  There has been increased interest in using prior information in statistical
analyses. For example, in rare diseases, it can be difficult to establish
treatment efficacy based solely on data from a prospective study due to low
sample sizes. To overcome this issue, an informative prior for the treatment
effect may be elicited. We develop a novel extension of the conjugate prior of
Chen and Ibrahim (2003) that enables practitioners to elicit a prior prediction
for the mean response for generalized linear models, treating the prediction as
random. We refer to the hierarchical prior as the hierarchical prediction
prior. For i.i.d. settings and the normal linear model, we derive cases for
which the hyperprior is a conjugate prior. We also develop an extension of the
HPP in situations where summary statistics from a previous study are available,
drawing comparisons with the power prior. The HPP allows for discounting based
on the quality of individual level predictions, having the potential to provide
efficiency gains (e.g., lower MSE) where predictions are incompatible with the
data. An efficient Markov chain Monte Carlo algorithm is developed.
Applications illustrate that inferences under the HPP are more robust to
prior-data conflict compared to selected non-hierarchical priors.
