Faithfulness in Chain Graphs: The Gaussian Case

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the regular Gaussian distributions that factorize with respect to a chain graph $G$ with $d$ parameters have positive Lebesgue measure with respect to $\mathbb{R}^d$, whereas those that factorize with respect to $G$ but are not faithful to it have zero Lebesgue measure with respect to $\mathbb{R}^d$. This means that, in the measure-theoretic sense described, almost all the regular Gaussian distributions that factorize with respect to $G$ are faithful to it.

💡 Research Summary

The paper investigates the relationship between chain graphs (CGs) and regular Gaussian distributions under the classic Lauritzen‑Wermuth‑Frydenberg (LWF) interpretation. A chain graph combines undirected edges, which encode symmetric conditional independences, with directed edges, which encode asymmetric (causal‑like) relationships. Under the LWF semantics a probability distribution is said to factorize with respect to a chain graph G if it can be written as a product of conditional Gaussian densities that respect the graph’s undirected cliques and directed parent sets. The authors focus on the notion of faithfulness: a distribution is faithful to G when every conditional independence implied by the distribution is exactly captured by the separation criteria of G, and conversely, no independence not implied by G holds in the distribution.

The main contribution consists of two measure‑theoretic theorems. First, the set Θ_G of all regular Gaussian parameters (means, covariances, and regression coefficients) that factorize according to G forms an open subset of ℝ^d, where d is the number of free parameters dictated by the graph. Consequently, Θ_G has positive Lebesgue measure. Second, the subset Θ_G^¬faith ⊂ Θ_G consisting of those parameters that yield a distribution not faithful to G has Lebesgue measure zero. In other words, almost every Gaussian distribution that respects the factorization constraints of G is also faithful to G.

The proof proceeds by explicitly parametrising the Gaussian model associated with G. For each undirected clique C, the covariance sub‑matrix Σ_C must be positive‑definite; for each directed edge, the conditional mean of a node is a linear function of its parents, with regression coefficients collected in a matrix B. These quantities are treated as independent real variables, subject only to the positive‑definiteness constraints, which are open conditions in ℝ^d. Hence Θ_G is an open set with non‑zero measure.

To address faithfulness, the authors translate any violation of the graph’s separation statements into algebraic equations. For example, a conditional independence X ⊥ Y | Z that is not implied by the graph translates into a determinant condition det(Σ_{X∪Z, Y∪Z}) = 0, which is a non‑trivial polynomial in the parameters. The set of solutions to a non‑identically‑zero polynomial in ℝ^d is a proper algebraic variety of dimension at most d − 1, and such a variety has Lebesgue measure zero. By enumerating all possible “extra” independences, the authors show that the union of the corresponding zero‑measure varieties is still measure‑zero. Hence Θ_G^¬faith is a null set.

The paper also discusses the concept of genericity: the faithful distributions form a dense open subset of Θ_G, reinforcing the measure‑theoretic result from a topological perspective. This mirrors earlier results for purely directed acyclic graphs (DAGs) and undirected Markov random fields, extending them to the richer mixed‑graph setting of chain graphs.

Practically, the result justifies a common assumption in structure‑learning algorithms (e.g., PC‑algorithm, Greedy Equivalence Search) that the underlying distribution is faithful to the true graph. Under Gaussianity, the assumption holds for “almost all” parameter values, providing a solid theoretical foundation for the consistency of such algorithms when applied to data generated by chain‑graph models.

The authors acknowledge limitations: the theorems rely on Gaussianity and the LWF interpretation; extensions to non‑Gaussian continuous families, mixed discrete‑continuous models, or alternative chain‑graph semantics (e.g., AMP) remain open problems. They suggest future work on Bayesian priors that avoid the measure‑zero non‑faithful region, empirical studies of how often finite‑sample data appear to violate faithfulness, and investigations of robustness when the true distribution lies near the boundary of the faithful set.

In summary, the paper establishes that within the parameter space of regular Gaussian distributions that factorize according to a chain graph G, the subset that fails to be faithful to G is negligible in the Lebesgue sense. Consequently, almost every such Gaussian distribution is faithful, strengthening the theoretical underpinnings of chain‑graph modeling and providing reassurance for statistical methods that depend on the faithfulness assumption.