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 discounting parameter. When the discounting parameter is modelled as random, the normalized power prior is recommended. In this work, we prove that the marginal posterior for the discounting parameter for generalized linear models converges to a point mass at zero if there is any discrepancy between the historical and current data, and that it does not converge to a point mass at one when they are fully compatible. In addition, we explore the construction of optimal priors for the discounting parameter in a normalized power prior. In particular, we are interested in achieving the dual objectives of encouraging borrowing when the historical and current data are compatible and limiting borrowing when they are in conflict. We propose intuitive procedures for eliciting the shape parameters of a beta prior for the discounting parameter based on two minimization criteria, the Kullback-Leibler divergence and the mean squared error. Based on the proposed criteria, the optimal priors derived are often quite different from commonly used priors such as the uniform prior.

翻译：幂先验是一类流行的信息性先验，用于整合历史数据信息。它通过将历史数据的似然函数提升至某次幂来发挥作用，该幂次作为折扣参数。当折扣参数被建模为随机变量时，推荐采用归一化幂先验。本文证明，对于广义线性模型，若历史数据与当前数据存在任何不一致，折扣参数的边际后验分布将收敛于零处的点质量；而当两者完全兼容时，它不会收敛于1处的点质量。此外，我们探讨了归一化幂先验中折扣参数最优先验的构建。具体而言，我们旨在实现双重目标：在历史数据与当前数据兼容时促进借用，在两者冲突时限制借用。基于两个最小化准则——Kullback-Leibler散度和均方误差，我们提出了直观的方法来设定折扣参数贝塔先验的形状参数。根据所提准则推导出的最优先验通常与常用先验（如均匀先验）存在显著差异。