Normalized Power Prior Bayesian Analysis

The elicitation of power prior distributions is based on the availability of historical data, and is realized by raising the likelihood function of the historical data to a fractional power. However, an arbitrary positive constant before the like- lihood function of the historical data could change the inferential results when one uses the original power prior. This raises a question that which likelihood function should be used, one from raw data, or one from a su±cient-statistics. We propose a normalized power prior that can better utilize the power parameter in quantifying the heterogeneity between current and historical data. Furthermore, when the power parameter is random, the optimality of the normalized power priors is shown in the sense of maximizing Shannon's mutual information. Some comparisons between the original and the normalized power prior approaches are made and a water-quality monitoring data is used to show that the normalized power prior is more sensible.