It's Bayes's rule, not Bayes' rule
This follows from Rule No. 1 in The Elements of Style by Strunk and White, still upheld as the standard (cf. "Apostrophe---Punctuation Guide" or even The New York Times). Yet in the economics literature, "Bayes's rule" is an endangered species, crowded out by "Bayes' rule" and "Bayes' formula." I am inclined to blame that on the publishers that have cornered the economics journals market. Their monopolistic advantage is unlikely to give them much incentive to retain good copy editors. Recently it took me quite a while to revert all the "Bayes' rule" changes that their copy editor made on my draft.