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 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.

Automated summary

Power priors are used to incorporate historical data into Bayesian analyses, but a new theoretical result shows that when the current data perfectly mirror the historical data and both sample sizes become large, the marginal posterior distribution of alpha does not converge to a point mass at alpha=1, but approaches the prior distribution instead. This implies that complete pooling of historical and current data is impossible when using a power prior with a beta prior for alpha.