Towards Optimal Learning of Chain Graphs

In this paper, we extend Meek’s conjecture (Meek 1997) from directed and acyclic graphs to chain graphs, and prove that the extended conjecture is true. Specifically, we prove that if a chain graph H is an independence map of the independence model induced by another chain graph G, then (i) G can be transformed into H by a sequence of directed and undirected edge additions and feasible splits and mergings, and (ii) after each operation in the sequence H remains an independence map of the independence model induced by G. Our result has the same important consequence for learning chain graphs from data as the proof of Meek’s conjecture in (Chickering 2002) had for learning Bayesian networks from data: It makes it possible to develop efficient and asymptotically correct learning algorithms under mild assumptions.

💡 Research Summary

The paper extends Meek’s conjecture—originally formulated for directed acyclic graphs (DAGs)—to the broader class of chain graphs (CGs), which may contain both directed and undirected edges. The original conjecture states that if two DAGs G and H satisfy the inclusion I(H) ⊆ I(G) of their induced independence models, then G can be transformed into H by a finite sequence of edge additions and covered edge reversals, while preserving the inclusion after each step. Chickering (2002) proved this conjecture for DAGs, providing the theoretical foundation for asymptotically correct score‑based structure learning algorithms such as GES.

The authors introduce two novel operations that are specific to CGs: Feasible Split (FbSplit) and Feasible Merge (FbMerge). FbSplit takes a block K of a CG and a subset L ⊆ K, adds the minimal set of edges required to make the split feasible, and then replaces the undirected edges between L and K\L with directed edges from the parents of L to the nodes in K\L. FbMerge performs the inverse: given two blocks L and R, it adds the minimal set of edges so that all directed edges from L to R can be replaced by undirected edges, effectively merging the two blocks. Both operations preserve the chain‑graph property and keep the independence model unchanged (I(H) = I(G) after each operation).

The central algorithm, Method B3, uses these operations to convert any CG G into the minimal I‑map Gα of its induced independence model relative to a given chain α (an ordered partition of the vertex set). Method B3 proceeds in three stages:

1. Construct β – builds an auxiliary chain β that is consistent with G and as close as possible to α. It repeatedly extracts terminal components of the current graph, ordering them by the rightmost node in α, and inserts them as blocks into β. This step ensures that β reflects the current block ordering while respecting the constraints of G.

2. Iterative block alignment – processes the blocks of α from right to left. For each block C of α, it finds the leftmost block K of β that intersects C, defines L = K ∩ C, and applies FbSplit(K, L, G) to separate L from the rest of K. β is updated accordingly. If the right neighbor R of L in β is not positioned to the right of L in α, FbMerge(L, R, G) is applied, merging L and R and updating β. This loop continues until β coincides with α.

3. Termination – when β = α, the algorithm has produced Gα, the unique minimal CG that is an I‑map of the original independence model and is consistent with α.

The authors prove several lemmas that guarantee correctness:

• Lemma 1 shows the existence and uniqueness of the minimal I‑map Gα for any graphoid independence model and any chain α.
• Lemma 2 establishes that for any component C of G, there is a unique maximal component of H containing a descendant of C, which is crucial for defining feasible splits/merges without violating independence.
• Lemma 3 links the ordering of descendants in G with the ordering imposed by α, ensuring that no descendant appears to the left of its ancestor in the target chain.
• Lemma 4 proves that Method B3 terminates after a finite number of operations and that each intermediate graph maintains the inclusion I(H) ⊆ I(G_t).

Consequently, the paper proves the extended Meek conjecture for chain graphs: if I(H) ⊆ I(G) for two CGs G and H, then G can be transformed into H by a finite sequence of directed/undirected edge additions together with feasible splits and merges, while preserving the inclusion after each operation. This result mirrors the original DAG case and provides a constructive algorithm for the transformation.

The practical implication is that score‑based learning algorithms for Bayesian networks can now be generalized to learn chain graphs efficiently and asymptotically correctly under mild assumptions. The feasible split/merge operations enable block‑wise modifications rather than edge‑wise tweaks, potentially reducing computational overhead in high‑dimensional settings. Moreover, because chain graphs subsume both DAGs and undirected graphical models, the work opens a pathway to learning richer mixed‑type dependency structures that capture both causal (directed) and symmetric (undirected) relationships within a unified framework.

In summary, the paper delivers a rigorous theoretical foundation for optimal learning of chain graphs, extending a seminal conjecture, introducing novel graph‑transform operations, and presenting an algorithm (Method B3) that is provably correct, finite, and suitable for building efficient, consistent structure‑learning procedures for mixed graphical models.