# Bayesian/Graphoid intersection property for factorisation spaces

## TL;DR

This paper generalizes the intersection property in Bayesian networks to factorisation spaces, providing new proofs and extending classical theorems like Hammersley-Clifford to broader settings, including non-finite graphs.

## Contribution

It introduces a general intersection property for factorisation spaces, offers a novel proof of existing results, and extends the Hammersley-Clifford theorem to non-finite graphs.

## Key findings

- Generalized intersection property for factorisation spaces
- New proof of the Hammersley-Clifford theorem
- Extension of decomposition into interaction subspaces to vector spaces

## Abstract

We remark that Pearl's Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of a general intersection property, in the sense of intersection of coverings, for factorisation spaces, also coined as factorisation models, factor graphs or by Lauritzen in his reference book 'Graphical Models' as hierarchical model subspaces. A particular case of this intersection property appears in Lauritzen's book as a consequence of the decomposition into interaction subspaces; the novel proof that we give of this result allows us to extend it in the most general setting. It also allows us to give a direct and new proof of the Hammersley-Clifford theorem transposing and reducing it to a corresponding statement for graphs, justifying formally the geometric intuition of independency, and extending it to non finite graphs. This intersection property is the starting point for a generalization of the decomposition into interaction subspaces to collections of vector spaces.