  The ongoing replication crisis in science has increased interest in the
methodology of replication studies. We propose a novel Bayesian analysis
approach using power priors: The likelihood of the original study's data is
raised to the power of $\alpha$, and then used as the prior distribution in the
analysis of the replication data. Posterior distribution and Bayes factor
hypothesis tests related to the power parameter $\alpha$ quantify the degree of
compatibility between the original and replication study. Inferences for other
parameters, such as effect sizes, dynamically borrow information from the
original study. The degree of borrowing depends on the conflict between the two
studies. The practical value of the approach is illustrated on data from three
replication studies, and the connection to hierarchical modeling approaches
explored. We generalize the known connection between normal power priors and
normal hierarchical models for fixed parameters and show that normal power
prior inferences with a beta prior on the power parameter $\alpha$ align with
normal hierarchical model inferences using a generalized beta prior on the
relative heterogeneity variance $I^2$. The connection illustrates that power
prior modeling is unnatural from the perspective of hierarchical modeling since
it corresponds to specifying priors on a relative rather than an absolute
heterogeneity scale.
