A mistake in probability theory in David Hume’s “Of Miracles”

When should a rational individual believe in a miracle?

David Hume, the great skeptical philosopher, answered: practically never. His argument ran as follows: Miracles are extremely rare events and thus have a very low prior probability. On the other hand,  people can be misled rather easily either by their own senses or by other people. Therefore, the rational reaction to hearing a miracle story is to reject it, except the evidence supporting it is overwhelming. “Extraordinary events require extraordinary evidence” became a popular summary of Hume’s point of view.

Here is a famous passage from Hume’s “Of Miracles” explaining the point:

When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle.

This argument sounds intuitively plausible and compelling, but it is mistaken. In fact Hume is committing an elementary error in probability theory, which shouldn’t be held against him since “Of Miracles” predates the writings of Bayes and Laplace.

In the language of modern probability theory, Hume asks us as to compare the prior probability that miracle X occurred,  \displaystyle Pr(X) , to the probability of seeing the evidence Y supporting miracle X even though X did not in fact occur, i.e. the conditional probability of Y given the negation of X,  \displaystyle Pr(Y | \neg X). Econometricians would call the latter the likelihood of Y under the hypothesis not-X. If  \displaystyle Pr(X) < Pr(Y | \neg X),  Hume says we should reject X in favor of not-X.

But this inference is unwarranted. What a rational observer ought to ask is: Given the evidence Y, is it more likely that X occurred or that it didn’t occur? We are looking for the posterior odds of X conditional on Y:  \displaystyle \frac{ Pr(X | Y)} { Pr(\neg X | Y) }. 

Bayes’ theorem immediately gives us what we are looking for:  \displaystyle  \frac{ Pr(X | Y)} { Pr(\neg X | Y) } = \frac{ Pr(Y | X) }{Pr(Y | \neg X) } \frac{ Pr(X) }{Pr(\neg X)} 

This equation makes it clear that even if Hume’s inequality  \displaystyle Pr(X) < Pr(Y | \neg X), holds, it is possible that the posterior odds of X are greater than 1. All we need for such as result is that the likelihood of having evidence Y under the hypothesis that X occurred is sufficiently higher than the likelihood of Y under the alternative hypothesis that X did not occur. In econometric terms, the likelihood ratio must exceed a critical value which depends on the prior odds against the miracle:  \displaystyle \frac{ Pr(Y | X) }{ Pr(Y | \neg X) } > \frac{ Pr(\neg X) }{ Pr(X) }. 

To conclude: A rational observer is justified in believing a miracle if the evidence for it is sufficiently more likely under the hypothesis that the miracle really did occur than under the hypothesis that it didn’t so as to offset the low prior odds for the miracle. Just comparing the low prior probability of a miracle to the probability of receiving false evidence in favor of it is not enough and can be misleading.