We propose a general class of prior distributions for arbitrary
regression models. We discuss parametric and semiparametric models. The prior
specification for the regression coefficients focuses on observable quantities
in that the elicitation is based on the availability of historical data $D_0$
and a scalar quantity $a_0$ quantifying the uncertainty in $D_0$. Then $D_0$
and $a_0$ are used to specify a prior for the regression coefficients in a
semiautomatic fashion. The most natural specification of $D_0$ arises when the
raw data from a similar previous study are available. The availability of
historical data is quite common in clinical trials, carcinogenicity studies,
and environmental studies, where large data bases are available from similar
previous studies. Although the methodology we present here is quite general, we
will focus only on using historical data from similar previous studies to
construct the prior distributions. The prior distributions are based on the
idea of raising the likelihood function of the historical data to the power
$a_0$, where $0 \le a_0 \le 1$. We call such prior distributions power prior
distributions. We examine the power prior for four commonly used classes of
regression models. These include generalized linear models, generalized linear
mixed models, semiparametric proportional hazards models, and cure rate models
for survival data. For these classes of models, we discuss the construction of
the power prior, prior elicitation issues, propriety conditions, model
selection, and several other properties. For each class of models, we present
real data sets to demonstrate the proposed methodology.