# Optimal Priors for the Discounting Parameter of the Normalized Power Prior

## TL;DR

This paper investigates the behavior of the discounting parameter in normalized power priors for generalized linear models, proving convergence properties and proposing methods to construct optimal priors that balance borrowing and conflict.

## Contribution

It provides theoretical insights into the marginal posterior of the discounting parameter and introduces procedures for eliciting optimal beta priors based on divergence and error minimization.

## Key findings

- Posterior for the discounting parameter converges to zero with data discrepancy.
- Optimal priors differ significantly from uniform priors.
- Proposed priors effectively balance borrowing and conflict.

## Abstract

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.