Statistical inference with partial prior information

Statistical inference procedures are considered when less complete prior information is available than usually considered. For the purposes of this paper, the prior information is taken to be the specification of a set of probability measures
\cal P
. With any one prior probability measure the corresponding Bayes' estimate may be found; the recommended inference procedure when a whole set of prior probabilities
\cal P
is available is to find the whole set of estimates corresponding to
\cal P
--this is called the set of feasible estimates
\^{\Theta}
. The procedure is shown to have some justification on philosophical grounds. Practical justification is also given in that finding
\^{\Theta}
is computationally feasible in particular cases--those cases investigated here include median, minimum mean square error (MMSE), and maximum {\em a posteriori} probability (MAP) estimation.