Test martingales, bayes factors and p-values

Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, Vladimir Vovk

Research output: Contribution to journalArticlepeer-review

50 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)84-101
Number of pages18
JournalStatistical Science
Issue number1
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


  • Bayes factors
  • Evidence
  • Hypothesis testing
  • Martingales
  • p-values


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