Bayesian-style belief updating involves changing probabilistic beliefs in response to evidence. Bayes’ rule is the theorem that indicates how much we should change our minds when we learn a new fact or observe new evidence.
Bayes’ rule also tells us how strong a piece of evidence has to be in order to support a given hypothesis. Extraordinary claims require extraordinary evidence. The theorem lets us see whether a piece of evidence is sufficient, or insufficient, to drive the probability of a hypothesis to over 50%. That is: an ”extraordinary claim” is one with a low prior probability in advance of considering the evidence, and “extraordinary evidence” is evidence with an extreme likelihood ratio favoring the claim over its alternatives.
A corollary to extraordinary claims require extraordinary evidence is “ordinary claims require only ordinary evidence”.