Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information
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Bayesian inference quantifies uncertainty directly and formally using classical probability theory, while frequentist inference does so indirectly and informally through the use of procedures with error rate control. Both have merits in the appropriate context, but the context isn't binary. There's an entire spectrum of contexts depending on what, if any, partial prior information is available to incorporate, with "Bayesian" and "frequentist" sitting on opposite ends of the spectrum corresponding to a complete, fully precise prior distribution and a vacuous, fully imprecise prior distribution, respectively. Common examples between the two extremes include those high-dimensional problems where, e.g., sparsity assumptions are relevant but fall short of determining a complete prior distribution. This paper ties the two frameworks together by treating those cases where only partial prior information is available using the theory of imprecise probability. The end result is a unified framework of (imprecise-probabilistic) statistical inference with a new validity condition that implies both frequentist-style error rate control for derived procedures and Bayesian-style no-sure-loss properties, relative to the given partial prior information. This theory contains both the classical "Bayesian" and "frequentist" frameworks as special cases, since they're both valid in this new sense relative to their respective partial priors. Different constructions of these valid inferential models are considered, and compared based on their efficiency.
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