Primary Decomposition for the Intersection Axiom

Consider the discrete conditional independence model M given by {X1 ⊥ X2 | X3, X1 ⊥ X3 | X2}. The intersection axiom for conditional independence can be applied with the statements of M as premises to derive the conclusion X1 ⊥ (X2, X3). But the independence axiom only holds in general when X is in the interior of the probability simplex, and it’s a natural question to ask what can be inferred about X when it may lie on the boundary, that is, what the primary decomposition of IM is. A conjecture of Dustin Cartwright and Alexander Engström recorded in [2, p. 152], our Theorem 2.1, characterises the minimal primary components of M for discrete distributions at the set-theoretic level, in terms of subgraphs of a complete bipartite graph. We state and prove this conjecture in Section 2. Then in Section 3 we discuss the ideal-theoretic question.