Jeffreys' paradigm of hypothesis testing

Harold Jeffreys based his views on the foundations of statistics on an objective Bayesian account of the theory of probability (see Bayesian interpretation of probability). He agreed with the Neyman-Pearson paradigm that one should always test a null hypothesis $H_0$ against an alternative hypothesis $H_1$ (unlike in Fisher’s paradigm, for instance).

Given such hypotheses and data $X$, Jeffreys wanted to

• Compute the Bayes factor $K={\mathbb{P}}_{H_0}(X)/{\mathbb{P}}_{H_1}(X)$, where each $H_0$ and $H_1$ are given prior probability 1/2 (this is what made Jeffreys an objective Bayesian, as opposed to a subjective Bayesian, who thought the priors should be calculated based on one’s own beliefs).
• Reject $H_0$ if $K\le 1$
• As evidence, report the posterior probabilities $\mathbb{P}(H_0|X)$ and $\mathbb{P}(H_1|X)$. (Recall that in the Bayesian interpretation the hypotheses are considered random objects so $\mathbb{P}(H_0|X)$ is well-defined).