Mathematical Foundations for a Theory of Confidence Structures

Area of Science:

• Statistics
• Mathematical Theory

Background:

• Current statistical methods face limitations in holistically representing inferential uncertainty.
• Bridging the gap between Bayesian and frequentist inference remains a challenge.

Purpose of the Study:

• Introduce a new mathematical object, the confidence structure.
• Develop a unified theory for statistical inference that combines Bayesian and frequentist approaches.

Main Methods:

• Enhancing confidence distribution theory with Dempster-Shafer evidence theory.
• Utilizing random set theory for mathematical proofs.
• Defining belief functions commensurate with Neyman-Pearson confidence.

Main Results:

• A novel mathematical object, the confidence structure, is defined.
• Confidence structures can be propagated through functions, enabling valid output structures.
• Mathematical proofs confirm the operative properties of confidence structures.

Conclusions:

• The theory of confidence structures offers a unified framework for statistical inference.
• This approach achieves the holistic goals of Bayesian inference while preserving frequentist empirical rigor.