Bayes’ Theorem and Extraordinary Claims

I came across the following articles which I believe make for an interesting juxtaposition:

1) Evidence, Miracles and the Existence of Jesus


2) The Argument from Miracles

The first is by Stephen Law and suggests that there’s good reason to at least remain skeptical about the existence of Jesus. I’m less interested in this thesis than in a thought experiment that Law introduces in the course of his argument. If you skip down to the subheading “The Ted and Sarah Case”, Law relates a hypothetical scenario in which Ted and Sarah, two close friends 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 explicit the analogy between the Ted and Sarah case and the gospels:

“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.”

Law also adds “the fact that it remains blankly mysterious why such reports would be made if they were not true does not provide us with very much additional reason to suppose that they are true.” He then derives a variation of the principle “extraordinary claims require extraordinary evidence.” I am on record as having expressed dissatisfaction with the way this principle is formulated. Law, for the purposes of this essay at least, is content to leave it vague.

That brings us to the second essay. This one is by Daniel Bonevac and is an argument for the rationality of belief in miracles. Following in the tradition of others, such as Swinburne, Bonevac takes a Bayesian approach to miracles. He argues that miracle claims are credible in Bayesian terms if certain conditions are met. Specifically, the number of witnesses matters. Take the following set-up. 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 miracle claim if it did not occur is .1. If we only have one witness, on Bayes’ theorem, the odds that a resurrection occurred are one in a billion. The skeptic appears to be justified. But Bonevac contends that it doesn’t take many more witnesses to drastically increase the odds. Given the numbers, it only takes 10 witnesses to bring the probability up to .5 and twelve to make it highly likely (.9888). Using slightly 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 six witnesses to make it a near certainty. The application to Law’s hypothetical scenario are clear. It’s also much less clear that ‘extraordinary claims require extraordinary evidence.’ According to Bayes’ theorem, Bonevac argues that ordinary evidence will suffice.

Now there are some issues one could raise against Bonevac’s methodology. For example, one might quibble with the priors he assigns. As Law says, the fact that we don’t know why people report a miracle when none has occurred doesn’t necessarily raise our credence in the claim as Bonevac seems to assume. Also, 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. Of course, a series of miracles raising our credence in subsequent miracles only works 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 it would be question begging to assume all of that. In the absence of iron-clad evidence for these earlier miracles, the series of miracles reported may serve to decrease our credence in these reports. For example, if the probability of miracle #1 is .5, the probability of miracle #1 and #2 is .25. Furthermore, the probability of miracles 1,2 and 3 is (roughly) .13. This is the problem of dwindling probabilities. I think this is partly what Law is trying to articulate when he says that the admixture of ‘extraordinary claims’ into the gospel narratives decreases our overall credence in those accounts. Finally, one might also question the extent to which the gospels are independent and to what extent, if any, the gospels are eyewitness accounts. Nonetheless, Bonevac’s calculations are enough to warrant caution in accepting the intuition behind Law’s thought experiment and the principle he derives from it. Again, we need a clearer understanding of the maxim ‘extraordinary claims require extraordinary evidence.’



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4 responses to “Bayes’ Theorem and Extraordinary Claims

  1. This is what I called a Beautiful Mind. Thank you for wonderful thoughts. Just to add on ‘extraordinary claims require extraordinary evidence.’ I think the term extraordinary is unclear and could be subjective/relative from one person to another.

    For a Naturalist, something that is outside the natural box is indeed ‘extraordinary’ but to a non-Naturalist, it simply ordinary. Thus I think this also depend on the worldview one holds. Thus what is extraordinary to a Naturalist could be simply be ordinary to a non-Naturalist. The problem is Naturalists assume that their position is a default thus evidence given should also be extraordinary also to a non-Naturalist.

    Another angle, even if true that Naturalism is a default position, I do not think ‘extraordinary claims require extraordinary evidence.’ If person A gives an extraordinary claim that she won 10 times in a go world wide lottery, does she need an extraordinary evidence? I do think so. I believe she simply need to show the ordinary ten receipts she won her lottery with.

    Thank you Dude Ex Machina.

    • Hi Prayson,

      Thanks for the comment. I agree that ‘extraordinary’ is usually unclear and subjective and this often results in both sides talking past each other. I’ve been trying to nail down a clearer, more rigorous formulation of the slogan ‘extraordinary claims require extraordinary evidence.’ I also find that many who use this phrase in the free-thought community aren’t aware of Bayes’ Theorem. As a result, they are basically relying on intuitions and often make mistakes. I’m trying in a small way to remedy that.

      I also agree that worldview differences play a part in assigning probabilities to something like a resurrection. The naturalist and the theist are going to have different background assumptions which means, in Bayesian terms, that they will assign different antecedent probabilities to the resurrection. I try to make as few assumptions as possible when assigning the priors. Basically, I assign a pretty low antecedent probabilities because resurrections — if they occur at all — don’t occur frequently. So I think Bonevac’s estimate of 1 in 10 billion (estimated on the basis of the entire population of the world throughout history and the frequency of resurrection reports) is probably fair.

      However, this is where most naturalists will stop. They’ll say something like “a miracle by definition is the least probable explanation. The odds against it are too high.” But, of course, we have to consider more than the antecedent probabilities — we also have to consider the specific evidence that may raise the overall probability over .5. We have to ask, “what are the odds that we’d have the evidence that we do if x didn’t occur?” Again, as Bonevac says, it doesn’t really take many witnesses to raise the probability significantly. This isn’t necessarily extraordinary evidence, unless by ‘extraordinary’ you mean ‘able to raise the overall probability above .5’. I think Law relies too much on our intuitions here and doesn’t do the math. When he says that the addition of 3 more witnesses in the Ted and Sarah case wouldn’t significantly raise the credibility of the testimony, he’s appealing to our intuitions which are simply wrong.

      Anyways, this is a long-winded comment so I’ll just close by saying that I’m not convinced by the resurrection or other miracle claims. But the problem is not that they lack ‘extraordinary’ evidence. Rather, I think I disagree with theists about what putative ‘facts’ we can allow as evidence and then plug into Bayes’ Theorem. But ironically I often find myself agreeing with theists, against naturalists, in saying that one cannot rule out miracles simply on the basis of low antecedent probability. Determining the antecedent probability is only the first step in assessing the overall probability, not the last.

      Thanks again for your comment.

      • Wow, your reasoning blows my mind. If I may add, probability of x is relative to background information y. I think, the background informations surrounding resurrection makes its increases the probability that indeed the first claim, namely God rised Jesus from the dead more probable.


        Calculating the Probability of the Resurrection Pr (R/B&E):
        B = Background knowledge E = Specific evidence (empty tomb, postmortem appearances, etc.) R = Resurrection of Jesus. Pr (R/B) = the intrinsic probability of the resurrection. Pr (E/B&R) = the explanatory power.

        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)]

        From this I believe one can see that given background information and evidences surround resurrection, then its more probable God rose Jesus from the dead.

        Many scholars, including Hume and Erham, mistakes Pr (R/ B & E) with Pr (R/B). Hume can be excused since this field of probability was not in his time, but I wonder how Erham still fail to see his mistake.

        Thank you once again.


  2. “Wow, your reasoning blows my mind.”

    I’ll take that as a compliment! 😉

    “Many scholars, including Hume and Erham, mistakes Pr (R/ B & E) with Pr (R/B).”

    Yes, that’s what I was getting at in my reply when I said that most naturalists only consider the antecedent probability or Pr (R/B). But you also have to consider the specific evidence as you say. Ehrman is one of the worst offenders on this point and refuses to be corrected on it. Perhaps this is because he lacks the philosophical background to grasp the point.

    I don’t disagree with the Christian theist on Bayesian grounds. I disagree on empirical/historical grounds. As I say in my reply, I disagree about the putative facts or what data we can allow as specific evidence.

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