Understanding posterior contraction behavior in Bayesian hierarchical models is of fundamental importance, but progress in this question is relatively sparse in comparison to the theory of density estimation. In this paper, we study two classes of hierarchical models for grouped data, where observations within groups are exchangeable. Using moment tensor decomposition of the distribution of the latent variables, we establish a precise equivalence between the class of Admixture models (such as Latent Dirichlet Allocation) and the class of Mixture of products of multinomial distributions. This correspondence enables us to leverage the result from the latter class of models, which are more well-understood, so as to arrive at the identifiability and posterior contraction rates in both classes under conditions much weaker than in existing literature. For instance, our results shed light on cases where the topics are not linearly independent or the number of topics is misspecified in the admixture setting. Finally, we analyze individual documents' latent allocation performance via the borrowing of strength properties of hierarchical Bayesian modeling. Many illustrations and simulations are provided to support the theory.

翻译：理解贝叶斯层次模型中的后验收缩行为具有基础重要性，但相较于密度估计理论，该问题的研究进展相对有限。本文研究两类用于分组数据的层次模型，其中组内观测可交换。通过隐变量分布的矩张量分解，我们建立了混合模型类（如隐狄利克雷分配）与多项分布乘积混合模型类之间的精确等价关系。该对应使我们能够利用后一类模型（已有较深入理解）的结果，从而在比现有文献更弱的条件下，获得两类模型的参数可识别性与后验收缩速率。例如，我们的结果揭示了混合设定中主题非线性独立或主题数量设定错误的情形。最后，我们通过层次贝叶斯建模的借力特性分析了单个文档的隐分配性能。文中提供了大量示例与仿真以支撑理论。