Foundations of Structural Statistics: Topological Statistical Theory
This foundational work addresses the need for a modern mathematical framework in statistics to handle advanced AI models, though it appears incremental in reformulating classical theory.
The paper tackles the problem of integrating complex structured model spaces, such as deep-learning models, into statistical theory by developing Structural Statistics, which emphasizes structural assumptions and transformations, and aims to bridge gaps with topology and differential geometry.
Topological statistical theory provides the foundation for a modern mathematical reformulation of classical statistical theory: Structural Statistics emphasizes the structural assumptions that accompany distribution families and the set of structure preserving transformations between them, given by their statistical morphisms. The resulting language is designed to integrate complicated structured model spaces like deep-learning models and to close the gap to topology and differential geometry. To preserve the compatibility to classical statistics the language comprises corresponding concepts for standard information criteria like sufficiency and completeness.