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Between Bayesian and frequentist inference, it's commonly believed that theformer is for cases where one has a prior and the latter is for cases where onehas no prior. But the prior/no-prior classification isn't exhaustive, and mostreal-world applications fit somewhere in between these two extremes. Thatneither of the two dominant schools of thought are suited for theseapplications creates confusion and slows progress. A key observation here isthat ``no prior information'' actually means no prior distribution can be ruledout, so the classically-frequentist context is best characterized as everyprior. From this perspective, it's now clear that there's an entire spectrum ofcontexts depending on what, if any, partial prior information is available,with Bayesian (one prior) and frequentist (every prior) on opposite extremes.This paper ties the two frameworks together by formally treating those caseswhere only partial prior information is available using the theory of impreciseprobability. The end result is a unified framework of (imprecise-probabilistic)statistical inference with a new validity condition that implies bothfrequentist-style error rate control for derived procedures and Bayesian-stylecoherence properties, relative to the given partial prior information. This newtheory contains both the Bayesian and frequentist frameworks as special cases,since they're both valid in this new sense relative to their respective partialpriors. Different constructions of these valid inferential models areconsidered, and compared based on their efficiency.

Valid and efficient imprecise-probabilistic inference with partial priors, I. First results