Is the Bayes Factor a Misnomer?
I often see the Bayes Factor (BF) defined as the ratio of two quantities: the probability of the data given one statistical hypotheses, and the probability of the data given the alternative hypothesis. Thus, BF = P(data | H) / P(data | H). But why is this Bayesian? Wouldn't a true Bayesian want to know the ratio of the posterior probabilities--i.e., the ratio of the probability of H given the data, and the probability of H given the data--such that The_Real_BF = P(H) | data) / P(H | data). After all, isn't Bayes' theorem's principle accomplishment to provide a way to derive posterior probabilities of hypotheses?
-- Richard Anderson