Normalized power priors always discount historical data

Power priors are used for incorporating historical data in Bayesian analyses by taking the likelihood of the historical data raised to the power $\alpha$ as the prior distribution for the model parameters. The power parameter $\alpha$ is typically unknown and assigned a prior distribution, most commonly a beta distribution. Here, we give a novel theoretical result on the resulting marginal posterior distribution of $\alpha$ in case of the the normal and binomial model. Counterintuitively, when the current data perfectly mirror the historical data and the sample sizes from both data sets become arbitrarily large, the marginal posterior of $\alpha$ does not converge to a point mass at $\alpha = 1$ but approaches a distribution that hardly differs from the prior. The result implies that a complete pooling of historical and current data is impossible if a power prior with beta prior for $\alpha$ is used.