  Bayesian inference requires specification of a single, precise prior
distribution, whereas frequentist inference only accommodates a vacuous prior.
Since virtually every real-world application falls somewhere in between these
two extremes, a new approach is needed. This series of papers develops a new
framework that provides valid and efficient statistical inference, prediction,
etc., while accommodating partial prior information and imprecisely-specified
models more generally. This paper fleshes out a general inferential model
construction that not only yields tests, confidence intervals, etc.~with
desirable error rate control guarantees, but also facilitates valid
probabilistic reasoning with de~Finetti-style no-sure-loss guarantees. The key
technical novelty here is a so-called outer consonant approximation of a
general imprecise probability which returns a data- and partial prior-dependent
possibility measure to be used for inference and prediction. Despite some
potentially unfamiliar imprecise-probabilistic concepts in the development, the
result is an intuitive, likelihood-driven framework that will, as expected,
agree with the familiar Bayesian and frequentist solutions in the respective
extreme cases. More importantly, the proposed framework accommodates partial
prior information where available and, therefore, leads to new solutions that
were previously out of reach for both Bayesians and frequentists. Details are
presented here for a wide range of examples, with more practical details to
come in later installments.
