标题： 后验预测$p$-值在修正的 Kolmogorov-Smirnov 检验下的渐近良好校准 显示英文标题

标题： Asymptotic well-calibration of the posterior predictive $p$-value under the modified Kolmogorov-Smirnov test

摘要： 后验预测的 $p$-值是贝叶斯模型检验中广泛使用的一种工具。然而，在大多数检验统计量下，其渐近零分布比均匀分布更集中在 1/2 附近。因此，它的有限样本行为难以解释，并且往往缺乏检验功效，这是实践中众所周知的问题。一种常见的检验统计量选择是带嵌入估计器的 Kolmogorov-Smirnov 检验。它为实值观测提供了一个全局的模型-数据偏差度量，并对模型误设敏感。在这项工作中，我们证明了在这种检验统计量下，后验预测 $p$-值在零假设下依分布收敛到均匀分布。我们进一步通过数值实验表明，该 $p$-值在有限样本中表现良好，并能有效检测广泛的替代模型。 显示英文摘要

摘要： The posterior predictive $p$-value is a widely used tool for Bayesian model checking. However, under most test statistics, its asymptotic null distribution is more concentrated around 1/2 than uniform. Consequently, its finite-sample behavior is difficult to interpret and tends to lack power, which is a well-known issue among practitioners. A common choice of test statistic is the Kolmogorov-Smirnov test with plug-in estimators. It provides a global measure of model-data discrepancy for real-valued observations and is sensitive to model misspecification. In this work, we establish that under this test statistic, the posterior predictive $p$-value converges in distribution to uniform under the null. We further use numerical experiments to demonstrate that this $p$-value is well-behaved in finite samples and can effectively detect a wide range of alternative models.