  The elicitation of power priors, based on the availability of historical
data, is realized by raising the likelihood function of the historical data to
a fractional power {\delta}, which quantifies the degree of discounting of the
historical information in making inference with the current data. When {\delta}
is not pre-specified and is treated as random, it can be estimated from the
data using Bayesian updating paradigm. However, in the original form of the
joint power prior Bayesian approach, certain positive constants before the
likelihood of the historical data could be multiplied when different settings
of sufficient statistics are employed. This would change the power priors with
different constants, and hence the likelihood principle is violated. In this
article, we investigate a normalized power prior approach which obeys the
likelihood principle and is a modified form of the joint power prior. The
optimality properties of the normalized power prior in the sense of minimizing
the weighted Kullback-Leibler divergence is investigated. By examining the
posteriors of several commonly used distributions, we show that the discrepancy
between the historical and the current data can be well quantified by the power
parameter under the normalized power prior setting. Efficient algorithms to
compute the scale factor is also proposed. In addition, we illustrate the use
of the normalized power prior Bayesian analysis with three data examples, and
provide an implementation with an R package NPP.
