  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 a discounting parameter. When the
discounting parameter is modeled as random, the normalized power prior is
recommended. Bayesian hierarchical modeling is a widely used method for
synthesizing information from different sources, including historical data. In
this work, we examine the analytical relationship between the normalized power
prior (NPP) and Bayesian hierarchical models (BHM) for \emph{i.i.d.} normal
data. We establish a direct relationship between the prior for the discounting
parameter of the NPP and the prior for the variance parameter of the BHM. Such
a relationship is first established for the case of a single historical
dataset, and then extended to the case with multiple historical datasets with
dataset-specific discounting parameters. For multiple historical datasets, we
develop and establish theory for the BHM-matching NPP (BNPP) which establishes
dependence between the dataset-specific discounting parameters leading to
inferences that are identical to the BHM. Establishing this relationship not
only justifies the NPP from the perspective of hierarchical modeling, but also
provides insight on prior elicitation for the NPP. We present strategies on
inducing priors on the discounting parameter based on hierarchical models, and
investigate the borrowing properties of the BNPP.
