The consistency of posterior distributions in density estimation is at the core of Bayesian statistical theory. Classical work established sufficient conditions, typically combining KL support with complexity bounds on sieves of high prior mass, to guarantee consistency with respect to the Hellinger distance. Yet no systematic theory explains a widely held belief: under KL support, Hellinger consistency is exceptionally hard to violate. This suggests that existing sufficient conditions, while useful in practice, may overlook some key aspects of posterior behavior. We address this gap by directly investigating what must fail for inconsistency to arise, aiming to identify a substantive necessary condition for Hellinger inconsistency. Our starting point is Andrew Barron's classical counterexample, the only known violation of Hellinger consistency under KL support, which relies on a contrived family of oscillatory densities and a prior with atoms. We show that, within a broad class of models including Barron's, inconsistency requires persistent posterior concentration on densities with exponentially high likelihood ratios. In turn, such behavior demands a prior encoding implausibly precise knowledge of the true, yet unknown data-generating distribution, making inconsistency essentially unattainable in any realistic inference problem. Our results confirm the long-standing intuition that posterior inconsistency in density estimation is not a natural phenomenon, but rather an artifact of pathological prior constructions.

翻译：密度估计中后验分布的一致性构成了贝叶斯统计理论的核心。经典研究建立了充分条件——通常将KL支持与高先验质量筛的复杂度界限相结合——以保证关于Hellinger距离的一致性。然而，现有理论尚未系统解释一个被广泛持有的观点：在KL支持下，违反Hellinger一致性是极其困难的。这表明现有的充分条件虽在实践中有效，但可能忽略了后验行为的某些关键方面。我们通过直接探究不一致性产生所必须满足的条件来填补这一空白，旨在为Hellinger不一致性确立一个实质性的必要条件。我们的研究起点是Andrew Barron的经典反例——这是KL支持下唯一已知的Hellinger一致性违反案例，其依赖于精心构造的振荡密度族和具有原子性的先验分布。我们证明，在包含Barron反例的广泛模型类中，不一致性要求后验持续集中于具有指数级高似然比的密度函数。这种行为反过来要求先验分布编码了对真实（但未知）数据生成分布难以置信的精确认知，使得不一致性在任何现实的推断问题中本质上无法实现。我们的结果证实了长期存在的直觉：密度估计中的后验不一致性并非自然现象，而是病态先验构造的产物。