Title: Posterior Consistency in Parametric Models via a Tighter Notion of Identifiability Show Chinese title

Title: 后验一致性在参数模型中通过一种更严格的可识别性概念

Abstract: We study Bayesian posterior consistency in parametric density models with proper priors, challenging the perception that the problem is settled. Classical results established consistency via MLE convergence under regularity and identifiability assumptions, with the latter taken for granted and rarely examined. We refocus attention on identifiability, showing that inconsistency arises only when the true distribution coincides with a weak limit of model densities in a way that violates identifiability. While such failures occur naturally in nonparametric settings, they are implausible and effectively self-inflicted in parametric models. Our analysis shows that classical regularity conditions are unnecessary: a mild strengthening of identifiability suffices to ensure consistency in parametric models, even when the MLE is inconsistent. We also demonstrate that parametric inconsistency requires carefully engineered, oscillatory model features aligned with the true distribution, which is unlikely to occur without adversarial design. Our findings also clarify the distinct mechanisms behind Bayesian and frequentist inconsistency and advocate for separate theoretical treatments. Show Chinese abstract

Abstract: 我们研究了在具有适当先验的参数密度模型中的贝叶斯后验一致性问题，挑战了这一问题已经解决的观点。经典结果通过正则性和可识别性假设下的最大似然估计（MLE）收敛证明了一致性，其中后者被视为理所当然且很少被检验。我们将注意力重新集中在可识别性上，表明只有当真实分布与模型密度的弱极限一致，并且这种一致性违反了可识别性时，才会出现不一致性。虽然在非参数设置中此类失败会自然发生，在参数模型中它们是不合理的且基本上是自找的。我们的分析表明，经典的正则条件是不必要的：只需稍微加强可识别性即可确保参数模型中的一致性，即使MLE是一致的。我们还证明了参数模型中的不一致性需要精心设计的、与真实分布一致的振荡模型特征，这在没有对抗性设计的情况下不太可能发生。我们的发现还澄清了贝叶斯和频率学派不一致背后的不同机制，并主张对两者进行独立的理论处理。