Inferential models and possibility measures

TL;DR

This paper introduces a new formulation of inferential models using possibility measures, simplifying the theory and clarifying their relationship with Fisher's fiducial inference, while maintaining validity and error control.

Contribution

It develops an alternative, simpler formulation of inferential models based on possibility measures, enhancing understanding and practical application.

Findings

The possibility measure formulation is practically equivalent to the original IM framework.

This new perspective clarifies the connection between IMs and Fisher's fiducial inference.

The approach aids in constructing optimal inferential models.

Abstract

The inferential model (IM) framework produces data-dependent, non-additive degrees of belief about the unknown parameter that are provably valid. The validity property guarantees, among other things, that inference procedures derived from the IM control frequentist error rates at the nominal level. A technical complication is that IMs are built on a relatively unfamiliar theory of random sets. Here we develop an alternative -- and practically equivalent -- formulation, based on a theory of possibility measures, which is simpler in many respects. This new perspective also sheds light on the relationship between IMs and Fisher's fiducial inference, as well as on the construction of optimal IMs.