Information and Set Algebras: Interpretation and Uniqueness of Conditional Independence

TL;DR

This paper introduces a new axiomatic framework for information algebras, demonstrating their embedding into set algebras where conditional independence has a clear set-theoretic interpretation, establishing equivalence with previous formulations.

Contribution

It presents a novel axiomatic formulation of information algebras and shows how they can be embedded into set algebras, clarifying the concept of conditional independence.

Findings

Embedding of information algebras into set algebras

Conditional independence interpreted via set partitions

Equivalence of the new axiomatic formulation with prior work

Abstract

A new seemingly weak axiomatic formulation of information algebras is given. It is shown how such information algebras can be embedded into set (information) algebras. In set algebras there is a natural relation of conditional independence between partitions. Via the embedding of information algebras this relation carries over to information algebras. The new axiomatic formulation is thereby shown to be equivalent to the one given in arXiv:1701.02658. In this way the abstract concept of conditional independence in information algebras gets a concrete interpretation in terms of set theoretical relations.