A Study on the Power Parameter in Power Prior Bayesian Analysis

TL;DR

This paper investigates the behavior of the power parameter in power prior Bayesian analysis, revealing potential issues with the scaling factor that can affect inference, especially with improper initial priors.

Contribution

It uncovers conditions under which the scaling factor becomes infinite, highlighting implications for posterior inference and the importance of careful selection of the power parameter.

Findings

Scaling factor may be infinite for some positive 4 with common priors.

Improper initial priors can jeopardize posterior inference.

Recommendations for choosing the power parameter near zero.

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 {\delta}. 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 paper, we show that the scaling factor might be infinite for some positive {\delta} 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 paper 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.