Which statistical hypotheses are afflicted with false confidence?

TL;DR

This paper demonstrates that many data-driven uncertainty quantification methods can falsely assign high confidence to incorrect hypotheses, especially non-convex ones, raising concerns about their reliability.

Contribution

It proves the false confidence theorem for general methods and identifies convex hypotheses as less susceptible, highlighting limitations of current uncertainty quantification techniques.

Findings

False confidence affects many hypotheses, especially non-convex ones.

Belief function-based methods are immune to false confidence.

Convex hypotheses are less affected by false confidence.

Abstract

The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines new light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone is insufficient. Towards a partial answer to the title question, I show that, roughly, complements of convex hypotheses are afflicted by false confidence.