The Probability of a Miracle

It’s the time of year when Christians celebrate the central miracle of their faith: the resurrection of Jesus. Of course, the resurrection has always had its share of skeptics, many of them philosophers. Perhaps the most famous is the 18th century philosopher David Hume, who argued that a miracle is, by definition, a highly improbable event. In fact, it is always more probable that eyewitnesses to an alleged miracle are lying or mistaken than that a law of nature has been violated. Therefore, eyewitness testimony cannot provide us with sufficient evidence to overturn our uniform experience of nature’s regularity. According to Hume and like-minded skeptics, a miraculous explanation is always less probable than even the least probable naturalistic explanation. But Hume lived before Bayes’ Theorem and arguably didn’t have the tools to give a rigorous account of the relevant probabilities. So it might be worth revisiting the question: how probable is a miracle? I don’t think there’s any general answer to this question; the probability will depend on the particular claim. For the sake of convenience, then, let’s stick with the alleged resurrection of Jesus of Nazareth.

First, however, a few preliminary words about skepticism. I’m skeptical of miracle claims and regard skepticism as the default position. There are a lot of miracle claims out there, many of which are unimpressive. I don’t regard the image of Christ on a grilled cheese sandwich as anything other than pareidolia. Also, the many alleged cases of miraculous healing are unlikely to convince those of a skeptical disposition. I’m very happy when someone’s cancer goes into remission, but attributing it to divine agency is a bit hasty because we know that cancer sometimes goes into remission spontaneously or in response to conventional treatment. These stories remind me of an anecdote, possibly apocryphal, about Emile Zola of “J’accuse!” fame. Zola, a renowned atheist, upon witnessing the many crutches abandoned at Lourdes observed that there were no wooden legs. The question “Why doesn’t God heal amputees?” has become an internet meme in the atheist community. For reasons I’ve mentioned elsewhere, I don’t think it’s a knock-down atheistic argument, I understand the sentiment behind it.

So, I’m sympathetic to skepticism, at least as a starting point, with respect to miracle claims. To paraphrase Hume, it would take a great deal of evidence to believe in a miracle. This Humean point has been paraphrased as another free-thought mantra: “extraordinary claims require extraordinary evidence.” As a rhetorical device, it does its job, however, it’s seldom defined with any degree of philosophical rigor. Since I don’t want to get too technical, I’ll simply agree that a resurrection qualifies as an extraordinary claim. But what is meant by ‘extraordinary evidence’? As I mentioned, Hume lived before Bayes’ Theorem, so we can sharpen some of this terminology in light of advances in probability theory. If we think along Bayesian lines, the amount of evidence necessary to render an extraordinary claim probable would be the amount of evidence necessary to raise the probability of the claim above .5 (with 1 being certainty). So ‘extraordinary evidence’ for our purposes is simply evidence sufficient to raise the probability of the claim above .5.

But here’s where things get complicated. Probability only makes sense on the basis of background knowledge. For example, we have to assign each claim a prior probability (henceforth ‘prior’) based on our background knowledge. When skeptics allege that a miracle is, by definition, an improbable event, they usually mean that it has a low prior based on our background knowledge. However, the theist and atheist are going to disagree about our background knowledge and, thus, about the prior. Importantly, background knowledge is not worldview neutral. The theist, who takes God’s existence as part of his background knowledge, is going to have a higher prior for miracles (maybe as high as .5). After all, if God created the entire universe out of nothing, a resurrection isn’t such a big deal. The atheist, however, does not take the existence of God as part of his background knowledge and, therefore, will assign a much lower prior (close to 0). Is there a way around this impasse? Well, for my part, I’m happy to assign a low prior (see below) because resurrections, if they happen at all, happen infrequently. In other words, if there is a God, he’s stingy with resurrections. I would agree that an event like a resurrection, even on theism, has a very low prior (but still not 0).

Of course, we have to consider more than the prior. We also have to consider specific evidence that might raise the posterior probability above .5. In other words, although an event may have a very low prior probability (like a man walking on the moon) there might be specific evidence (eyewitnesses, documentation, film footage, etc.) that raises the overall probability above .5. Therefore, determining the prior is only the first step in determining overall probability, not the last. Incidentally, Hume is often accused of considering only the prior probability, i.e. our background knowledge, in his critique of miracles [Pr (R/B) instead of Pr (R/B&E)*]. But we have another problem here: Christians and atheists are going to disagree about the strength of the specific evidence in the case of the resurrection. Pretty much all of the evidence that we have for the resurrection (empty tomb, postmortem appearances) comes from alleged eyewitness testimony in the four gospels and Paul’s letters. Christians typically think this testimony is trustworthy; skeptics do not. Once again, it seems like someone will have to make a concession.

One skeptic who is willing to play along and assume eyewitness testimony for the sake of argument is Stephen Law. However, he’s unimpressed with the power of eyewitness testimony to overcome very low priors. He uses a hypothetical scenario called the “Ted and Sarah case” to make his point. Suppose two close friends, Ted and Sarah, who are generally reliable, not given to practical jokes, etc. tell you that a man named Bert “flew around their sitting room by flapping his arms, died, came back to life again, and finished by temporarily transforming their sofa into a donkey.” Law concludes that he is not justified in believing that his friends have witnessed a miracle. Although his friends’ testimony provides some evidence, it is by no means sufficient. Law then makes the analogy between the Ted and Sarah case and the gospels explicit:

Of course, we should acknowledge there are differences between the historical evidence for the miracles of Jesus and the evidence provided by Ted and Sarah that miracles were performed in their sitting room. For example, we have only two individuals testifying to Bert’s miracles, whereas we have all four Gospels, plus Paul, testifying to the miracles of Jesus. However, even if we learn that Ted and Sarah were joined by three other witnesses whose testimony is then added to their own, surely that would still not raise the credibility of their collective testimony by very much.

Of course, this all sounds intuitively plausible. Unfortunately, our intuitions regarding probability are often unreliable. What does a Bayesian approach say about the number of witnesses needed to raise the probability of an extraordinary claim?

Another philosopher named Daniel Bonevac has addressed this question. Bonevac disagrees with Law’s assertion that increasing the number of witnesses only negligibly raises the probability. He sets the problem up this way: Let’s assume that the probability of a resurrection is 1 in 10 billion. Let’s further assume that the probability that someone would report a miracle if it occurred is .99. Finally, let’s suppose that the probability that someone would report a resurrection if it did not occur is .1. If we only have one witness, on Bayes’ Theorem, the odds that a resurrection occurred are 1 in a billion. So far, so good for the skeptic. But Bonevac contends that a few more witnesses drastically increase the odds. Given the numbers, it only takes 10 witnesses to bring the probability up to .5 and 12 (apostles?) to make it highly likely (.9888). Using less conservative probability estimates (.999 and .01 in place of .99 and .1) he argues that it only takes 5 witnesses (the gospels and Paul, we might say) to bring the probability of the resurrection up to .5 and 6 witnesses to make it a near certainty. If Bonevac is right, his conclusion clearly has a bearing on Law’s dismissal of eyewitness testimony.

Of course, there are some issues one could raise against Bonevac’s methodology. For example, one might raise the problem of dwindling probabilities. Later in the essay, Bonevac suggests that a series of miracles might be more credible than one miracle in isolation. But a series of miracles only raises our credence in subsequent miracles if we already know earlier miracles occurred. In other words, if we already know that Jesus did in fact turn water into wine, feed the five thousand, and raise Lazarus from the dead, then his own resurrection becomes more probable. But we can’t simply assume all of that. Ironically, in the absence of hard evidence for these earlier miracles, the series of reported miracles may serve to decrease our credence in these reports. In other words, if the probability of miracle #1 is .5, the probability of miracles #1 and #2 is .25, and the probability of miracles #1, #2 and #3 is .125. This is the problem of dwindling probabilities. Of course, one might also question the extent to which the gospels are independent and to what extent, if any, they are eyewitness accounts. Nonetheless, Bonevac’s calculations are enough to warrant caution in accepting the intuition behind Law’s thought experiment. When it comes to probability, our intuitions are usually wrong; we have to do the math.

After all that, I’m afraid my conclusion is going to be rather anticlimactic. Obviously, I don’t think we can prove a miracle or even show that it’s more probable than not. I think even the most charitable assessment of the evidence only puts the probability at .5. Strictly speaking, we’re left with agnosticism. However, I do think that we can at least show that Hume’s easy dismissal is premature. At the very least, we might have to modify our intuitions about the value of eyewitness testimony in assessing extraordinary claims. I also think this case is a good reminder that reason is not neutral with respect to one’s worldview, which will largely determine the priors one assigns. In other words, believers are not necessarily irrational in affirming the resurrection. Skeptics often rely upon the prima facie incredulity of such a claim, but, as we’ve seen, it’s quite a bit more complicated. On the basis of the prior alone, the skeptic is quite right: the odds are infinitesimal. However, if one accepts specific evidence (testimony, empty tomb, postmortem appearances), then the belief that something extraordinary happened is not flatly irrational. Naturally, the skeptic will reject this evidence and we may be left with a stalemate. Nevertheless, if a believer can show that he’s not irrational in his belief, that’s no small accomplishment.

*William Lane Craig used the following Bayesian formulation in a debate with Bart Ehrman.

B = Background Knowledge

E = Specific Evidence (testimony, empty tomb, postmortem appearances)

R = Resurrection

Pr (R/B&E)= [Pr (R/B) × Pr (E/B&R)]/ [Pr (R/B) × Pr (E/B&R) + Pr (not-R/B) × Pr (E/B& not-R)]

Is the Resurrection the Best Evidenced Miracle Claim?

Christian apologists often claim that the alleged resurrection of Jesus of Nazareth is the best evidenced miracle claim. It’s sometimes unclear to me what they mean by this. Consider the following two claims:

1. The resurrection of Jesus is the best evidenced miracle claim.

2. The resurrection of Jesus is the best evidenced miracle claim on historical grounds alone.

Which of these claims are apologists making? If the first, I’m not convinced it’s true. The evidence for the resurrection that we have is basically documentary and testimonial, at best. We don’t have any physical traces of the resurrection.* In the case of other miracle claims, such as some of the Lourdes cases, we have evidence like medical records, X-rays, photographs, etc. These were not available in the ancient world. Hence, it seems to be the case that modern miracle claims are better evidenced than the resurrection. So I’m not confident that 1 is true.

What about about the more qualified second statement? Is the resurrection the best evidenced miracle on historical grounds alone? I’m not sure 2 is true either. Again, the type of evidence that we have is basically documentary and testimonial. We have documents, the gospels and some of Paul’s epistles, which give testimony to the resurrection. It’s debatable to what extent the gospels provide eyewitness testimony (of the four canonical gospels, only two even claim to be written by eyewitnesses) but let’s assume for the sake of argument that they do. We don’t have expert testimony in the case of the gospels like we do in the case of some modern miracles claims (doctor’s testimony, for example). We also don’t have any of the original documents in the case of the gospels, but even if we did, the gospels are a genre of ancient literature. Even if one assumes that the gospels are examples of ancient biographies, they’re not disinterested biographies in the modern sense. The gospel writers are evangelists, after all. One could argue that it’s unfair to hold the gospels to modern standards. But I’m not asking whether the gospels provide documentation comparable to other ancient sources. Rather, the question is whether the resurrection is the best evidenced miracle claim on historical grounds.

I’m not convinced that it is. Compare, for example, two miracle claims I mentioned in my last post, the alleged healing of Mr. Savini by Padre Pio in 1949, and the alleged Miracle of Calanda in 1640. In both cases, we have expert witness testimony and medical documentation. In the latter case, we also have extant original documents that are signed and notarized. We simply don’t have such evidence in the case of the resurrection. To say that such evidence wasn’t available in the ancient world is beside the point.** Again, apologists don’t claim that the resurrection is the best evidenced miracle claim of the ancient world. At least that doesn’t seem to be their position and if it is they should be more clear about it. So I’m not convinced that 2 is true either. Where does this leave the apologist’s claim? Note that nothing I’ve said is incompatible with the resurrection having happened. But in that case, one has to ask: if the resurrection happened, and it’s Christianity’s central miracle, why wouldn’t it be the best evidenced miracle?

* Of course, one may count the Shroud of Turin as authentic or think that the Church of the Holy Sepulcher is the actual (vacant) burial place of Jesus; however, most Protestant apologists don’t argue in favor of either.

** While it’s true that modern medical evidence wasn’t available, I’m not so sure about signed and notarized documents being uncommon in the ancient world. The Romans were judicious record-keepers.

A Rigorous Formulation of “Extraordinary Claims Require Extraordinary Evidence”

I came across this rigorous formulation of the free thought mantra ‘Extraordinary claims require extraordinary evidence.’ I’ve always taken the phrase to mean something like this, but I’ve never worked out those intuitions with any precision. Thanks to Mr McIntosh — a Christian theist no less– for doing what the free thought community should have done but hasn’t: offer a formalized statement of their well-worn slogan.

To summarize, an ‘extraordinary claim’ is defined as ‘a claim with a very low intrinsic probability.’ Another way of saying this is that the prior probability of the claim being true based on our background knowledge is very low (but presumably not zero). So the evidence for this claim must be strong enough to bring the posterior probability of the claim up to .5 or better. This is indeed the strategy adopted by people like Richard Swinburne and Lydia and Timothy McGrew in their Bayesian arguments for the Resurrection of Jesus. They argue that even if the initial probability is low, this antecedent improbability can be counterbalanced by the specific evidence in this case. When one considers this evidence, the probability of the claim’s being true goes up well over .5 (I think both Swinburne and the McGrew put it at over .9).

If this is what the mantra basically means, then it is relatively uncontroversial. Theists, at least in the cases I mentioned, believe that they can meet their burden of proof here; non-theists will disagree. The non-theists will simply deny that the amount or kind of specific evidence in this case is sufficient to overcome the very low prior probability. But here is where the issue gets a bit complicated. What evidence could overcome this low intrinsic probability? As Mr McIntosh asks, ‘Would it be a lot of evidence that cumulatively confirms c (ignoring the problem of dwindling probabilities), or just one piece of evidence that highly confirms c, or possibly a combination?’ The answer here is not clear.

Another problem: theists and non-theists are unlikely to agree on what constitutes ‘background knowledge.’ Some theists say that if God exists, then we have to start with higher priors than we otherwise would. If the existence of God is one of our background beliefs, so it goes, the prior probability of the claim goes up. The claim would no longer be so extraordinary, nor would it need extraordinary evidence. Perhaps good evidence — testimonial evidence of the kind we would accept in mundane matters — would suffice. This seems to be the way William Lane Craig argues. I’ve heard him say in the context of debates that the free thought mantra is demonstrably false. I think he takes it to mean, following Hume, that the only relevant factor to consider when assessing a miracle claim is the prior intrinsic probability. However, we also have to consider our background beliefs and the posterior probability on the evidence that we have. Obviously we do have to consider those factors as McIntosh’s formulation notes.

So let’s take the issue of prior probability on our background beliefs. The legitimacy of granting a claim, like the Resurrection, a higher antecedent probability depends on the theist’s dialectical strategy. In Craig’s case, he argues for the Resurrection only after he believes he’s made a good case for the existence of God from natural theology. He then thinks it’s not overwhelmingly improbable that God would raise Jesus from the dead. Swinburne seems to argue similarly. The McGrews, however, are arguing for the existence of God from this particular miracle claim. As such, they can’t help themselves to higher priors. They assume low priors, but contend that the evidence brings the posterior probability up. They also argue that we can use this evidence to retroactively update our priors. The non-theist will remain unimpressed by this, I think. Obviously theists and non-theists in this debate disagree about the success of natural theology and the quality and/or amount of evidence we have in this particular case. There is also the problem of dwindling probabilities that Plantinga points out. Even if we assume for the sake of argument that it’s very probable, say .9, that God exists, that probability dwindles the more claims we add. Thus, the probability that God exists might be high, but the probability that he’d become incarnate is necessarily lower and the probability that he would die and rise again lower still, until we’re below .5. Both Swinburne and the McGrews have responded to Plantinga. Because of the technical nature of these replies, I can’t comment in detail. In any case, it’s a bit of a moot point. The non-theist is not going to grant the claim ‘God exists’ a probability exceeding .5. So agreeing on the background beliefs seems like a non-starter.

What about the quality or quantity of evidence in the case of the Resurrection? Again, the theist and non-theist are going to disagree about it’s strength. The theist will say that it succeeds in overcoming the antecedent improbability — and is thus ‘extraordinary’ — while the non-theist is going to deny this. So it seems we’re stuck with a stalemate. We’re left with the question of the merit of the slogan ‘extraordinary claims require extraordinary evidence.’ Provided the terms are rigorously defined, the slogan is admissible. However, I still doubt that it will be dialectically useful. It seems to me that we’re unlikely to reach consensus on what beliefs should form our background knowledge and, therefore, how to assign the priors. However, it seems to me that we have to assign low priors, even if we’re willing to concede for the sake of argument that God exists. After all, God, if he exists, is stingy with resurrections, as I think even the theist will have to admit. Therefore, on their alleged frequency alone, we should assign a low prior probability. The theist and non-theist will also disagree about whether the evidence is ‘extraordinary’ enough to overcome the low antecedent probability and raise the posterior probability above .5.

Nonetheless, a more rigorous formulation of this mantra is valuable. It helps us avoid semantic dead ends and focus on the argument in more detail. It also suggests that theists (at least evidentialist ones) and non-theists might be able to agree in principle that ‘extraordinary claims require extraordinary evidence’ even while disagreeing in practice when it comes to a particular claim like the Resurrection. However, such agreement on methodology is no small matter. Indeed, it might be the best we can hope for in theist/non-theist debate.