Properties of semi-elementary imsets as sums of elementary imsets

TL;DR

This paper investigates the structure of semi-elementary imsets, showing they can be expressed as sums of elementary imsets and analyzing the relations among their multiple representations.

Contribution

It introduces a recursive method to represent semi-elementary imsets as sums of elementary imsets and studies the relations connecting different representations.

Findings

Any semi-elementary imset can be expressed as a sum of elementary imsets.

All representations of a semi-elementary imset are interconnected through relations among four elementary imsets.

The semi-graphoid axioms translate into simple identities among semi-elementary imsets.

Abstract

We study properties of semi-elementary imsets and elementary imsets introduced by Studeny (2005). The rules of the semi-graphoid axiom (decomposition, weak union and contraction) for conditional independence statements can be translated into a simple identity among three semi-elementary imsets. By recursively applying the identity, any semi-elementary imset can be written as a sum of elementary imsets, which we call a representation of the semi-elementary imset. A semi-elementary imset has many representations. We study properties of the set of possible representations of a semi-elementary imset and prove that all representations are connected by relations among four elementary imsets.