The abandonment of the concept of prior probability by the frequentists had an important theoretical consequence. One could no longer talk about the probability of a hypothesis given particular data, the posterior probability (also called the "inverse probability"), since there was no longer a prior probability available from which it could be calculated. Instead, the frequency interpretation forces one to deal with the probability of the data given a particular hypothesis, that is, the likelihood (also called the "direct probability"). In formal terms, the Bayesians focused on p(H\D), the inverse probability, whereas the frequentists focused on p(D\H), the direct probability, because the latter asserted that in reality one could seldom derive the prior probability of a hypothesis from data.

Sir Ronald A. Fisher Influenced by the criticism of the early frequentists, Fisher rejected prior probabilities and refused to commit himself to a definition of probability that was subjective. There was, however, an essential tension in Fisher's writings. On one hand, as a frequentist, he held a lifelong opposition to the very idea of inverse probabilities, that is, to probabilities of hypotheses given data. On the other hand, the notion of inverse probability crept into various parts of his writings, as he himself was willing to admit (Aeree, 1978, pp. 119-123). And it was just this element that made his theory attractive to psychologists.