Game-theoretic statistics and safe anytime-valid inference

Aaditya Ramdas, Peter Grünwald, Vladimir Vovk, Glenn Shafer

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Safe anytime-valid inference (SAVI) provides measures of statistical evidence and certainty—e-processes for testing and confidence sequences for estimation—that remain valid at all stopping times, accommodating continuous monitoring and analysis of accumulating data and optional stopping or continuation for any reason. These measures crucially rely on test martingales, which are nonnegative martingales starting at one. Since a test martingale is the wealth process of a player in a betting game, SAVI centrally employs game-theoretic intuition, language and mathematics. We summarize the SAVI goals and philosophy, and report recent advances in testing composite hypotheses and estimating functionals in nonparametric settings.
Original languageEnglish
Pages (from-to)576 -601
Number of pages26
JournalStatistical Science
Issue number4
Publication statusPublished - 6 Nov 2023


  • Test martingales
  • Ville’s inequality
  • universal inference
  • reverse information projection
  • e-process
  • optional stopping
  • confidence sequence
  • nonparametric composite hypothesis testing

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