Monday, July 1, 2013

Reader question about Bayes's Theorem and the alleged resurrection of Jesus

I got the following email today from a fellow named Thomas. He wants to know about Bayes's theorem and its applications to historical questions, notably the alleged resurrection of Jesus. Thomas writes:
Dear Mr Lindsay
I wonder whether I can ask you a question about the application of Bayes' theorem to miracles. I shall start by saying that I certainly don't believe in miracles. However, it seems to me that there is a problem in using Bayes' theorem to argue against miracles. Consider, for example, the probability of having the evidence that we do concerning Jesus' alleged resurrection if Jesus wasn't resurrected. It seems to me that the probability of having that evidence is infinitesimal. The reason is that the probability of any event in history is infinitesimal. If history could be rewound to 100BC and replayed a billion times there would be no Christianity, no belief in a resurrection and, in fact, no Jesus. The same could be said about the Second World War. If you could rewind history to 1800 and replay it a billion times there would be no Adolf Hitler. By that I don't mean that there would be Adolf Hitler but he would live a different life, I mean that there would be no such individual at all.
Is this the right way of looking at it, or have I missed something?
I'm putting this here because I wrote Thomas a rather long response that actually covers a topic I've been meaning to blog about for a while concerning Bayes's theorem. My response follows the dashed lines.

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Hi Thomas,
In general, I'd say that Bayesian reasoning has stood up quite well in practice, so there is some pretty solid reasoning behind it and its application. Applying Bayes's Theorem to the historical process is a reasonable thing to do, but it has to be done carefully.

Let me start with one of your assumptions (I get the impression you've been reading Richard Carrier, maybe?). "The probability of any event in history is infinitesimal." This is problematic on a number of fronts.

First, infinitesimals are kind of a fringe branch of mathematics except as a formalism. In other words, I don't know if they mean anything, really. This may or may not be a huge strike against using them in Bayesian reasoning. If it is a strike against them, I will have to deal with the fact that it does some damage to one of my arguments about God's non-existence, which is fine. This question is a rather big one for me lately, and I don't think it's resolved in general yet. The problem could actually be with Bayesian reasoning in certain cases, though, which is another matter.

Second, which probability are you talking about? In Bayesian reasoning, you're having to consider two separate probabilities (really, for hypotheses like this, plausibility is the right word, not probability): the prior and the posterior. The prior plausibility is how likely we estimate the hypothesis to be just on whatever background knowledge we have, whereas the posterior plausibility is how likely we estimate it to be after we consider the evidence for or against it. Which, the prior or the posterior, are you saying is infinitesimal? It seems you're talking about the prior.

Third, this raises the question about how we assign prior plausibilities in the first place. Ultimately, they don't matter, and so a decent case can be made that in situations where we are trying to answer questions about one-time events or broad hypotheses, we simply shouldn't assign a prior at all. All we need to do is examine the evidence and see where it is leading the posterior. In the case of the resurrection of Jesus, the evidence points almost uniformly away from such an occurrence, so whatever prior plausibility we assign it, unless we assume it is certainly true or false (or all but, i.e. infinitesimally different from one or zero), will be swamped by the evidence that we have that suggests that no such thing ever occurred. Some of this is by examining what evidence we have, much of it is examining expected evidence we do not have, and still a lot is based upon Hume's standard for such claims. The famous maxim now known as "Hitchens's razor: That which can be asserted without evidence can be dismissed without evidence" could be rendered in Bayesian speak to capture this very notion along with your observation. Essentially, what it's saying is that the prior plausibility for certain kinds of hypotheses (e.g. historical events, one-time occurrences, ontological hypotheses) is ultimately completely arbitrary and therefore may be meaningless.

Fourth, according to Bayesian reasoning, whatever Richard Carrier and a few others assert, assigning an infinitesimal prior is probably not allowed. In this case, this fact renders your claim incoherent. You simply cannot claim that the (prior) plausibility of any event in history is infinitesimal. This opens wide vistas of questions regarding how we are to handle hypotheses with Bayes's Theorem since there are infinitely many potential hypotheses. This book, I'm told, is quite good for this problem, but I haven't done more than skim it:

Fifth, the entire way that most people (including me in the past) use Bayes's Theorem to evaluate hypotheses is probably incorrect. This is not the same as saying that it is incorrect to engage in Bayesian reasoning, but it urges us to be more cautious in how we do it. Frequently, we're given an equation that allows us to calculate an estimate of the posterior probability by considering three numbers: (1) the prior, (2) the likelihood we would see the evidence we see given the validity of the hypothesis in question, (3) the likelihood we would see the evidence we see under alternative hypotheses. In practice, here's the problem: (1) the prior itself may be meaningless in many cases, see above. (2) How can we estimate this number except in relatively simple cases? Particularly, we should expect a lot of error in this number. (3) This number is probably actually impossible to estimate since there are infinitely many potential alternative hypotheses, each with their own plausibilities--and these may be problematic in a deeper sense (next paragraph). Since (1) is subjective and ultimately pretty unimportant, (2) is error-prone particularly to biases like confirmation bias, and (3) is probably impossible to estimate and might actually be meaningless, this application of Bayes's theorem (popular with Richard Carrier, blogged about by me here before following Carrier) is probably not very informative at all.

The deeper problem alluded to in (3) above is that we could have two different hypotheses that have enormous amounts of explanatory power. For example, consider the hypothesis "general relativity is correct." What are the alternative hypotheses here? What are their plausibilities? What if the alternative hypothesis subsumes general relativity in the same way general relativity subsumed Newtonian mechanics? What if we reach a point with our theoretical frameworks, e.g. with hypotheses like string theory, etc., where we are no longer able to experimentally verify that one has more explanatory and predictive power than another. I don't care about the physics here--my question is how do we estimate the probability required for (3)? I don't think we can do it at all.

A better way to use Bayes's theorem, which I've been meaning to blog about for months, is to directly compare two hypotheses, using (admittedly subjective) estimates for the prior and degree to which the evidence is predicted by that hypothesis. This gives us a sense of the relative weights of two hypotheses proposed to explain the same phenomenon. For example, let's consider the resurrection of Jesus as one of our hypotheses, and let's consider "his desperate followers lied, hallucinated, or otherwise mistook the situation as is typical in cult situations."

If we use Bayes's theorem on both of these hypotheses and then divide one by the other, we can see their relative weights. I put "Resurrection" on top and "Cult Behavior" on bottom and see it like this:

P(Resurrection given evidence)     Prior(Resurrection)     P(Evidence on Resurrection) 
---------------------------------------------- = ----------------------------- x --------------------------------------------
P(Cult behavior given evidence)    Prior(Cult Behavior)    P(Evidence on Cult Behavior)

Now, while I don't necessarily think applying a prior probability to a hypothesis like "God exists" is valid, doing so regarding the alleged resurrection of Jesus is probably not only valid but strongly supported on "both sides" of the discussion. I don't think it's controversial in the least to suggest that (if we're talking in general) that the prior plausibility of the resurrection is very low. Indeed, the potency of the Christian story depends upon that fact, so we can all reasonably agree that it is very unlikely that someone tortured to death on a cross rose from the dead. This kind of thing just never apparently happens. I don't think it's the least bit controversial, given the number of them that have and still arise exhibiting identical forms of behavior, to suggest that the prior plausibility of cult behavior is very high (nearly 1).

The examination of evidence is trickier. My general contention is that faith is a cognitive bias that distorts how we evaluate evidence, shifting our evaluation to support what we already believe. The most argumentation is needed here, particularly since we have already presented reasons that people may try to dismiss the already overwhelmingly damning case that the prior plausibilities present. I would argue this: had there been a genuine resurrection, Jesus' alleged promises would be recognizably verified, i.e. there would be unambiguous evidence in favor of Christianity. That such evidence does not exist is powerfully damaging to the case for the resurrection. Furthermore, the overwhelming fracturing of the Christian faith into over 40,000 denominations suggests strongly that there's more not to believe than there is to believe with this whole story. Furthermore, this is exactly what we expect to see in cult behavior that takes on a life of its own outside of the original charismatic groups and leaders: the untenable, unsupported claims lead to differences in interpretation and thus schisms. On that, without trying to estimate either number in the fraction on the far right, I suggest that the number on bottom must be considerably larger than the number on top, not least because everything we see in light of the resurrection can be equally well explained in light of cult behavior, but not vice-versa.

Given that, we have that the comparative weight of the two hypotheses in question tilts very, very heavily toward "cult behavior" and away from "resurrection." If we admit the priors in this case, one in which background knowledge is pretty extensive (we've seen no one else, literally no one, not any animal even, come back to life as in the Jesus story), then the question is almost beneath consideration since the relative prior for the resurrection against the prior for cult behavior is necessarily pitifully low and everything else in the equation lowers it.

To really beat this dead horse, the entire Christian story hinges upon the existence of God. There, I do not think we can assess priors meaningfully, but the evidence of the world has consistently pointed away from any god other than some decreasingly interesting Deist God (current state of affairs: "God" is the underlying field of quantum states and the fluctuation that allowed the Big Bang to occur). This is, very importantly, not the God of Christianity. Since the entire Christian story hinges upon a specific God hypothesis that isn't this one, and since the evidence has uniformly pointed away from that God (with "superstitious people exist and believe superstitions" as a reasonable comparative hypothesis) for as long as we've been able to collect evidence, the whole house of cards has really fallen on Christianity--without having to examine even a single one of its questionable, reprehensible, and bankrupt supposedly divine teachings.

I hope that answers your question. Since I put a lot of effort into this and don't think it belittles you in any way, I intend to publish it on my blog. Just wanted to let you know. Thank you for the question and your patience with my long reply.

1 comment:

  1. James,
    >> "but the evidence of the world has consistently pointed away from any god other than some decreasingly interesting Deist God (current state of affairs: "God" is the underlying field of quantum states and the fluctuation that allowed the Big Bang to occur). This is, very importantly, not the God of Christianity. "

    As you might guess, I would say the evidence doesn't do this. I wouldn't try to overstate my case and say the evidence only points to the God of Christianity. I would say that the evidence is seemingly consistent with the God of Christianity. You disagree, why? What evidence is *necessarily* inconsistent with Christianity - and by that I mean common core Christian theology rather than interdenominational disagreements that I couldn't care less about defending?