The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

As we saw, Bayes’s rule is formally an elementary consequence of his definition of conditional probability. But epistemologically, it is far from elementary. It acts, in fact, as a normative rule for updating beliefs in response to evidence. In other words, we should view Bayes’s rule not just as a convenient definition of the new concept of “conditional probability” but as an empirical claim to faithfully represent the English expression “given that I know.” It asserts, among other things, that the belief a person attributes to S after discovering T is never lower than the degree of belief that person attributes to S AND T before discovering T. Also, it implies that the more surprising the evidence T—that is, the smaller P(T) is—the more convinced one should become of its cause S. No wonder Bayes and his friend Price, as Episcopal ministers, saw this as an effective rejoinder to Hume. If T is a miracle (“Christ rose from the dead”), and S is a closely related hypothesis (“Christ is the son of God”), our degree of belief in S is very dramatically increased if we know for a fact that T is true. The more miraculous the miracle, the more credible the hypothesis that explains its occurrence. This explains why the writers of the New Testament were so impressed by their eyewitness evidence.