Test Martingales, Bayes Factors and p-Values

A nonnegative martingale with initial value equal to one measures evidence against a probabilistic hypothesis. The inverse of its value at some stopping time can be interpreted as a Bayes factor. If we exaggerate the ev- idence by considering the largest value attained so far by such a martingale, the exaggeration will be limited, and there are systematic ways to eliminate it. The inverse of the exaggerated value at some stopping time can be in- terpreted as a p-value. We give a simple characterization of all increasing functions that eliminate the exaggeration.

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