  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.
