We define an extension of the posterior predictive $p$-value for multiple test statistics and establish a bound on its frequency under the assumption of model correctness. We argue that the conservativity of the posterior predictive $p$-value increases with model dimension, and we demonstrate the ability of the joint $p$-value to overcome this problem in many cases. We also compare the joint $p$-values to other alternative $p$-values designed to have higher power and show that the joint $p$-value can achieve similar performance for model rejection while maintaining more favorable computational and interpretive properties.

翻译：本文定义了多重检验统计量的后验预测$p$-值的推广形式，并在模型正确性假设下建立了其频率界限。我们论证了后验预测$p$-值的保守性随模型维度增加而增强，并展示了联合$p$-值在许多情况下克服该问题的能力。我们还将联合$p$-值与旨在提升功效的其他备选$p$-值进行了比较，结果表明联合$p$-值在模型拒绝方面可达到相似性能，同时保持更优的计算与解释特性。