A mathematical characterization of confidence as valid belief

TL;DR

This paper proposes a mathematical interpretation of confidence as belief functions, demonstrating how valid inferential models can be constructed to align confidence regions with belief plausibility, enhancing statistical understanding.

Contribution

It introduces a complete-class theorem linking confidence regions with valid belief/plausibility functions via the IM framework.

Findings

Valid belief functions can be constructed for any confidence region.

The IM framework provides a complete characterization of confidence as belief.

Implications for statistical interpretation and communication are discussed.

Abstract

Confidence is a fundamental concept in statistics, but there is a tendency to misinterpret it as probability. In this paper, I argue that an intuitively and mathematically more appropriate interpretation of confidence is through belief/plausibility functions, in particular, those that satisfy a certain validity property. Given their close connection with confidence, it is natural to ask how a valid belief/plausibility function can be constructed directly. The inferential model (IM) framework provides such a construction, and here I prove a complete-class theorem stating that, for every nominal confidence region, there exists a valid IM whose plausibility regions are contained by the given confidence region. This characterization has implications for statistics understanding and communication, and highlights the importance of belief functions and the IM framework.