Open Access
September, 1994 Posterior Predictive $p$-Values
Xiao-Li Meng
Ann. Statist. 22(3): 1142-1160 (September, 1994). DOI: 10.1214/aos/1176325622


Extending work of Rubin, 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 $\alpha$-level posterior predictive test is often close to but less than $\alpha$ and will never exceed $2\alpha$.


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Xiao-Li Meng. "Posterior Predictive $p$-Values." Ann. Statist. 22 (3) 1142 - 1160, September, 1994.


Published: September, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0820.62027
MathSciNet: MR1311969
Digital Object Identifier: 10.1214/aos/1176325622

Primary: 62F03
Secondary: 62A99

Keywords: $p$-value , Bayesian $p$-value , Behrens-Fisher problem , Discrepancy , multiple imputation , nuisance parameter , pivot , significance level , tail-area probability , test variable , Type I error

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 3 • September, 1994
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