The Rule of Three, its Variants and Extensions

The Rule of Three (R3) states that 3/ n  is an approximate 95% upper limit for the binomial parameter, when there are no events in  n  trials. This rule is based on the one-sided Clopper–Pearson exact limit, but it is shown that none of the other popular frequentist methods lead to it. It can be seen as a special case of a Bayesian R3, but it is shown that among common choices for a non-informative prior, only the Bayes–Laplace and Zellner priors conform with it. R3 has also incorrectly been extended to 3 being a "reasonable" upper limit for the number of events in a future experiment of the same (large) size, when, instead, it applies to the binomial mean. In Bayesian estimation, such a limit should follow from the posterior predictive distribution. This method seems to give more natural results than—though when based on the Bayes–Laplace prior technically converges with—the method of prediction limits, which indicates between 87.5% and 93.75% confidence for this extended R3. These results shed light on R3 in general, suggest an extended Rule of Four for a number of events, provide a unique comparison of Bayesian and frequentist limits, and support the choice of the Bayes–Laplace prior among non-informative contenders.