  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.
