On the normalized power prior

The power prior is a popular tool for constructing informative prior distributions based on historical data. The method consists of raising the likelihood to a discounting factor in order to control the amount of information borrowed from the historical data. However, one often wishes to assign this discounting factor a prior distribution and estimate it jointly with the parameters, which in turn necessitates the computation of a normalizing constant. In this article, we are concerned with how to approximately sample from joint posterior of the parameters and the discounting factor. We first show a few important properties of the normalizing constant and then use these results to motivate a bisection-type algorithm for computing it on a fixed budget of evaluations. We give a large array of illustrations and discuss cases where the normalizing constant is known in closed-form and where it is not. We show that the proposed method produces approximate posteriors that are very close to the exact distributions and also produces posteriors that cover the data-generating parameters with higher probability in the intractable case. Our results suggest that the proposed method is an accurate and easy to implement technique to include this normalization, being applicable to a large class of models. They also reinforce the notion that proper inclusion of the normalizing constant is crucial to the drawing of correct inferences and appropriate quantification of uncertainty.