We introduce a joint posterior $p$-value, an extension of the posterior predictive $p$-value for multiple test statistics, designed to address limitations of existing Bayesian $p$-values in the setting of continuous model expansion. In particular, we show that the posterior predictive $p$-value, as well as its sampled variant, become more conservative as the parameter dimension grows, and we demonstrate the ability of the joint $p$-value to overcome this problem in cases where we can select test statistics that are negatively associated under the posterior. We validate these conclusions with a pair of simulation examples in which the joint $p$-value achieves substantial gains to power with only a modest increase in computational cost.

翻译：我们提出了一种联合后验$p$值，它是针对多个检验统计量的后验预测$p$值的扩展，旨在解决现有贝叶斯$p$值在连续模型扩展设定中的局限性。具体而言，我们证明后验预测$p$值及其抽样变体随着参数维度的增加而变得更加保守，并展示了联合$p$值在能够选择后验下负相关的检验统计量时克服这一问题的能力。我们通过两个模拟实例验证了这些结论，其中联合$p$值在仅适度增加计算成本的情况下实现了显著的统计功效提升。