Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information

Abstract

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.

Uncertainty quantification in logistic regression using random fuzzy sets and belief functions

“…In the robust Bayes approach, prior information is represented by a set of probability distributions, an approach also advocated by Martin [39] in the context of Inferential Models.…”

Fiducial inference viewed through a possibility-theoretic inferential model lens

“…First I have to stress that there is no prior knowledge about Θ assumed here, i.e., the prior about Θ is vacuous. While I don't believe a vacuous-prior-knowledge assumption is realistic in most applications (Martin 2022a), this is the typical starting point in the literature. This means that neither ordinary Bayesian inference with a single prior distribution (Berger 1985;Bernardo and Smith 1994;Ghosh et al 2006) nor generalized Bayesian inference with a proper subset of all prior distributions (Augustin et al 2014;Walley 1991) are viable options to achieve the desired goal.…”

On Processing Heterogeneous Sources of Limited Data for Uncertainty Quantification in a Possibilistic Framework

“…On the far end of the spectrum of available priors [8] lies the case of complete epistemic uncertainty, i.e. the case of an unknown distribution within certain limits.…”