Mathematical foundations for a theory of confidence structures

Abstract

This paper introduces a new mathematical object: the confidence structure. A confidence structure represents inferential uncertainty in an unknown parameter by defining a belief function whose output is commensurate with Neyman–Pearson confidence. Confidence structures on a group of input variables can be propagated through a function to obtain a valid confidence structure on the output of that function. The theory of confidence structures is created by enhancing the extant theory of confidence distributions with the mathematical generality of Dempster–Shafer evidence theory. Mathematical proofs grounded in random set theory demonstrate the operative properties of confidence structures. The result is a new theory which achieves the holistic goals of Bayesian inference while maintaining the empirical rigor of frequentist inference.