question about Michael Tooley’s use of the Principle of Indifference where the PI says that if you have no information that would allow you to say which of n exclusive and exhaustive options will come true, you should assign each a probability of 1/n. This principle, if correct, would allow one to obtain knowledge of probabilities from the fact that one is ignorant. He uses this principle in two arguments to show that the probability of God’s existence is less than ½:

1) He argues that the prior probability of theism is less than half because there are three kinds of divine being that could exist:

Omnipotent, Omniscient, Omnibenevolent
Omnipotent, Omniscient, Perfectly Evil,
Omnipotent, Omniscient, Morally Indifferent

2) His version of the problem of evil is similar in that he lists more than two possibilities that exclude the existence of morally sufficient reasons with respect to the evil in the world, and concludes that the existence of God is less than half.

My first thought is that the PI only gives meaningful probabilities for things that exhibit a mechanical symmetry structure (i.e. dice, playing cards). However, PI doesn’t apply in the case of different combinations of divine attributes, and right-making and wrong-making properties because they involve people. In any case, I am not very confident in my attempt at a defeater, so I would really appreciate to hear your thoughts on Tooley here.

Thank you,

Lydia McGrew said...

Kevin, my suggestion is that you do a quick googling, find Tim's e-mail address that way (it shouldn't be hard), and send him that question in an e-mail. As he said, often these things are better discussed in e-mail than in comboxes.

Truth Seeker said...


I was wondering where recent work in cognitive psychology (i.e. confirmation bias, cognitive dissonance, eyewitness memory, percentage of population who have reported experiencing hallucinations, etc.) would factor in in Bayes theorem? I suspect that it depends on how the information is being used. For example, somebody might say that the origin of the disciples belief can be explained by an appeal to cognitive dissonace in which case this would factor into the right hand of the denominator in Bayes. However, what seem more common these days (and perhaps more lazy) is for 'skeptics' to just lump of all this information into the prior probability in Bayes theorem and declare victoriously that it is very likely on our background knowledge that people would believe in a resurrection even if there wasn't one in first-century Palestine. Now, I know how to deal with those who might try to use cogn. psychology to explain away something like the origin of the disciples belief in the resurrection, but I am uncertain as to how to respond to people who just want to throw all this information into the prior probability and declare victory. Does this information affect the prior probability of identifying a miracle? If so, then do we simply use the odd form of Baye's and show a disanalogy say between what we know about cognitive dissonance and the SPECIFIC circumstances surrounding the origin of the discples belief?

Thank you,

Lydia McGrew said...

I admit that I have trouble seeing how all of that would figure into the prior probability of the resurrection's having actually occurred.

It seems possible that any skeptic who tries to do that is confusing the prior probability of the resurrection with the prior probability of the evidence.

The prior probability of the evidence will be the combination of its probability given H and ~H (in this case R and ~R).

These considerations are obviously intended to make the evidence appear more probable given ~R. This will raise the prior probability of the evidence--in the ordinary form of Bayes's Theorem that is the bottom term on the right-hand side of the equation. When one is using the odds form, the prior probability of the evidence as such doesn't appear as a term. Rather, one looks at the ratio P(E|H)/P(E|~H), and this sort of evidence would make the P(E|~H) higher.

In neither case would such evidence, as I'm understanding what you're describing, make the resurrection *less probable*--that is, lower the prior probability of the resurrection.

Truth Seeker said...

Thanks, that is helpful. I wonder how we can distinguish between what counts as part of our background knowledge and what counts as specific evidence. This seems to be crucial when it comes to plugging in evidence into Bayes. Could we say that the evidence for the empty tomb, post-mortem appearances, and the origin of the disciples belief is our specific evidence. When it comes to our background knowledge I am sure that something like cognitive dissonance belongs there but then I would want to say that this doesn't do a good job in explaining the origin of the disciples belief for the various reasons N.T. Wright responds with (i.e. resurrection was well-defined, that there were other concepts that would have worked to exalt Jesus, that when an alleged Messiah died his followers either went home or got a new Messiah, etc.). So, would our cog. sci. data go on the left side of this factor:

Pr(not-R/B) × Pr (E/B& not-R)

Whereas N.T. Wright's responses would go into the 'E' on the right side of the above factor?

Lydia McGrew said...

As long as we know very clearly what we are doing, it isn't crucial that this or that definitely go into E as opposed to B. Everything is supposed to be conditioned on in the end. However, some ways of dividing it up are more helpful and make more sense than others.

My inclination would be to put everything you've just named into B. Ideally, what we want to do is to interpret the impact of the specific evidence, such as the disciples' testimony of the appearances, by way of a Bayes factor. It seems to me that everything you have named is relevant to helping to evaluate how strongly that E favors R over ~R. The skeptic raises some consideration that is supposed to lessen the impact of E (the appearance reports), lessen the magnitude of the Bayes factor for E, and Wright and others respond with considerations that are meant (whether they put it this way or not) to show that, no, there isn't really a good explanation for the appearance reports on ~R after all.

Now, I understand how this is a little confusing if we imagine that a consideration Wright is raising is something a given skeptic didn't know before. How can we be treating this consideration as part of B (background) if the skeptic had never heard it before?

Strictly speaking, if we are modeling the diachronic development of the skeptic's knowledge base, what we could do is to label a consideration of that sort as some separate piece of evidence--some En *other than* the specific evidence (such as the appearance reports). Then we could regard the skeptic as conditioning anew on this En, which should cause him to reevaluate R.

However, I wouldn't do it that way. For clarity's sake, in making the argument, I think we want to put those things into our data *before* conditioning on the specific evidence, so that we can really get a good fix on the magnitude of the Bayes factor for the specific evidence.

I therefore would say that even if the skeptic actually learns about some consideration Wright raises after learning about the disciples' reports, we should model it as though the information came in the opposite order so that the considerations Wright proposes are already in B for purposes of evaluating the Bayes factor for the appearance reports.

There is a whole literature on order variance in Bayesian conditioning. The short version is that if you regard yourself as always conditioning on uncertain evidence, the order of updates will affect the final probability.

As a foundationalist, however, I regard statements of non-certain propositions as shorthand for the much more cumbersome foundational sensory and memory reports that actually undergird my knowledge of those non-certain propositions. When you are conditioning on certainties, you have order invariance.

Really, what all this amounts to is this: When we talk about "our background knowledge B" we're really imagining some sort of composite, knowledgeable subject who has this vast amount of background information that a given ordinary subject, be he skeptic or Christian, probably doesn't actually have. Then, many of our arguments take the form of pointing out, drawing attention to, making _actually_ known, items in this "bag," as it were, of cumulative background knowledge about history, cognitive psychology, Jewish customs and beliefs, etc., etc., that the person arguing at the moment believes should influence the Bayes factor for the salient E, the item of interest at the moment, such as the reports of the resurrection experiences.

There's nothing wrong with this at all, as long as we realize what is going on.

Truth Seeker said...

Okay, great. That makes a lot of sense. I have one more pressing issue that I have been encountering recently. With respect to the 'multiplication rule,' some people have pointed out that in order to make a case for the resurrection there are a number of steps involved (establishing the wording of the original text, applying the criteria of authenticity, establishing the facts that go into the case, inferring the best explanation, etc.). But then, even if we assume each of the steps is highly likely to have arrived at something true(.9) we encounter a dwindling of probabilities. Now, this sounds like a hi-jacked version of Plantinga, and I have read Tim's response to Plantinga as well. However, although I 'sort of' understand the conceptual error lurking here (having to do primarily with not allowing for bayesian updating?) I am unable to apply it in practical terms to the various historical realities that go into making a case for the resurrection; especially since different people with chop up the stages 'required' for the case for the resurrection in different ways. I mean, where does a legitimate application of the multiplication rule begin and end (esp. with respect to making a case for the resurrection? It seems like anybody can chop up any number of steps in any manner they want, and then declare victory without even doing the hard work of looking at the specific evidence at each 'stage.'

Lydia McGrew said...

I don’t think it’s helpful to try to do a “multiplication rule” of that sort and think of it in terms of “stages.”

Here’s a much clearer way (in my opinion) to look at it: Look at our argument in the Blackwell anthology. Look at the Bayes factor we come up with. Now look at the items that figure into that Bayes factor–for example, that each of thirteen witnesses was willing to suffer death for testifying that he had seen Jesus after his resurrection in various detailed settings, including group settings, that Paul converted abruptly and said that it was because of a vision of Jesus, and the like.

Note that when we use Bayes’s Theorem, we treat some given piece of evidence E as if it were certain, even though this is to some extent an epistemological fiction for the sake of calculation. So the statement that the disciples claimed such-and-such (for example) doesn’t literally have probability 1.

Now, look back at the “uncertainties” that are being brought up and ask yourself which of them actually affect the probability of the items being treated as evidence and, in good judgment, _how much_ they really lower the probability of those things. For example, you mention “establishing the wording of the original text.” It would be a real error of judgment to think that any actual textual difficulties about exact wording make it anything other than highly, highly probable that the disciples claimed this. Real, specific textual issues just aren’t all that great and don’t cast any doubt on these facts. (Bart Ehrman is quite misleading about these sorts of things.) If by “authenticity” you mean to ask whether the gospels really do give us close-to-the-events, credible records of what the eyewitnesses claimed, yes, that is a big topic, but New Testament scholarship has been very messed up for a long time and has seriously exaggerated the uncertainties about the source and historicity of the gospels. In particular, the dismissal of patristic evidence on this topic has been unconscionable and has proceeded on the basis of very poor arguments. Here I suggest you find and listen, as much as you have time for, to Tim’s on-going series of lectures, on Youtube from Apologetics315, on the historicity of the Gospels and Acts. I cannot recall how many are up–two or three. There are six planned.

Note: The whole issue about authenticity is really part and parcel of “establishing the facts of the case.” (You see that I’m going through your examples individually.) The reason we’re concerned with the authenticity of the gospels and Acts is _because_ we are concerned that if they were written by people who weren’t close to the events or don’t reflect eyewitness testimony, then our various bits of E–E1, E2, etc.–didn’t really happen and shouldn’t be conditioned on. So those two are really bound together. If we have doubts about authenticity, that will be reflected in the probability we actually give to our E’s. (continued below)

Lydia McGrew said...

At that point, here’s a kind of rough cut thing to do: Pick a sample prior probability for R, the resurrection. Make it something that you think makes sense based on other evidence, other than what we bring up in the paper. Then do the odds form of Bayes’s Theorem as we give it in the paper, using our cumulative Bayes factor. See what you come up with as the odds for the posterior. It will (I predict) be extremely high, unless the prior has been jerry-rigged to offset it. Then just knock off a bit from that to take into account the fact that the bits of evidence don’t strictly speaking have probability 1 and therefore that it also doesn’t have probability 1 that all of them are true but that they _are_ exceedingly well-established. And you’ll still have something very high for the posterior of R. I admit that that isn’t a precise calculation. But I think it’s not misleading. I could at this point go into a disquisition on the Jeffrey Conditioning formula for updating on uncertain evidence, but I actually think that would not cast as much light as this rough sketch.

Note: “Inferring the best explanation” for the pieces of evidence (your last item) does not cast doubt on the probability of any of the E’s, because it is a matter of explaining the E’s, not a matter of deciding whether the E’s happened. It is tied up intimately with our argument in the Blackwell piece itself–that is, it’s related directly to the Bayes factor we give. So there you have to look at our Bayes factor and decide whether it seems like a fair statement of the weight of the case *given* these pieces of evidence.

Truth Seeker said...

Okay, thanks. Now, I listened to your interview on CommonSenseAtheism, and you gave the illustration having to do with a girl trying to figure out her chances of going on a date with a boy. I was wondering if you have any other analogies to illustrate the point that a chain argument as you call it, doesn't really model the reasoning we are using in everyday reasoning, or when constructing a case for the resurrection? I think a really solid and well-thought out illustration would help a lot people with this issue, and I am probably not bright enough or well read enough to come up with one on my own, so I was hoping that you and/or Tim had developed one...

Thank you.

Lydia McGrew said...

I think the "chain argument" models the way we think about how things _might_ happen or _could just possibly have happened_ but not the way that we are justified in believing that something actually did happen.

You can make up lots of examples similar to mine about the girl getting a date. Imagine any event that might happen in the future but that you don't have any very strong reason to believe will happen. Then spin out a causal chain that could bring it about, where every event in the causal chain depends on the previous one. You'll see that by the time you are done, this doesn't give you any very strong reason to expect the event. It's just a "possible way things might go." It may serve a purpose in showing that such-and-such isn't _wildly_ implausible or that it _could_ happen that way, but that's not anything like as strong as the evidence you get when the event actually happens.

What it really just amounts to is a way of getting a handle on the prior probability of the event, which isn't the same as specific evidence.

So suppose I've just filled out paperwork for part-time work as a store clerk. The manager has told me that I'll just work fill-in hours if someone else is on vacation or off sick, because he doesn't have a regular job for me.

So I'm trying to think about whether I'm going to get any hours next week. I might think to myself, "Well, I saw one of the workers today at that store, and he was sneezing and coughing. Maybe he'll get sick enough that he can't work next week, and then maybe there won't be any other person on regular duty who can fill in his hours, and then maybe I'll get called in."

Now, that _could_ happen. It's not wildly implausible. It just doesn't do a huge amount to raise the probability that I'm going to get called in. There are several things that could go wrong with it. Maybe this guy will get better quickly. Maybe he's incredibly tough and motivated and won't take time off even if he feels pretty lousy. Maybe there is someone else who will get his hours if he has to take time off.

(There are also other ways that I could get called in, of course. Maybe someone is about to go on vacation that I don't know about. So this little story about the person getting sick doesn't give me the _whole_ of my prior probability. But let's say it's the only thing I actually know about, and I'm trying to encourage myself by dredging up all the reasons why I might get hours next week.)

Lydia McGrew said...

Chain reasoning like that can also come up in spinning out possible scenarios regarding the past. Here's a real-world one.

When I was in graduate school I had a professor I really liked a lot, but he had a bee in his bonnet about trying to argue that Edmund Spenser, the 1500's English Renaissance poet, was influenced by the Greek Fathers--the ancient Greek Orthodox Christian writers.

He brought up several passages in Spenser's magnum opus The Faerie Queene where he thought he detected influence by the Greek Fathers. I was totally unconvinced. There simply was nothing in those passages that seemed to me all that remarkable in a Renaissance work or to require special explanation in terms of esoteric knowledge of the Greek Fathers.

So, to try to bolster his argument, he pointed out that the particular patristic writings he was interested in had been translated into English in the 1500's not very long prior to The Faerie Queene and (I seem to recall) that a friend of Spenser had a copy in his library.

It was amazing how much he thought this _added_ to the case, even though the actual textual evidence was just incredibly thin. He couldn't understand why I was unimpressed.

But of course the fact was that the mere possibility that Spenser saw the book in his friend's library (for which we had no other evidence), read it, and decided to include _very_ subtle allusions to it in his long poem was just that: a possibility. Whoop-de. The fact that the book was translated and might have been available to Spenser (of course, we don't actually know how much time Spenser spent browsing his friend's library) raised *slightly* the prior of the hypothesis that Spenser alluded to it. Better than if Spenser had had to dig up a copy and read it in Greek. And the translation does show somewhat greater interest in the Greek fathers in Spenser's time than if they had not been translated, so maybe someone would have recommended the book to Spenser or something. But this is all highly conjectural and does very little indeed to make up the deficiencies of the actual textual argument for an actual allusion.

You can see how something like this is a chain argument:

If Spenser saw the book in his friend's library and if he read it and if he read the particular passage in question then he might have been influenced to put an allusion to it into his poem.

Just doesn't amount to much.

I would imagine that there are many such "arguments" (just to be a bit controversial) in evolutionary biology: If this mutation took place, and if it was conserved even though it wasn't advantageous by itself, and if this other mutation took place, and if it was conserved, etc., etc., then eventually after a bunch of such steps that all just happened to work out just so we might have gotten the structure we presently have.

Truth Seeker said...

Thank you Lydia, I really appreciate your time on this. After reading, and re-reading your comments, I think something dawned on me. Does Plantinga's Dwindling Probabilities' argument only affect the prior probability of a historical case for the resurrection? Is it true of the 'multiplication rule' in general that it doesn't include the specific evidence at all, but rather, only has to do with the prior probability of some claim? If this is true, then I think I finally understand what is wrong this type of argument.

Thank you,

Lydia McGrew said...

"Does Plantinga's Dwindling Probabilities' argument only affect the prior probability of a historical case for the resurrection?"

Yes, that's correct. The error in Plantinga's argument is a very simple one: He confuses a prior probability with a posterior probability.

To get a little more technical, he sets the _prior_ probability of Theism as an upper bound on the _posterior_ probability of the specifics of Christianity, including the resurrection. He understands correctly that within any synchronic probability distribution the specific claims of Christianity cannot be more probable than the more general claim of theism. That's true. However, he simply blunders by using the prior probability of theism (before taking into account specific evidence that gives us strong reason to think that the resurrection actually took place) for setting that upper bound. But obviously, if you came to have strong evidence for the resurrection, the probability of theism itself should go up. So even if you were an agnostic before, if you came to have good evidence for R, you shouldn't be an agnostic anymore.

Lydia McGrew said...

"Is it true of the 'multiplication rule' in general that it doesn't include the specific evidence at all, but rather, only has to do with the prior probability of some claim?"

That's a slightly different question.

Multiplication of probabilities comes into play any time we have a conjunction of independent propositions. There could be contexts in which we should multiply probabilities or can model inference usefully by multiplying probabilities that don't just have to do with priors. For example, you can see in our Blackwell article that we multiply our Bayes factors for the various pieces of evidence. Then we have an argument that, even if some of the pieces of evidence are not independent, this doesn't mean that we're overestimating the strength of our case (but may actually be underestimating it) by multiplying the Bayes factors.

You might have a situation where an hypothesis was very ad hoc and had to include multiple independent parts in order to take account of the evidence. So, for example, a skeptic who wants to explain the testimony that the tomb was empty may hypothesize that grave robbers stole the body. But this does nothing to explain the evidence of, say, Paul's conversion, so he must hypothesize as an independent proposition that Paul had some sort of fit or hallucination that caused his conversion. The grave robber hypothesis also does very little to raise the probability that the disciples would have these incredible, detailed, multi-person hallucinations, so that's another partial explanation that is largely independent and has to be brought into play to explain all the evidence. Generally, the "penalty" for such an ad hoc multiplication of independent partial explanations can be modeled by multiplying Bayes factors. There is also a way of looking at it as reflected in a low prior probability of the ad hoc hypothesis, because the ad hoc hypothesis is this cobbled-together entity involving multiple independent events. It isn't just "The Resurrection didn't happen" but something much more complex than that.

So multiplication in the very broad sense comes up in a lot of places in Bayesian reasoning. The weird thing is that Plantinga's phrase "principle of dwindling probabilities" has grabbed people's imagination to such an extent that they think that if one ever uses multiplication in any circumstances in probability theory, one must acknowledge that there is "something to" the use Plantinga makes of it in Warranted Christian Belief as a supposed problem for evidentialist apologetics.

I hope all of that is clarifying rather than confusing.

Truth Seeker said...


I was wondering how to respond to someone who alleges that those who only argue that the specific evidence for the resurrection is really strong may be committing the base rate fallacy in general, and for two reasons in particular: 1) human testimony to the occurence of miralces is notoriously unreliable (Lourdes France), and 2) given that over 100 billion people have lived on Earth, and none of them have been resurrected, it is very unlikely that Jesus was resurrected. Basically, the prior probability of the resurrection is so low (I think this is synonympus with 'base rate') that even if the specific evidence for the resurrection is high, we shouldn't believe it.

Second, is the resurrection hypothesis ad hoc since in addition to the existence of God, we would need to show with some probability that God wanted to raise Jesus from dead right?

Thank you vor your time on this

Lydia McGrew said...

There is no reason to suspect that a person who argues that the specific evidence for the resurrection is strong is committing anything like a base rate fallacy unless that person seems to be saying that automatically the posterior probability is high given that the specific evidence is strong. We certainly do _not_ do this. Instead, we point out how low of a prior probability can be overcome by evidence of the strength that we do have. We consider that this puts the onus on the skeptic to show why the prior probability should be considered to be that low.

A straight induction concerning the number of people who have been raised from the dead is in any event an extremely crude and misleading way of trying to calculate a prior probability. The prior probability for the resurrection of Jesus should primarily focus on the existence of the Judaic God whom Jesus claimed to represent and the probability that Jesus would rise from the dead given that God does exist. Nobody claims that God randomly raises 1/n people in the world, so simply counting up the number of resurrections in the world is a ridiculous approach.

Moreover, in saying that _no one_ other than Jesus has been resurrected, the skeptic is assuming without argument that other reported resurrections are all false, which we shouldn't simply grant. For example, I don't claim that the evidence for the resurrection of Lazarus is as strong nor nearly as strong as the evidence for the resurrection of Jesus, but I'm also not going to grant without argument that no one other than Jesus has ever been resurrected in the history of mankind.

As for the reliability of witness testimony, simply grabbing some other situation, such as Lourdes, where miracles are reported and using this to try to come up with a generic “reliability number” for “witness testimony to the resurrection” is an extremely misguided approach. The Bayes factor for a given piece of witness testimony to the miraculous will depend on the particulars of the situation, not on some general “reliability quotient” for all witnesses to the miraculous. Tim's and my recently released paper in the Oxford volume _Probability in the Philosophy of Religion_ is all about this very issue. Here is the link to the volume at Amazon.

Our paper is on pp. 46-63 and unfortunately is not available on-line, but perhaps you can get the book through interlibrary loan if your library does not have a copy. You can read parts of it, though not every page, using the “look inside” function at Amazon.

See also our section on Hume and the “trial by proxy” in our long article on the resurrection in the Blackwell anthology edited by Craig and Moreland. Here, in case you don't have it handy, is the link to the preprint version of that article.

That section is entitled “Knavery, Folly, and the Love of Wonder.”

Lydia McGrew said...

On your last question, I cannot at all see why the issue you raise would make the resurrection hypothesis ad hoc. The resurrection is hypothesized as the best explanation of specific evidence, which is exactly the opposite of an ad hoc hypothesis. My own position on ad hocness is that one makes an ad hoc move when one tries to counter evidence that disconfirms one's preferred hypothesis by adding bells and whistles to that hypothesis to make a pretense of avoiding disconfirmation. There is no apparently disconfirmatory evidence in view here, nor does the question about Divine intentions constitute some evidence that appears to disconfirm the resurrection.

As for the specific question about divine intentions, there are a number of points to be made in answer. First, as I stated in the previous comment, it is certainly useful in thinking about the prior probability that Jesus would rise from the dead to examine the probability that Jesus would rise given that the Judaic God exists. This may or may not take the form of specific evidence showing that Jesus was, say, the "kind of person" whom the Judaic God would be likely to raise from the dead. If no such evidence happens to be available, that doesn't create any specific problem. However, if there were specific evidence that Jesus was a wicked person, that would of course be problematic, as God would not be likely to raise a wicked person from the dead. Naturally, I don't concede that any evidence exists that Jesus was a wicked person.

We do have some evidence relevant to the prior probability here: that is the evidence that Jesus fulfilled prophecy in his life and death, which is evidence that Jesus is the Messiah. And if Jesus is the Messiah, then there is some reason to believe that he would be raised from the dead, especially given that Jesus himself prophesied his own resurrection. I have an article that discusses this point coming out probably in 2013 in a special issue of Philosophia Christi. My article is on a handful of prophecies that Jesus appears to have fulfilled, and it also includes a section that shows how this is relevant to the prior probability of the resurrection.

Another point to consider, however, comes at this from a different angle. Suppose that we are asking whether the evidence for the resurrection confirms the existence of God. In that case what we need to know is that the _comparative_ probability for that evidence is higher given that the Judaic God exists than given that he does not exist. This does not require specific evidence about divine desires and intentions, since it is a comparative probability. It's especially important to bear in mind that we don't have to have a _specific_ probability that God "would want to" raise Jesus from the dead, nor do we have to have a particularly _high_ probability that God "would want to" raise Jesus from the dead. I have an article that should be coming out any day entitled "History and Theism" in the Routledge Companion to Theism that argues this very point concerning divine intentions and desires. A different article that discusses the same point at a more popular level is available here:

Lydia McGrew said...

The evidence for Jesus' own miracles and also the Liar, Lunatic, or Lord argument are relevant to the question of whether God would raise Jesus from the dead.

Truth Seeker said...

Thank Lydia,

You are awesome :)

Truth Seeker said...


This question is a bit different, but I was wondering how you might respond. My atheist professor deems the whole project of natural theology as a failure since what he calls the big 5 attributes of God (singular, personal, omnipotent, omniscient, essentially good) are underdetermined by all the arguments from natural theology. Is there anyway to build a rational bridge from the hypothesis of an omni-God and the arguments from natural theology?

Thank you

Lydia McGrew said...

A couple of points: I often find it interesting that professors say this kind of thing but don't therefore seem interested in actually granting the conclusions of natural theology. I mean, if they are so unthreatening to their worldview, and if the arguments actually work, why not believe the conclusion? Would it really be so unobjectionable to them to believe in many gods or in gods with a morally mixed nature who were the First Cause of the universe? Somehow I doubt it.

I say this as someone who isn't really a natural theology person. That is to say, I do my work in what Richard Swinburne (I believe) has taken to calling "ramified natural theology." I work in the areas of Bayesian analysis of things that actually purport to be revelatory, so naturally, if they can be supported, they are going to leap right over your professor's objections.

As far as the attributes listed, he's *completely* wrong as far as "personal" is concerned. Some sort of surd impersonal force is simply not going to do the work needed as a First Cause for arguments like the argument from mind, the Kalam cosmological argument, or the teleological argument. After all, to take just one of these as an example, the whole point of the teleological argument is _precisely_ that the explanation must lie in a Being who is capable of causing *real teleology*, which means a personal entity.

As far as the other attributes (omnibenevolence, omniscience, singularity, etc.), as you know, there is an entire literature out there on this very subject. Somehow, I'd be willing to bet more on St. Thomas Aquinas, William Lane Craig, or Richard Swinburne any day than on a professor who can say something that flippant. However, I don't claim to mastery of that natural theology literature and would encourage you to go for it yourself and immerse yourself in it if interested in un-ramified natural theology.

But I'll bet you anything that your professor has never considered either a) the way that the argument from miracles, prophecy, and the like is utterly untouched by his point and b) the fact that such ramifications are complementary to traditional natural theology. The two can work together.

Truth Seeker said...


I was recently reading, but not fully understanding Thomas Crisp's article in the book Analytic Theology: new essays in the philosophy of theology (

In that chapter he considers the following set of premises to argue that the Bible is divinely inspired:

T: God exists.
A: God intervenes in history to provide a propositional revelation about
B: Jesus’s teachings were such that they could be plausibly interpreted as
implying that he intended to found a church that would function for a
long period time as an authoritative source of information about him.
C: Jesus rose from the dead.
D: In raising Jesus from the dead, God declared his approval of Jesus’s
E: The Church that, by the start of the Wfth century, had pronounced on which books were divinely inspired, is a legitimate successor—the ‘closest
continuer’—of the church founded by Jesus.
If so, then the strongest case for IB will be compromised by an undermining objection.

As far as I can tell, the undermining objection has to do with certain branches of the lattice Tim has in his article where he responds to Plantinga's PDP argument except that Crisp thinks that these other branches are either too low or inscrutable and that this results in a Dwindling of Probabilities though for different reasons than Platinga argues. I use similar premises for thinking that the Bible is inspired (though shorted in number), but still, if Crisp is correct, it seems I cannot use arguments and evidence to believe the Bible is inspired any longer. What do you think?

Thank you,

Lydia McGrew said...

I've seen that article. The fact that he in talks about the lattice device that Tim invented really is pretty irrelevant. Tim was explicit (as I recall) that that device is merely a tool for modeling the Theorem on Total Probabilities and should *not* be thought of as modeling inference per se. Crisp's use of it seems to me merely a distraction.

In the middle of his article Crisp makes a fairly serious technical error, an error that he should have avoided, given his inclination towards the use of technical devices and given the fact that he evidently thinks rather highly of himself in that regard. Since I will be urging one or more young philosophers to take the opportunity to tear into this one--it would make a pretty sweet publication--I am more than slightly disinclined to explain that error in detail here. Suffice it to say that Crisp does not in fact understand comparative inference and the evaluation of Bayes factors, though he quite evidently thinks he does. Because he does not understand comparative inference and the evaluation of Bayes factors, he says some airy and dismissive things that are badly misguided and that are quite important for his argument. Sorry not to be able to be more explicit.

Lydia McGrew said...

I have to add: Since I infer that Thomas Crisp is not a Roman Catholic, I don't quite know where some of the premises in that argument are coming from. All that stuff about the authoritative church throughout time is not what I would say about arguing that the Scriptures are divinely inspired. But that's neither here nor there as far as the error I alluded to in the previous comment.

Anonymous said...

Thank you so much for all your work in this area Lydia. I am completely new to probability theory however and I wondered if you could recommend a good place to begin reading? Thanks.

Lydia McGrew said...

It wd. depend entirely on your background and on what you have already read and are already familiar with. Do you have any math background, for example? Any philosophy background? You can find my e-mail address at What's Wrong With the World on my author page and write me with more details, if you'd like.

Truth Seeker said...


I lost your other contact information so I am using this medium instead:

As I understand the 'normalizability objection' if we could offer some good theoretical reasons for making the unbounded strengths and quantities of the constants finite, we could arrive at meaningful probabilities. Robin Collins attempts to do this with his notion of an epistemically illuminated range and the restricted principle of indifferece. Do you think this is a satisfactory defeater?

Thank you,
Kevin V

Truth Seeker said...

Hello again,

Ok, I found Tim's reply to Robin on this and so I wanted to offer an independent reason like Tim suggests, "If we had some independent reason to think that either a designer or (in some sense) the random selection of a universe would be limited to the range of possibilities that we can probe with our current theories at the dawn of the 21st century, then we might be able to make progress."

What if we defend and adopt the position that the laws of nature are metaphysically necessary? Then we would have independent reason to think that a desginer would be limited to the range of possibilities that we can probe at the dawn of the 21st century, no?

Thank you,
Kevin V

Gary said...

When a Christian starts using complex mathematical formulas and philosophical theories to defend his belief in first century corpse reanimation-transformation (aka:
resurrections)…I yawn.

I yawn because it is soooo silly.

I know for a fact that if a Muslim attempted to use these same ploys to defend the veracity of Islam’s claim that Mohammad flew to heaven on a winged horse, the very same Christians would snicker and hand-wave away these arguments without giving them a second thought, believing that these tactics are nothing more than an obvious, desperate attempt to dress up a superstition as believable reality.

Tim said...

Thanks for sharing your unargued opinion, Gary.