Conditional independence on semiring relations

TL;DR

This paper introduces a semiring-based framework for analyzing conditional independence, unifying concepts across database, probability, and information theories, and identifies key semiring properties that preserve independence properties.

Contribution

It extends conditional independence analysis to semiring relations, revealing how semiring properties influence axiomatic and decomposition results in various domains.

Findings

Positivity and multiplicative cancellativity are crucial semiring properties.

The framework generalizes independence testing methods from databases and information theory.

Relationships between different independence notions are characterized via model theory.

Abstract

Conditional independence plays a foundational role in database theory, probability theory, information theory, and graphical models. In databases, conditional independence appears in database normalization and is known as the (embedded) multivalued dependency. Many properties of conditional independence are shared across various domains, and to some extent these commonalities can be studied through a measure-theoretic approach. The present paper proposes an alternative approach via semiring relations, defined by extending database relations with tuple annotations from some commutative semiring. Integrating various interpretations of conditional independence in this context, we investigate how the choice of the underlying semiring impacts the corresponding axiomatic and decomposition properties. We specifically identify positivity and multiplicative cancellativity as the key semiring properties that enable extending results from the relational context to the broader semiring framework. Additionally, we explore the relationships between different conditional independence notions through model theory, and consider how methods to test logical consequence and validity generalize from database theory and information theory to semiring relations.