Abstract
Bayesian inference under a weakly informative prior density is a key research subject for pursuing the construction of widely applicable Bayesian procedures under various practical situations. A non-informative prior density can be defined as a limit of a sequence of weakly informative prior densities, and reversely such a sequence can connect a proper prior density with a weakly informative prior density. The behavior of a Bayesian procedure under a weakly informative prior density is an important issue, though the current theory of the empirical Bayes method often treats it as a least favorable case. This article reviews inferential procedures under the assumption of a family of prior densities connecting from a degenerated density on a known point to a non-informative density. Then, we emphasize the need for introducing suitable definitions of Bayesian likelihoods. Lindley's paradox and the actual difficulty in defining a non-informative prior density are referred to as two attractive implications.
