  Posterior probabilistic statistical inference without priors is an important
but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of
belief functions, and Bayesian inference with default priors are attempts to
achieve this goal but, to date, none has given a completely satisfactory
picture. This paper presents a new framework for probabilistic inference, based
on inferential models (IMs), which not only provides data-dependent
probabilistic measures of uncertainty about the unknown parameter, but does so
with an automatic long-run frequency calibration property. The key to this new
approach is the identification of an unobservable auxiliary variable associated
with observable data and unknown parameter, and the prediction of this
auxiliary variable with a random set before conditioning on data. Here we
present a three-step IM construction, and prove a frequency-calibration
property of the IM's belief function under mild conditions. A corresponding
optimality theory is developed, which helps to resolve the non-uniqueness
issue. Several examples are presented to illustrate this new approach.
