Historical information is often available in clinical trials, environmental studies, and other applied settings, but the statistical value of such information depends on how compatibility between historical and current data is handled. This talk discusses the normalized power prior (NPP) as a coherent Bayesian framework for adaptive historical borrowing. The starting point is the original power prior, by Ibraham and Chen (1998), which discounts the historical likelihood through a power parameter δ ∈ [0, 1]. While this formulation is appealing for a fixed δ, treating δ as random through the joint power prior can violate the likelihood principle, because equivalent likelihood representations may induce different posteriors. The normalized power prior resolves this issue by incorporating the normalizing factor that depends on δ, thereby producing a coherent posterior formulation.

The talk first explains the motivation for adaptive borrowing and the role of the discounting parameter. It then develops the normalized power prior from both a logical and a theore tical perspective, including its interpretation through expected weighted Kullback-Leibler divergence. The behavior of the method is illustrated in basic models, including Bernoulli and normal linear settings, where one can see how posterior borrowing responds to compatibility between historical and current information. Applications are discussed. Furthermore, practical computation is also addressed through the identity for log C(δ), which underlies numerical evaluation of the normalizing factor.