Bayesian/Graphoid intersection property for factorisation spaces

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.

💡 Research Summary

The paper revisits Judea Pearl’s intersection axiom—a cornerstone of graphoid theory that states if a random variable X is conditionally independent of Y given (Z,W) and also independent of W given (Z,Y), then X is independent of the pair (Y,W) given Z. While this property is traditionally presented within the language of conditional independence relations, the author shows that it is in fact a special case of a much broader “intersection property” that lives in the realm of factorisation spaces (also known as factor graphs, hierarchical model subspaces, or factorisation models).

The author first formalises factorisation spaces. Given a finite index set I and a family of sets (E_i){i∈I}, a partial covering A⊆𝒫(I) is a collection of subsets of I. For each a∈A, the product space E_a=∏{i∈a}E_i is defined, and the factorisation space F_A consists of all strictly positive functions f on the full product E_I that can be written as a product of functions f_a that depend only on the coordinates in a. This definition is then extended to infinite coverings by introducing the lower set ˆA={b⊆I | ∃a∈A, b⊆a} and defining F_A:=F_{ˆA}.

Next, the paper builds a lattice structure on the collection of partial coverings. A preorder ≤ is introduced: A≤B iff every element of A is contained in some element of B. An intersection operation ⊓ is defined by taking pairwise intersections of elements from two coverings. Quotienting by the natural equivalence relation (A∼B iff A≤B and B≤A) yields a genuine poset 𝒫₂(I) that carries the familiar lattice identities (commutativity, distributivity of ⊓ over ∪, etc.).

The central result, Theorem 4.1 (Intersection property for factorisation spaces), states that for any family (A_j)_{j∈J} of elements of 𝒫₂(I),

  ⋂{j∈J} F{Â_j} = F_{⋂_{j∈J} Â_j}.

In words, the intersection of a collection of factorisation spaces is itself a factorisation space whose covering is the intersection of the corresponding lower‑set coverings. This theorem generalises a result that appears in Lauritzen’s “Graphical Models” as a corollary of the decomposition into interaction subspaces, but the proof given here does not rely on orthogonal decompositions; it follows directly from the lattice properties of partial coverings. Consequently, the map A↦F_A is a lattice morphism that preserves both joins (unions) and meets (intersections).

Armed with this intersection property, the author provides a new, conceptually transparent proof of the Hammersley–Clifford theorem. For a finite undirected graph G=(I,A), let C be the set of cliques, L(G) the set of distributions satisfying the local Markov property, and P(G) those satisfying the pairwise Markov property. By expressing L(G) and P(G) as factorisation spaces over the lower‑set coverings ˆA_L and ˆA_P, and observing that both lower‑set coverings equal the clique covering C (Equation 1.17), the intersection property yields

  P(G) ⇔ L(G) ⇔ F_C,

which is precisely the statement that a strictly positive distribution is Markov with respect to G if and only if it factorises over the cliques of G. This proof reduces the classical theorem to a purely combinatorial statement about coverings and avoids the traditional reliance on the Möbius inversion or explicit construction of potentials.

Beyond the immediate applications, the paper highlights that the lattice isomorphism between partial coverings and factorisation spaces provides a unified algebraic framework for “direct‑sum decompositions” of vector spaces. The author points to a forthcoming work