Decision theory via model-free generalized fiducial inference

TL;DR

This paper introduces a decision-theoretic framework based on model-free generalized fiducial inference, establishing consistency and connecting it with existing theories, thus advancing finite-sample valid predictive decision-making tools.

Contribution

It develops an MFGF-based decision theory approach, linking fiducial inference, conformal prediction, and imprecise probability, with proven consistency and nonasymptotic bounds.

Findings

Established pointwise and uniform consistency of MFGF upper risk function.

Derived nonasymptotic concentration bounds for the risk approximation.

Laid groundwork for future decision-theoretic analyses using MFGF.

Abstract

Building on the recent development of the model-free generalized fiducial (MFGF) paradigm (Williams, 2023) for predictive inference with finite-sample frequentist validity guarantees, in this paper, we develop an MFGF-based approach to decision theory. Beyond the utility of the new tools we contribute to the field of decision theory, our work establishes a formal connection between decision theories from the perspectives of fiducial inference, conformal prediction, and imprecise probability theory. In our paper, we establish pointwise and uniform consistency of an {\em MFGF upper risk function} as an approximation to the true risk function via the derivation of nonasymptotic concentration bounds, and our work serves as the foundation for future investigations of the properties of the MFGF upper risk from the perspective of new decision-theoretic, finite-sample validity criterion, as in Martin (2021).