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The power prior is a useful general class of priors that can be used for arbitrary classes of regression models, including generalized linear models, generalized linear mixed models semiparametric survival models with censored data, frailty models, multivariate models, and nonlinear models. The power prior specification for the regression coefficients focuses on observable quantities in that the elicitation is based on historical data, D₀, and a scalar quantity, a₀, quantifying the heterogeneity between the current data, D, and the historical data D₀. The power prior distribution is then constructed by raising the likelihood function of the historical data to the power a₀, where 0 ≤ a₀ ≤ 1. The scalar a₀ is a precision parameter that can be viewed as a measure of compatibility between the historical and current data. In this article we give a formal justification of the power prior and show that it is an optimal class of …