Monday, December 11, 2006

A Bayesian model for the historical case for Christianity

This is the first post in what I hope will be a series on the historical evidence for Christianity. The views on this are enormous in variation, from those who think Christianity completely unsupported by history to those who believe that the historical evidence for it is so strong that only someone suppressing the truth could remain an unbeliever. My view is in the middle. There is as positive historical case to be made for Christianity, but it depends on what's in the rest of a person's belief system whether that case is sufficient.

The model I’d like to use to discuss this is one using Bayes’ Theorem. I had trouble showing BT on my blog, so I am expressing it here.

P(B|A) =
P(B) x P(A|B)
Over P(A)

Let’s take B to be the thesis that the founding of Christianity involved action by God or some other powerful supernatural agent. Not-B, on the other hand, would be the view that Christianity was founded without the aid of any beings of superhuman power, that ordinary natural causation produced all the events which resulted in the spread of Christianity on earth. A would be the various pieces of historical evidence which can be brought forward to bear on this issue.

P(B) in this theorem would be the initial probability that the founding of Christianity was miraculous, before we examine the specific historical evidence. The problem here is that I know of no way to objectively measure the antecedent probability of anything. In fact, I am pretty much a subjectivist Bayesian. I think that people have personal prior probabilities and they ought to alter those initial probabilities as evidence comes in , but I don’t know of any way to actually prove that one set of initial probabilities is correct and another is not. Some people have maintained that it is possible to go from how frequently an event has occurred in the past to how antecedently likely it is to occur now, but the problem is that every singe event falls under a range of classes. Hume didn’t use Bayes’ theorem, but if he did he would have said that miracles are event-types that occur so infrequently in experience that the prior probability for them so low as to make belief unreasonable no matter what the other figures are. C. S. Lewis’s book Miracles was a book called Miracles: A Preliminary Study, meaning that his argument was designed to show that the antecedent probability for the miraculous should not be vanishingly low. I am going to presuppose that different people will have different priors for miracles.

P(A/B) is the likelihood that the pieces of evidence should exist on the assumption that God was involved in the founding of Christianity. If God were miraculously involved in, say the resurrection of Jesus, should we expect to find a new movement arising based on the claim, would it make sense of a Jewish group arising that changed the Sabbath from Saturday to Sunday, etc.

P(A) is how likely it is that these event should have occurred whether or not there was any miraculous involvement. Is what happened in the life of Jesus and the founding of Christianity likely to have happened. Perhaps human gullibility and fallibility is such that people would have come up with something like this anyway, even without divine intervention.

I’m going to set aside the issue of prior probabilities and ask the question of whether P (A/B) is signiicantly greater that P/A. If it is, then there is a confirming argument for Christianity to be found in history, even if, according to many people’s credence function, it is insufficient to secure acceptance for Christianity. If it is not significantly greater, then there is no confirming argument for Christianity to he had from history.

This is a link to my Infidels essay on miracles, which should help understand the basic concepts.

Bill said...

Your project seems identical to mine at Debunking Christianity, although we are approaching this from completely different perspectives. I have been using Bayes’ Theorem to weigh the evidence for and against the resurrection. I would greatly appreciate your critique of my assessment of the priors and the evidences I have evaluated so far. Like you I plan to continue to include evaluations of the relevant evidences.

My initial post illustrated on the topic was to use Bayes' Theorem to assess an ESP claim here. My initial evaluations of the plausibility of the resurrection are here, here, and here.

By the way, if you use the preformatted text tag "pre", you can make Bayes' Theorem look fairly decent.

The Uncredible Hallq said...

I think that a strong case can be made that the probability of many facts about Christianity is greater on the hypothesis that it was of human origin: the fact that the resurrection appearances were private in contrast to Jesus' public ministry, Jesus' failure to fulfill then-current interpretations of messianic prophecy, the reports that Jesus' healings weren't always sucessful, the lack of modern miracles (and abundance of false claims of miracles), the fact that God supposedly gave the Great Commission to humans when he could have spread the world quicker with angels, etc. The thinness of the historical record means the evidence against must be circumstantial, but in this case it is quite strong.

Jason Pratt said...

Gosh. I just have picky logical concerns about the project from the outset. {g}

(Though Bill and Hallq do demonstrate the ambiguousness of the whole thing, too. When the whole project depends on intuitive estimates of liklihood at every level, then we might as well stick with reporting those estimates. By which I’m not saying anything against intuitive estimates of liklihood, btw. But at best this kind of formula only describes in algebraic shorthand why a particular person believes an inductive evaluation.)

Anyway, I spent waaaaaaaaaaaaaaaaaaaaaaaaaaaaaayyyyyyyyyyyyy too much of my ‘work’ work time yesterday composing a comment for elsewhere; so instead of going into every possible crit, I’ll stick with something that might be revisable. (But which will have to be revised, on peril of double-standard.)

{{P(B) in this theorem would be the initial probability that the founding of Christianity was miraculous, before we examine the specific historical evidence. The problem here is that I know of no way to objectively measure the antecedent probability of anything.}}

Which is a reasonably cautious qualification to make.

However: if you’re going to null P(B) out of the equation on this ground (and equations of this sort aren’t really intended to have elements nulled out, btw), then that sauce ought to be cooking P(A)’s gander, too.

This is because P(A) pretty much _has to be_ a prior probability estimate to be of any feasible use at all in the equation. (Not incidentally, the Bayesian element it’s replacing is avowedly a prior probability estimate. Even less incidentally, this formula has about as much resemblance to Bayes’ Theorem as Islam does to orthodox trinitarian Christianity. Making P(A) out to be something other than prior probability isn’t going to improve that comparison. But that’s another much larger-scale crit. {s})

If P(A) is supposed to be posterior probability, then I have at least two more problems. First, I have no idea what sort of probability P(B|A) is supposed to be, then. (Nominally it’s supposed to be posterior probability after A is given.) Second, though--which admittedly would get around the first--the _posterior_ liklihood estimate (or probability--in this case they’d be the same, though normally they’re not) of any non-repeatable historical set is 100%. (i.e. we’re not talking about a repeating process like throwing a ball on a plane and seeing where it lands from time to time and developing a ratio of expectation from that. Which is the kind of thing that BT was set up to evaluate probabilities of, btw. Not non-repeatable historical events, much less the liklihood of non-historical concepts being true. Good thing this isn’t really BT, then, hm? {g}) Dividing anything by 100% simply returns the element without alteration. (Divide any number by 1. You get that number.)

So even if P(A) _could_ (in some way unknown to me) be considered posterior probability, it might as well not even be there in this case, because due to the _type_ of thing being evaluated there it would carry no algebraic weight.

If P(A) is considered to be prior probability, though, then you’re either going to have to null it out on the same ground of principle for ignoring P(B); or else you’re going to have to include P(B) on the same ground you’re including P(A). (None of which even touches what kind of probability P(A|B) is supposed to be. Posterior probability of A given B?? Admittedly I’m not entirely sure. Originally it was supposed to represent the predictive power of a hypothesis and current evidence in regard to new evidence that has already shown up, although in that case it would still be a prior probability estimate. I mentioned that this wasn’t really Bayes’s Theorem, right? {s})

If the reply is that you’re okay with using subjective antecedent probabilities for P(A) but not for P(B)--due to P(B)’s sensitive topic, let’s say--then my reply is going to be that on _this_ ground P(A|B) ought to be even _more_ taboo than P(B), because _it_ depends on granting provisional _certainty_ (not a mere liklihood estimate) to B and evaluating A’s liklihood in that provisional light.

One way or another, you’re going to be back to including P(B). So you might as well go ahead and do it now. And if you think it’s too question-begging (or something like that) to include it--then I would say you should strenuously re-evaluate the feasibility of the project. (Or at least re-design the formula. {s})

Jason Pratt

Jason Pratt said...

Except, of course, I've mis-spelled "likelihood" every single time. {pounding forehead viciously!} Oh well, at least I was consistent... {self-critical g!}

JP

Anonymous said...

Here's some subjective intuitive likelihoods of my own:

-That a bunch of devout, monotheistic Jews would have included a crucified rabbi in the identity of God in less than 20 years after his death.

-Precisely the fact that this crucified rabbi failed to live up to popular messianic expectations...why did people continue to revere him, and not only revere him but give him honor and worship which in Jewish circles was meant for God only?

-Precisely the reports that Jesus' miracles weren't always successful, and the inclusion of other embarassing details in the Gospels which suggest that with them we have a sober attempt to hand down to us the whole truth of the matter, together with amazingly successful miracles and authoritative teaching.

-the abundance of modern miracles (I don't know where you get your information, hallq; you need to spend a few years working with missionaries in Africa and China)

All of these and more make it more likely to me that Christianity had a supernatural origin.

Anonymous said...

So apparently, if the miracles of Jesus were a failure or if the miracles of Jesus were a success, they count for evidence that Jesus was God?

In any case, the literal, historical Incarnation of Jesus makes little sense without the literal, historical Fall of Man. That alone pretty much places the Christian myth in the same category as the Greek myths as far as I'm concerned.
Sam

Anonymous said...

The early Christians would have been murdered within weeks if they had changed the Sabbath from Fri/Sat to Sat/Sun.

Anonymous said...

Anonymous,

I did not say that whether all the miracles of Jesus were false or all of them were true they are evidence for his divinity. I was only arguing that the fact that the evangelists were willing to incorporate otherwise embarassing elements in their accounts suggests that they are basically reliable sources for the miracles at which Jesus didn't fail at as well. And don't forget the circumstances in which he failed: because of people's unbelief.

And the literal, historic Fall only has consequences for one particular conception of the meaning of the Incarnation, that of Paul. With the Incarnation and the Resurrection as starting points, Paul felt compelled to interpret them in light of Scripture as he understood it. There is no reason why Christian theology, just like science, cannot formulate better models for understanding particular phenomena, all the more so in a case like the Incarnation, which is literally a mind-boggling mystery.

I'm not aware of any NT evidence that the Christians changed the day of the Sabbath. But I think eventually the two holidays (Sabbath and 'Lord's Day' or Sunday) became differentiated, to the point where a largely Gentile Christianity simply stopped observing the Sabbath. Pliny does say that Christians gathered on the first day of the week before dawn, which suggests that they were commemorating the Resurrection.

Anonymous said...

"And the literal, historic Fall only has consequences for one particular conception of the meaning of the Incarnation, that of Paul. With the Incarnation and the Resurrection as starting points, Paul felt compelled to interpret them in light of Scripture as he understood it. There is no reason why Christian theology, just like science, cannot formulate better models for understanding particular phenomena, all the more so in a case like the Incarnation, which is literally a mind-boggling mystery."

What's mind-boggling is how readily a Christian as yourself would be willing to throw out Paul's view of the Incarnation. Especially in light of the church's belief that Paul's theology is inspired by God.
Not sure why you would even need to bother with insisting on an actual, historical Incarnation if you take such a lax view toward the traditional teachings of the church. Perhaps a better model for the Incarnation is simply that it is an ahistorical metaphor of God's love for humankind:?
Sam

Anonymous said...

No, that's completely misguided, Anonymous. I am not so quick to 'throw out' traditional teachings and I do think St Paul's theology was inspired and insightful. But holding Paul's teaching in such high standing should not in the least preclude our ability to engage with the texts afresh in each generation. Our commitment to Scripture requires it, in fact. It's a false dichotomy that says either you stick completely and rigidly with a fixed, pre-defined understanding of Scripture and doctrine, or you just throw the tradition to the wind and pretty much invent your own theology. There's a fine balance to be drawn between conservatism and reformation.

Jason Pratt said...

Plus, there are ways of considering the Fall of Man to be historical while also considering the _anthropology_ of the Genesis 1-2 story to be given in something like visionary shorthand. (How to compress thousands of generations of history into something that brings matters up to a contemporary Abrahamic date as effeciently as possible? The result would have a mythical _look_ to it.)

Tim said...

Bill,

I had a look at your project over on DC. It looks to me like you're factoring in some of the considerations incorrectly, but it's a subtle issue and I don't want to say anything definite until I have more time to go over it. This is not just a quibble over your numbers. If you're dying to know what I have in mind, drop me a note and I'll sketch it for you.

Hallq's list of considerations that tell against Christianity is pretty uneven, and many of them strike me in exactly the opposite way. The resurrection appearances were public enough for Paul to state flatly that there were still hundreds of living witnesses (I Cor 15:6). If the healing comment is a reference to Mark 6:5 then it seems like a misinterpretation; but perhaps Hallq has something else in mind. The bit about messianic expectations actually tells in favor of its authenticity; if you're making something up, why not fit it into an existing framework? Lack of modern miracles does tell against a position which holds that they should proliferate today, but it doesn't put a dent in the historic Protestant position. Again, the bit about spreading the word with angels instead of with men has a point only on the assumption that God's only goal is to optimize the spreading of the word. But this is obviously not the NT position.

Bill said...

Tim,

Please do sketch out any mistakes in my approach. I would be grateful.

I will admit that I would be very surprised if you found an error in my applications of the numbers. In addition to using the formulation I presented in the blog, I used a logarithmic formulation of Bayes' Theorem and got the same results. The alternate equation I used was 4-11 in Jaynes book, "Probability Theory: The Logic of Science" found here.

There were some independence assuptions I made, but I did try to make them explicit.

Tim said...

Bill,

The problem is precisely in the independence assumptions. As you know, the simple Bayes factors can't just be multiplied unless the relevant screening relations hold. In Jaynes's book, equation 4.11 doesn't build in that assumption, but the assumption is made explicit in 4.12 and then applied to 4.11 to yield the simpler version in 4.13, which appears to be what you're actually using on the blog (without the log scaling factor).

Whether the assumption of independence is justified and what happens when it is not are issues that cannot be decided by the formal machinery alone. In the case of the calculation you're doing, if my first impression pans out, their failure will make a significant difference to the computation. And there is reason to suspect that the effect will snowball as you incorporate further data.

As I said, drop me a line and I'll try to explain my concern.

Tim said...

Bill,

Oops! You're working from the pdf file of Jaynes's book, whereas I'm working from the physical book. My references are right for the physical book, but in the pdf file you link to things are numbered differently. In terms of that numbering, my claim would be that you need to use something like 4-10 (with or without the scaling factors) rather than 4-11 when the condition at the bottom of p. 405 fails.

I apologize for any confusion caused by my citing the book rather than the file.

Bill said...

Tim,

I don’t see your email address on your profile, so I don’t how to drop you a line. I would be happy to continue the conversation in private if you would prefer. I gave some consideration to the assumption of epistemologically independent evidence. I briefly discussed in my post http://debunkingchristianity.blogspot.com/2006/09/evaluating-evidence-for-resurrection.html from which I quote:

Technical caveats: I am presuming that the evidences here are epistemologically independent. If the evidence E is comprised of several constituents (i.e. E = e1, e2, ..., en), by the product rule in probability P(E|H) = P(e1, e2, ... en|H) = P(e1|H) P(e2|e1,H) P(e3|e1e2,H) ... My analysis presumes that P(e2|e1,H) = P(e2|H), P(e3|e1e2,H)= P(e3|H), etc.

If I am not careful I could, in essence, count the same evidence more than once. Expressing the relations of the evidence in a Bayesian belief network may provide a much better assessment of the evidences. However, this increases the complexity of the analysis and if the epistemic couplings of the evidences are not strong, the overall results will not be affected significantly. Further, it seems to me that coupling the evidences would likely reduce the strength of the evidence for the resurrection.

You obviously take issue with my last statement so I will try to explain it a bit. It seemed to me that the evidences for the resurrection seemed strongly coupled. I listed 11 evidences for the resurrection http://debunkingchristianity.blogspot.com/2006/09/evidence-for-resurrection.html. If it were shown that Mark wrote legend and Paul’s conversion experience was based on a vision and the appearance to the 500 were also visionary, that would impact all 11 evidences I presented. The evidences against the resurrection don’t seem to share a common dependence.

That said I have not actually constructed a Bayesian network to express my beliefs. I am just beginning to learn about them. I would like to see how they could be applied to this analysis.