Abstract

The power prior and its variations have been proven to be a useful class of informative priors in Bayesian inference due to their flexibility in incorporating the historical information by raising the likelihood of the historical data to a fractional power δ. The derivation of the marginal likelihood based on the original power prior, and its variation, the normalized power prior, introduces a scaling factor $C(\delta )$ in the form of a prior predictive distribution with powered likelihood. In this article, we show that the scaling factor might be infinite for some positive δ with conventionally used initial priors, which would change the admissible set of the power parameter. This result seems to have been almost completely ignored in the literature. We then illustrate that such a phenomenon may jeopardize the posterior inference under the power priors when the initial prior of the model parameters is improper. The main findings of this article suggest that special attention should be paid when the suggested level of borrowing is close to 0, while the actual optimum might be below the suggested value. We use a normal linear model as an example for illustrative purposes.

摘要

幂先验及其变体已被证明是贝叶斯推理中一类有用的信息先验，因为它们通过将历史数据的可能性提高到分数幂 δ 来灵活地合并历史信息。基于原始功率先验的边际似然的推导，及其变化，归一化功率先验，引入比例因子$C(\delta )$以具有强力可能性的先验预测分布的形式。在这篇文章中，我们展示了对于具有常规使用的初始先验的某些正δ的比例因子可能是无限的，这将改变功率参数的可接受集。这个结果似乎在文献中几乎完全被忽略了。然后我们说明，当模型参数的初始先验不正确时，这种现象可能会危及幂先验下的后验推理。本文的主要发现表明，当建议借贷水平接近于 0 时应特别注意，而实际最优值可能低于建议值。为了便于说明，我们使用正态线性模型作为示例。