Test martingales, Bayes factors, and p-values

A nonnegative martingale with initial value equal to one measures the evidence against a probabilistic hypothesis. The inverse of its value at some stopping time can be interpreted as a Bayes factor. It can be shown that if we exaggerate the evidence by considering the largest value attained so far by such a martingale, the exaggeration will not be great, and there are systematic ways to eliminate it. The inverse of the exaggerated value at some stopping time can be interpreted as a p-value. We give a simple characterization of all increasing functions that eliminate the exaggeration.