Asymptotically informative prior for Bayesian analysis

In classical Bayesian inference the prior is treated as fixed, it is asymptotically negligible, thus any information contained in the prior is ignored from the asymptotic first order result. However, in practice often an informative prior is summarized from previous similar or the same kind of studies, which contains non-negligible information for the current study. Here, different from traditional Bayesian point of view, we treat such prior to be non-fixed. In particular, we give the data sizes used in previous studies for the prior the same status as the size of the current dataset, viewing both sample sizes as increasing to infinity in the asymptotic study. Thus the prior is asymptotically non-negligible, and its original effects are ressumed under this view. Consequently, Bayesian inference using such prior is more efficient, as it should be, than that regarded under the existing setting. We study some basic properties of Bayesian estimators using such priors under convex losses and the 0-1 loss, and illustrate the method by an example via simulation.