Large-Sample Theory for Inferential Models: A Possibilistic Bernstein–von Mises Theorem

The inferential model (IM) framework offers alternatives to the familiar probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. Allowing uncertainty quantification to be imprecise makes exact validity/reliability possible. But is imprecision and exact validity compatible with attainment of statistical efficiency? This paper gives 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, asymptotically, its credal set is the smallest that contains the Gaussian distribution with variance equal to the Cramér–Rao lower bound.