Bayesian hierarchical model (BHM) has been widely used in synthesizing information across subgroups. Identifying heterogeneity in the data and determining proper strength of borrow have long been central goals pursued by researchers. Because these two goals are interconnected, we must consider them together. This joint consideration presents two fundamental challenges: (1) How can we balance the trade-off between homogeneity within the cluster and information gain through borrowing? (2) How can we determine the borrowing strength dynamically in different clusters? To tackle challenges, first, we develop a theoretical framework for heterogeneity identification and dynamic information borrowing in BHM. Then, we propose two novel overlapping indices: the overlapping clustering index (OCI) for identifying the optimal clustering result and the overlapping borrowing index (OBI) for assigning proper borrowing strength to clusters. By incorporating these indices, we develop a new method BHMOI (Bayesian hierarchical model with overlapping indices). BHMOI includes a novel weighted K-Means clustering algorithm by maximizing OCI to obtain optimal clustering results, and embedding OBI into BHM for dynamically borrowing within clusters. BHMOI can achieve efficient and robust information borrowing with desirable properties. Examples and simulation studies are provided to demonstrate the effectiveness of BHMOI in heterogeneity identification and dynamic information borrowing.

翻译：贝叶斯层次模型（BHM）已广泛用于跨子组的信息综合。识别数据中的异质性并确定恰当的借取强度，一直是研究者追求的核心目标。由于这两个目标相互关联，我们必须将其统筹考虑。这种联合考虑带来了两个基本挑战：（1）如何在簇内同质性与通过借取获得的信息增益之间取得平衡？（2）如何在不同簇中动态确定借取强度？为应对这些挑战，我们首先发展了BHM中异质性识别与动态信息借取的理论框架。接着，我们提出了两种新颖的重叠指标：用于识别最优聚类结果的重叠聚类指数（OCI），以及为簇分配适当借取强度的重叠借取指数（OBI）。通过融入这些指标，我们开发了一种新方法BHMOI（含重叠指标的贝叶斯层次模型）。BHMOI包含一种新颖的加权K-Means聚类算法，通过最大化OCI获得最优聚类结果，并将OBI嵌入BHM以实现簇内动态借取。BHMOI能实现高效且稳健的信息借取，并具有令人满意的特性。通过实例和模拟研究，验证了BHMOI在异质性识别与动态信息借取中的有效性。