Power priors are used for incorporating historical data in Bayesian analyses by taking the likelihood of the historical data raised to the power $\alpha$ as the prior distribution for the model parameters. The power parameter $\alpha$ is typically unknown and assigned a prior distribution, most commonly a beta distribution. Here, we give a novel theoretical result on the resulting marginal posterior distribution of $\alpha$ in case of the the normal and binomial model. Counterintuitively, when the current data perfectly mirror the historical data and the sample sizes from both data sets become arbitrarily large, the marginal posterior of $\alpha$ does not converge to a point mass at $\alpha = 1$ but approaches a distribution that hardly differs from the prior. The result implies that a complete pooling of historical and current data is impossible if a power prior with beta prior for $\alpha$ is used.

翻译：幂先验通过将历史数据的似然函数取幂$\alpha$作为模型参数的先验分布，用于在贝叶斯分析中整合历史数据。幂参数$\alpha$通常未知，并被赋予一个先验分布，最常见的是贝塔分布。本文针对正态模型和二项模型，给出了$\alpha$边缘后验分布的一个新颖理论结果。与直觉相反，当当前数据与历史数据完美匹配且两个数据集的样本量任意大时，$\alpha$的边缘后验分布不会收敛到$\alpha = 1$处的点质量，而是趋近于一个几乎与先验分布无差异的分布。该结果表明，若使用带贝塔先验的幂先验，历史数据与当前数据的完全合并是不可能的。