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Zbl 0974.62021
Bayarri, M.J.; Berger, James O.
Quantifying surprise in the data and model verification. (With discussion). (English)
[A] Bernardo, J. M. (ed.) et al., Bayesian statistics 6. Proceedings of the 6th Valencia international meeting, Alcoceber near Valencia, Spain, June 6-10, 1998. Oxford: Clarendon Press. 53-82 (1999). ISBN 0-19-850485-3/hbk

Summary: $P$-values are often perceived as measurements of the degree of surprise in the data, relative to a hypothesized model. They are also commonly used in model (or hypothesis) verification, i.e., to provide a basis for rejection of a model or hypothesis. We first make a distinction between these two goals: quantifying surprise can be important in deciding whether or not to search for alternative models, but is questionable as the basis for rejection of a model. For measuring surprise, we propose a simple calibration of the $p$-value which roughly converts a tail area into a Bayes factor or `odds' measure. Many Bayesians have suggested certain modifications of $p$-values for use in measuring surprise, including the predictive $p$-value and the posterior predictive $p$-value. We propose two alternatives, the conditional predictive $p$-value and the partial posterior predictive $p$-value, which we argue to be more acceptable from Bayesisn (or conditional) reasoning.

MSC 2000:: *62F15 Bayesian inference

Keywords: Bayes factors; Bayesian p-values; Bayesian robustness; conditioning; model checking; predictive distributions

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