The Binomial Ideal of the Intersection Axiom for Conditional Probabilities

The binomial ideal associated with the intersection axiom of conditional probability is shown to be radical and is expressed as an intersection of toric prime ideals. This resolves a conjecture in algebraic statistics due to Cartwright and Engström. Conditional independence contraints are a family of natural constraints on probability distributions, describing situations in which two random variables are independently distributed given knowledge of a third. Statistical models built around considerations of conditional independence, in particular graphical models in which the constraints are encoded in a graph on the random variables, enjoy wide applicability in determining relationships among random variables in statistics and in dealing with uncertainty in artificial intelligence. One can take a purely combinatorial perspective on the study of conditional independence, as does Studený [9], conceiving of it as a relation on triples of subsets of a set of observables which must satisfy certain axioms. A number of elementary implications among conditional independence statements are recognised as axioms. Among these are the semi-graphoid axioms, which are implications of conditional independence statements lacking further hypotheses, and hence are purely combinatorial statements. The intersection axiom is also often added to the collection, but unlike the semi-graphoid axioms it is not uniformly true; it is our subject here. Formally, a conditional independence model M is a set of probability distributions characterised by satisfying several conditional independence constraints. We will work in the discrete setting, where a probability distribution p is a multi-way table of probabilities, and we follow the notational conventions in [1]. Consider the discrete conditional independence model M given by {X1 ⊥ X2 | X3, X1 ⊥ X3 | X2} where Xi is a random variable taking values in the set [ri]. Throughout we assume r1 ≥ 2. The set of distributions in the model M is the variety whose defining ideal IM ⊆ S = C[pijk] is IM = (pijkpi′j′k − pij′kpi′jk : i, i ′ ∈ [r1], j, j ′ ∈ [r2], k ∈ [r3]) + (pijkpi′jk′ − pijk′pi′jk : i, i ′ ∈ [r1], j ∈ [r2], k, k ′ ∈ [r3]). The intersection axiom is the axiom whose premises are the statements of M and whose conclusion is X1 ⊥ (X2, X3). This implication requires the further hypothesis that the distribution p is in the interior of the probability simplex, i.e. that no individual probability pijk is zero. It’s thus a natural question to ask what can 1 Department of Mathematics, University of California, Berkeley, [email protected].