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Posterior predictive p-values. (English) Zbl 0820.62027

Summary: Extending work of D. B. Rubin [ibid. 12, 1151-1172 (1984; Zbl 0555.62010)] this paper explores a Bayesian counterpart of the classical p-value, namely, a tail-area probability of a “test statistic” under a null hypothesis. The Bayesian formulation, using posterior predictive replications of the data, allows a “test statistic” to depend on both data and unknown (nuisance) parameters and thus permits a direct measure of the discrepancy between sample and population quantities. The tail- area probability for a “test statistic” is then found under the joint posterior distribution of replicate data and the (nuisance) parameters, both conditional on the null hypothesis. This posterior predictive p- value can also be viewed as the posterior mean of a classical p-value, averaging over the posterior distribution of (nuisance) parameters under the null hypothesis, and thus it provides one general method for dealing with nuisance parameters.

Two classical examples, including the Behrens-Fisher problem, are used to illustrate the posterior predictive p-value and some of its interesting properties, which also reveal a new (Bayesian) interpretation for some classical p-values. An application to multiple-imputation inference is also presented. A frequency evaluation shows that, in general, if the replication is defined by new (nuisance) parameters and new data, then the Type I frequentist error of an α-level posterior predictive test is often close to but less than α and will never exceed 2α.

MSC:
62F15Bayesian inference
62F03Parametric hypothesis testing
62A01Foundations and philosophical topics in statistics