The Annals of Statistics

Posterior Predictive $p$-Values

Xiao-Li Meng

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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 $2\alpha$.

Article information

Ann. Statist. Volume 22, Number 3 (1994), 1142-1160.

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


Primary: 62F03: Hypothesis testing
Secondary: 62A99: None of the above, but in this section

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


Meng, Xiao-Li. Posterior Predictive $p$-Values. Ann. Statist. 22 (1994), no. 3, 1142--1160. doi:10.1214/aos/1176325622.

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