Test Martingales, Bayes Factors and $p$-Values

Glenn Shafer; Alexander Shen; Nikolai Vereshchagin; Vladimir Vovk

doi:10.1214/10-sts347

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arXiv · 0912.4269

Test Martingales, Bayes Factors and $p$-Values

Glenn Shafer·Alexander Shen·Nikolai Vereshchagin·Vladimir Vovk

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Abstract

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 evidence 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 interpreted as a $p$-value. We give a simple characterization of all increasing functions that eliminate the exaggeration.

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Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, Vladimir Vovk. 2009-12-21. Test Martingales, Bayes Factors and $p$-Values. https://doi.org/10.1214/10-sts347

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