Test Martingales, Bayes Factors and p-Values

By Glenn Shafer, A. Shen, N. Vereshchagin, Vladimir G Vovk. 2009

188 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b6b2d4ddfe98305a00d18d1a602711e72089293e

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Citations in the corpus

False discovery rate control with e‐values (2020, methodology): “The fact that the above stopped processes yield valid e- and p-values is by the optional stopping theorem and Ville’s inequality (Shafer et al., 2011; Howard et al., 2020) respectively.”
Insuring against loss of evidence in game-theoretic probability (2010, methodology): “One similar measuretheoretic result, for the case where Sceptic’s strategy is known in advance, is proven in [5] using a simple method based on Lévy’s zero-one law.”
Admissible anytime-valid sequential inference must rely on nonnegative martingales. (2020, background): “Recently, another set of highly interrelated ideas has been put forward under a variety of names by authors such as Shafer, Vovk, Grünwald, and their collaborators: test martingales [20] (since the test statistics are sometimes martingales), e-values or sequential e-values [23, 24] (e for expectatio”
Safe Testing (2019, methodology): “Sequential testing (Lai, 2009; Siegmund, 2013), pioneered by Wald (1947) Related Work on Relating p -values and e -variables Shafer et al. (2011) and Shafer and Vovk (2019) give a general formula for calibrators f (see also Vovk and Wang (2021) and Vovk (1993); Sellke et al. (2001) for early work…”
Time-uniform central limit theory and asymptotic confidence sequences (2021, methodology): “(151) By far the most common method for deriving p-processes is by first deriving a so-called e-process — a stochastic process that is upper-bounded by a nonnegative supermartingale under the null P0 [40, 12, 17].”
E-values: Calibration, combination and applications (2019, unknown): “A simple characterization of the class of all calibrators was ﬁrst obtained in Shafer et al. [2011].”
Estimating means of bounded random variables by betting (2020, methodology): “Nonnegative martingales starting at one have been explicitly considered in some hypothesis testing regimes where they are sometimes referred to as test martingales [21, 22].”