Test Martingales, Bayes Factors and $p$-Values

Details Test Martingales, Bayes Factors and $p$-Values Test Martingales, Bayes Factors and $p$-Values

2009-12-21

arxiv.org/abs/0912.4269v3

Statistical Science 2011, Vol. 26, No. 1, 84-101

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/10-STS347 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M

Scientific paper

10.1214/10-STS347

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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