Categoroids: Universal Conditional Independence

TL;DR

This paper introduces categoroids, an algebraic framework unifying various structures of conditional independence, enabling universal representation and analysis across different models in AI and statistics.

Contribution

It proposes categoroids as a novel algebraic structure that captures the universal properties of conditional independence, linking different frameworks through functoroids and natural transformations.

Findings

Categoroids unify multiple conditional independence frameworks.

Functoroids preserve relationships between categoroids.

Universal representations of conditional independence are constructed.

Abstract

Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are defined as a hybrid of two categories: one encoding a preordered lattice structure defined by objects and arrows between them; the second dual parameterization involves trigonoidal objects and morphisms defining a conditional independence structure, with bridge morphisms providing the interface between the binary and ternary structures. We illustrate categoroids using three well-known examples of axiom sets: graphoids, integer-valued multisets, and separoids. Functoroids map one categoroid to another, preserving the relationships defined by all three types of arrows in the co-domain categoroid. We describe a natural transformation across functoroids, which is natural across regular objects and trigonoidal objects, to construct universal representations of conditional independence.. We use adjunctions and monads between categoroids to abstractly characterize faithfulness of graphical and non-graphical representations of conditional independence.