The relationship between the power prior and hierarchical models

The power prior has emerged as a useful informative prior for the incorporation of historical data in a Bayesian analysis. Viewing hierarchical modeling as the "gold standard" for combining information across studies, we provide a formal justification of the power prior by examining formal analytical relationships between the power prior and hierarchical modeling in linear models. Asymptotic relationships between the power prior and hierarchical modeling are obtained for non-normal models, including generalized linear models, for example. These analytical relationships unify the theory of the power prior, demonstrate the generality of the power prior, shed new light on benchmark analyses, and provide insights into the elicitation of the power parameter in the power prior. Several theorems are presented establishing these formal connections, as well as a formal methodology for eliciting a guide value for the power parameter $a_0$ via hierarchical models.