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Bayesian inference requires specification of a single, precise priordistribution, whereas frequentist inference only accommodates a vacuous prior.Since virtually every real-world application falls somewhere in between thesetwo extremes, a new approach is needed. This series of papers develops a newframework that provides valid and efficient statistical inference, prediction,etc., while accommodating partial prior information and imprecisely-specifiedmodels more generally. This paper fleshes out a general inferential modelconstruction that not only yields tests, confidence intervals, etc.~withdesirable error rate control guarantees, but also facilitates validprobabilistic reasoning with de~Finetti-style no-sure-loss guarantees. The keytechnical novelty here is a so-called outer consonant approximation of ageneral imprecise probability which returns a data- and partial prior-dependentpossibility measure to be used for inference and prediction. Despite somepotentially unfamiliar imprecise-probabilistic concepts in the development, theresult is an intuitive, likelihood-driven framework that will, as expected,agree with the familiar Bayesian and frequentist solutions in the respectiveextreme cases. More importantly, the proposed framework accommodates partialprior information where available and, therefore, leads to new solutions thatwere previously out of reach for both Bayesians and frequentists. Details arepresented here for a wide range of examples, with more practical details tocome in later installments.

Valid and efficient imprecise-probabilistic inference with partial priors, II. General framework