Large-sample theory for inferential models: a possibilistic Bernstein--von Mises theorem

TL;DR

This paper introduces a possibilistic Bernstein--von Mises theorem within the inferential model framework, showing that imprecise uncertainty quantification can be asymptotically efficient and reliable, paralleling classical Bayesian results.

Contribution

It establishes a new asymptotic result for inferential models, demonstrating their potential for exact validity and efficiency in statistical inference.

Findings

IM solutions are asymptotically efficient.

Credal sets converge to Gaussian distributions matching Cramér-Rao bounds.

Imprecise uncertainty quantification can be compatible with classical efficiency.

Abstract

The inferential model (IM) framework offers alternatives to the familiar probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. Allowing this uncertainty quantification to be imprecise makes it possible to achieve exact validity and reliability. But is imprecision and exact validity compatible with attainment of the classical notions of statistical efficiency? The present paper offers an affirmative answer to this question via a new possibilistic Bernstein--von Mises theorem that parallels a fundamental result in Bayesian inference. Among other things, our result demonstrates that the IM solution is asymptotically efficient in the sense that its asymptotic credal set is the smallest that contains the Gaussian distribution whose variance agrees with the Cramer--Rao lower bound.