Martingale testing in game-theoretic probability
Glenn Shafer
Rutgers Business School - Newark and New Brunswick
New Jersey, USA,

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Statistical testing can be framed as a repetitive game between two players, Forecaster and Skeptic. On each round, Forecaster sets prices for various gambles, and Skeptic chooses which gambles to make. If Skeptic multiplies by a large factor the capital he puts at risk, he has evidence against Forecaster's ability. His capital at the end of each round is a measure of his evidence against Forecaster so far. This can go up and then back down. If you report the maximum so far instead of the current value, you are exaggerating the evidence against Forecaster. The exaggeration corresponds to using a p-value rather than a Bayes factor for testing. There are simple ways to correct the exaggeration, which also apply to p-values.


Test martingales, Bayes factors, and p-values (with Alexander Shen, Nikolai Vereshchagin, and Vladimir Vovk). Statistical Science 26:84-101, 2011.

Insuring against loss of evidence in game-theoretic probability (with A. Philip Dawid, Steven de Rooij, Alexander Shen, Nikolai Vereshchagin, and Vladimir Vovk). Statistics and Probability Letters 81:157-162, 2011.

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