  This article illustrates how indirect or prior information can be optimally
used to construct a prediction region that maintains a target frequentist
coverage rate. If the indirect information is accurate, the volume of the
prediction region is lower on average than that of other regions with the same
coverage rate. Even if the indirect information is inaccurate, the resulting
region still maintains the target coverage rate. Such a prediction region can
be constructed for models that have a complete sufficient statistic, which
includes many widely-used parametric and nonparametric models. Particular
examples include a Bayes-optimal conformal prediction procedure that maintains
a constant coverage rate across distributions in a nonparametric model, as well
as a prediction procedure for the normal linear regression model that can
utilize a regularizing prior distribution, yet maintain a frequentist coverage
rate that is constant as a function of the model parameters and explanatory
variables. No results in this article rely on asymptotic approximations.
