Asymptotically informative prior for Bayesian analysis

In classical Bayesian inference the prior is treated as fixed and its effects are ignored asymptotically, and useful information, if any, is wasted. However, in practice often an informative prior is summarized from previous similar or the same kind of studies, which contains useful cumulative information for the current study. We treat such prior to be non-fixed, i.e., we give the data sizes in the prior studies similar status as the that of the current dataset. Under this formulation, the prior is asymptotically nonnegligible, and its original information is transferred to the new study. We explore some basic properties of Bayesian estimators under such prior formulation, and illustrate the method via simulation.