choice of weights is unimportant for the validity of the HMP under the null hypothesis, the weights influence the power of the procedure. Supplementary Methods §5C of and an online tutorial consider the issue in more detail. Bayesian interpretations of the HMP The HMP was conceived by analogy to Bayesian model averaging and can be interpreted as inversely proportional to a model-averaged Bayes factor when combining p-values from likelihood ratio tests. The harmonic mean rule-of-thumb I. J. Good reported an empirical relationship between the Bayes factor and the p-value from a likelihood ratio test. For a null hypothesis nested in a more general alternative hypothesis he observed that often, where denotes the Bayes factor in favour of versus Extrapolating, he proposed a rule of thumb in which the HMP is taken to be inversely proportional to the model-averaged Bayes factor for a collection of tests with common null hypothesis:For Good, his rule-of-thumb supported an interchangeability between Bayesian and classical approaches to hypothesis testing. Bayesian calibration of p-values If the distributions of the p-values under the alternative hypotheses follow Beta distributions with parameters , a form considered by Sellke, Bayarri and Berger, then the inverse proportionality between the model-averaged Bayes factor and the HMP can be formalized as where is the prior probability of alternative hypothesis such that is the expected value of under alternative hypothesis is the weight attributed to p-value incorporates the prior model probabilities and powers into the weights, and normalizes the weights. The approximation works best for well-powered tests (). The harmonic mean p-value as a bound on the Bayes factor For likelihood ratio tests with exactly two degrees of freedom, Wilks' theorem implies that , where is the maximized likelihood ratio in favour of alternative hypothesis and therefore , where is the weighted mean maximized likelihood ratio, using weights Since is an upper bound on the Bayes factor, , then is an upper bound on the model-averaged Bayes factor:While the equivalence holds only for two degrees of freedom, the relationship between and and therefore behaves similarly for other degrees of freedom. Under the assumption that the distributions of the p-values under the alternative hypotheses follow Beta distributions with parameters and that the weights the HMP provides a tighter upper bound on the model-averaged Bayes factor:a result that again reproduces the inverse proportionality of Good's empirical relationship. References Multiple comparisons Statistical hypothesis testing