Possibilistic inferential models: a review

TL;DR

This paper reviews recent developments in possibilistic inferential models (IMs), highlighting their construction, reliability, and connections to modern statistical methods like bootstrap and conformal prediction.

Contribution

It provides a comprehensive review of possibilistic IM theory, methods, computational tools, and introduces a generalization linking IMs to contemporary statistical techniques.

Findings

Possibilistic IMs are straightforward to construct and highly reliable.

They offer a fully conditional, Bayesian-like probabilistic reasoning.

Connections with bootstrap and conformal prediction are established.

Abstract

An inferential model (IM) is a model describing the construction of provably reliable, data-driven uncertainty quantification and inference about relevant unknowns. IMs and Fisher's fiducial argument have similar objectives, but a fundamental distinction between the two is that the former doesn't require that uncertainty quantification be probabilistic, offering greater flexibility and allowing for a proof of its reliability. Important recent developments have been made thanks in part to newfound connections with the imprecise probability literature, in particular, possibility theory. The brand of possibilistic IMs studied here are straightforward to construct, have very strong frequentist-like reliability properties, and offer fully conditional, Bayesian-like (imprecise) probabilistic reasoning. This paper reviews these key recent developments, describing the new theory, methods, and computational tools. A generalization of the basic possibilistic IM is also presented, making new and unexpected connections with ideas in modern statistics and machine learning, e.g., bootstrap and conformal prediction.