The power prior is a popular class of informative priors for incorporating information from historical data. It involves raising the likelihood for the historical data to a power, which acts as a discounting parameter. When the discounting parameter is modeled as random, the normalized power prior is recommended. Bayesian hierarchical modeling is a widely used method for synthesizing information from different sources, including historical data. In this work, we examine the analytical relationship between the normalized power prior (NPP) and Bayesian hierarchical models (BHM) for \emph{i.i.d.} normal data. We establish a direct relationship between the prior for the discounting parameter of the NPP and the prior for the variance parameter of the BHM. Such a relationship is first established for the case of a single historical dataset, and then extended to the case with multiple historical datasets with dataset-specific discounting parameters. For multiple historical datasets, we develop and establish theory for the BHM-matching NPP (BNPP) which establishes dependence between the dataset-specific discounting parameters leading to inferences that are identical to the BHM. Establishing this relationship not only justifies the NPP from the perspective of hierarchical modeling, but also provides insight on prior elicitation for the NPP. We present strategies on inducing priors on the discounting parameter based on hierarchical models, and investigate the borrowing properties of the BNPP.

翻译：幂先验是一类常用的信息性先验分布，用于整合历史数据信息。该方法通过将历史数据的似然函数取幂次，该幂次作为折扣参数。当折扣参数被建模为随机变量时，推荐使用归一化幂先验。贝叶斯层次模型是整合不同来源（包括历史数据）信息的广泛使用框架。本研究针对独立同分布正态数据，深入分析归一化幂先验与贝叶斯层次模型之间的解析关系。我们建立了归一化幂先验中折扣参数的先验分布与贝叶斯层次模型中方差参数的先验分布之间的直接对应关系。该关系首先针对单一历史数据集情形建立，随后推广至具有数据集特定折扣参数的多个历史数据集情形。针对多个历史数据集，我们发展并确立了与贝叶斯层次模型匹配的归一化幂先验理论，该理论通过建立数据集特定折扣参数之间的依赖关系，使得推断结果与贝叶斯层次模型完全等价。建立这种对应关系不仅从层次建模角度证明了归一化幂先验的合理性，还为该先验的 elicitation 提供了新见解。我们提出了基于层次模型构建折扣参数先验分布的策略，并深入研究了匹配先验的信息借用特性。