MosaCD: LLM-Assisted Causal Discovery

• MosaCD is a constraint-based causal discovery method that uses high-confidence seed sets derived from CI tests and LLM annotations to robustly orient causal edges from observational data.
• It addresses cascading errors common to traditional PC-style methods by employing a confidence-down propagation strategy that prioritizes non-collider evidence and orders decisions by statistical support.
• Empirical evaluations on BNLearn benchmarks demonstrate that MosaCD achieves higher F1 scores and lower false orientation rates compared to standard methods, reflecting its improved reliability.

Searching arXiv for the MosaCD paper and closely related causal-discovery references.

MosaCD is a constraint-based causal discovery method for learning causal structure from observational data. It is designed for the standard setting in which an unknown causal directed acyclic graph $G=(V,E)$ must be recovered, or approximated up to its Markov equivalence class, from conditional independence relations inferred from data. The method is introduced as a response to a specific weakness of classical PC-style workflows: they orient v-structures first and then propagate directions from those initial seeds, which can produce cascading orientation errors when conditional independence tests are noisy or separating-set search is incomplete. MosaCD addresses this by combining two ingredients: a high-confidence seed set derived from both conditional-independence evidence and LLM annotations, and a confidence-down propagation strategy that orients the most reliable edges first (Lyu et al., 28 Sep 2025). The paper does not define MosaCD as a variant of “Mosaic,” “MOS,” or “MOSAIK”; it presents MosaCD as a distinct causal-discovery method (Lyu et al., 28 Sep 2025).

1. Problem formulation and methodological setting

MosaCD operates in the classical constraint-based setting for causal discovery. The unknown target is a DAG

$G=(V,E),$

where $V$ is the set of observed variables and $E$ is the set of causal edges. As in standard constraint-based methods, the starting point is a skeleton learner that uses conditional independence tests to construct an undirected skeleton together with a record of separating sets. The paper explicitly positions MosaCD relative to algorithms such as PC, PC-stable, and CPC, and states that it can be integrated with any skeleton-based discovery method (Lyu et al., 28 Sep 2025).

The central technical motivation is that standard collider orientation relies on the absence of a middle node from recorded separating sets. For an unshielded triple $X-Z-Y$, the classical criterion is: $$Z \notin S \quad \text{for all } S \in \Sigma(X,Y),$$ which leads to the orientation

$X \to Z \leftarrow Y.$

MosaCD argues that this criterion is fragile in finite samples because a large $p$-value may reflect low power rather than true independence, and because separating-set search may be incomplete (Lyu et al., 28 Sep 2025). This shifts attention from collider-first orientation toward more conservative handling of seed directions.

A further aspect of the problem setting is the introduction of LLM-derived directional information. The paper treats LLM outputs not as authoritative causal judgments but as potentially useful priors that require filtering, because direct use of LLM answers is vulnerable to hallucination, inconsistency, and positional bias (Lyu et al., 28 Sep 2025). This places MosaCD at the intersection of constraint-based causal discovery and LLM-assisted orientation, while keeping the statistical skeleton-learning stage separate.

2. Core architecture of MosaCD

At a high level, MosaCD proceeds in five stages. First, it runs a skeleton search procedure to obtain an undirected skeleton $G_{\text{skel}}$ and a separating-set record $\Sigma$. Second, it constructs initial orientation seeds using repeated shuffled LLM queries together with CI/separating-set context. Third, it performs iterative orientation propagation using a confidence-down strategy. Fourth, it applies a least-conflict orientation step to remaining undirected edges. Fifth, if undirected edges still remain, it optionally aggregates LLM votes into a weighted directed graph and uses that graph to complete orientation (Lyu et al., 28 Sep 2025).

The method’s most distinctive design choice is that it does not simply append LLM answers to a conventional PC pipeline. Instead, it redesigns the orientation stage itself so that seed construction is conservative and propagation is ordered by reliability. This differs from prior LLM-assisted approaches that use LLMs before discovery as priors or after discovery for post hoc orientation, while leaving the core propagation logic unchanged (Lyu et al., 28 Sep 2025).

The paper states that the seed set is derived from both CI tests and LLM annotations, but it is also explicit that MosaCD does not introduce a new skeleton-learning procedure. Its contribution is primarily an orientation layer on top of skeleton-based discovery (Lyu et al., 28 Sep 2025).

3. Seed construction and reliable LLM priors

MosaCD’s seed construction stage is built around repeated LLM querying with shuffled answer orderings. For each undirected edge $G=(V,E),$0, the LLM receives variable names and descriptions together with CI-related information, including minimal separating sets and associated $G=(V,E),$1-values. The prompts may also include “neighbor chain(s) that must normally remain non-collider,” and they explicitly allow an undecided answer (Lyu et al., 28 Sep 2025).

The key filtering device is a shuffled-query procedure intended to exploit LLM positional bias. The method randomizes the order of candidate answers, repeats the query 5 times, and uses both answer orderings in which the two possible directions appear in swapped positions. An edge direction is retained only if it is the majority vote in both orderings after mapping answers back to the same semantic direction (Lyu et al., 28 Sep 2025). This is the paper’s operational definition of a high-confidence LLM seed.

A proposed seed $G=(V,E),$2 is discarded if it contradicts $G=(V,E),$3 on any unshielded triple or if it creates a directed or semi-directed cycle in the current partially directed graph. The resulting seed set is intentionally sparse and conservative: the method prefers abstention to injecting low-confidence orientation errors (Lyu et al., 28 Sep 2025).

The paper does not present a continuous probabilistic confidence score for seeds. Instead, confidence is procedural: consistency under shuffled repeated prompts, compatibility with the separating-set record, and acyclicity. This suggests that MosaCD treats seed quality as a structural filtering problem rather than a calibrated uncertainty-estimation problem.

4. Confidence-down propagation

The second defining component of MosaCD is its confidence-down propagation strategy. Standard PC-like methods orient colliders first and then apply propagation rules. MosaCD instead repeatedly applies three orientation steps, prioritizing stronger CI support and favoring non-collider evidence before collider evidence (Lyu et al., 28 Sep 2025).

The first rule is unsupervised acyclic propagation. If an undirected edge $G=(V,E),$4 exists and there is already a semi-directed path

$G=(V,E),$5

in the current PDAG $G=(V,E),$6, then MosaCD orients

$G=(V,E),$7

This is an acyclicity-preserving propagation rule closely related in spirit to Meek-style reasoning (Lyu et al., 28 Sep 2025).

The second rule is CI-supervised propagation on partially ordered triples. For each unshielded partially ordered triple

$G=(V,E),$8

triples are processed in descending order of the strongest available independence evidence, represented by the largest $G=(V,E),$9-value among recorded separating sets for $V$0 and $V$1. If $V$2 appears in all minimal separating sets in $V$3, the method orients

$V$4

If $V$5 appears in none of the minimal separating sets in $V$6, it orients

$V$7

The paper emphasizes that the first case is treated as more reliable than classical collider evidence, because inclusion of $V$8 in a separating set is positive evidence whereas exclusion may be accidental (Lyu et al., 28 Sep 2025).

The third rule is collider orientation on fully unordered triples. For each unshielded triple

$V$9

again sorted by descending $E$0 in $E$1, MosaCD orients

$E$2

if $E$3 appears in none of the minimal separating sets. Thus, collider orientation is not abandoned, but it is delayed and reliability-ordered (Lyu et al., 28 Sep 2025).

This ordering is the core of “confidence-down” propagation. It is not based on a learned confidence predictor; instead, it uses the strength of CI support and a structural preference for non-collider-supported decisions.

5. Least-conflict completion and optional final orientation

After iterative propagation, MosaCD applies a least-conflict orientation procedure to remaining undirected edges. For an undirected edge $E$4, it tests both candidate directions, closes the graph under the orientation rules, counts how many recorded statements in $E$5 are contradicted, and chooses the orientation with fewer conflicts. If the two choices are tied, the edge may remain undirected (Lyu et al., 28 Sep 2025).

The paper then describes an optional final orientation stage. If undirected edges still remain, MosaCD aggregates LLM votes into a weighted directed graph, removes weak edges to break cycles, derives a topological order, and orients the remaining edges accordingly (Lyu et al., 28 Sep 2025). The paper does not present this stage as the main contribution; rather, it functions as a cleanup step after the more principled seed-and-propagate process.

A plausible implication is that the method is designed to extract as much reliable orientation information as possible from CI structure before resorting to vote aggregation. This preserves the role of LLMs as filtered auxiliary experts rather than replacing statistical constraints with semantic priors.

6. Formal properties and relationship to classical methods

MosaCD’s theoretical statement is a correctness theorem under ideal assumptions. If the Causal Markov condition and Adjacency-Faithfulness hold, the CI oracle is perfect, the skeleton is correct, and the initial seed set is both $E$6-consistent and acyclic, then running the confidence-down propagation stage to convergence returns the CPDAG of the true DAG, and the least-conflict stage adds nothing further (Lyu et al., 28 Sep 2025).

An important special case is also stated: when the seed set is empty, MosaCD’s propagation stage returns the same PDAG as PC, PC-stable, and CPC (Lyu et al., 28 Sep 2025). This situates MosaCD as a conservative extension rather than a replacement of standard constraint-based orientation in the oracle regime.

The paper also introduces a stylized error-ratio analysis comparing collider-first and non-collider-first identification. It defines

$E$7

Under the paper’s asymptotic model, this ratio exceeds 1 for both PC and CPC. For PC, the expression is

$E$8

The interpretation given is combinatorial: there are more candidate conditioning subsets that do not contain the middle node than ones that do, making collider-first strategies more error-prone (Lyu et al., 28 Sep 2025).

This theoretical analysis does not amount to a finite-sample guarantee for MosaCD. Rather, it supports the paper’s design choice to prioritize non-collider evidence and to sort decisions by CI strength.

7. Empirical evaluation and observed behavior

The empirical study uses 10 BNLearn benchmark datasets: Cancer, Asia, Child, Insurance, Water, Mildew, Alarm, Hailfinder, Hepar2, and Win95pts, spanning 5 to 76 nodes. For simulated datasets, the paper uses 20,000 samples. Skeletons are generated with PC, PC-stable, and CPC, and the orientation baselines include PC, Meek, Shapley-PC, ILS-CSL, and SCP. All LLM-based methods use GPT-4o-mini and the same variable metadata (Lyu et al., 28 Sep 2025).

The evaluation metric is F1 score for detecting true edge orientations. With a PC skeleton, MosaCD achieves the best F1 on 9 of 10 datasets. Reported values include 0.90 on Child, 0.87 on Insurance, 0.90 on Mildew, 0.93 on Alarm, 0.72 on Hepar2, and 0.81 on Win95pts; Water is the one dataset where MosaCD is second-best, with 0.59 against 0.60 for ILS-CSL (Lyu et al., 28 Sep 2025). The performance pattern remains favorable with PC-stable and CPC skeletons.

The paper’s most direct evidence for its seed-construction strategy is the comparison of seed quality. Averaged across datasets and skeleton methods, MosaCD identifies $E$9 more true directions than PC seeding and yields only 4.8% false directions, compared with 26.7% for PC seeds (Lyu et al., 28 Sep 2025). Even on datasets with less informative descriptions, such as Hailfinder and Win95pts, MosaCD’s average false-seed rate is reported as 16.3%, versus 21.7% for PC (Lyu et al., 28 Sep 2025).

Ablations further support the two main components. When variable descriptions are made uninformative, performance declines and fewer true seeds are produced, but the method still generates only a small number of false seeds (Lyu et al., 28 Sep 2025). In propagation experiments that vary the number of true seeds or the proportion of false seeds, MosaCD consistently outperforms Meek’s propagation, which the paper interprets as evidence that confidence-ordered propagation is more robust to imperfect seeds (Lyu et al., 28 Sep 2025).

The method is also evaluated across several LLM backbones, including Claude-3.5-Haiku, GPT-4o-mini, GPT-5, Claude-Sonnet-4, and GPT-oss-120b. The paper reports consistent performance across these models, with GPT-oss-120b somewhat weaker (Lyu et al., 28 Sep 2025). This suggests that the shuffled-query filtering strategy is not tightly coupled to a single language-model family.

8. Scope, limitations, and interpretation

MosaCD is best understood as an orientation-focused extension of skeleton-based causal discovery. It is particularly suited to settings where the skeleton is reasonably good, CI tests are informative but imperfect, and variable names or descriptions carry enough semantics for LLMs to contribute filtered directional priors (Lyu et al., 28 Sep 2025).

The method does not solve adjacency recovery errors produced by the skeleton learner. If the underlying skeleton omits a true edge or includes a spurious one, MosaCD inherits that structural limitation (Lyu et al., 28 Sep 2025). Its theoretical result also assumes a DAG setting without latent confounding in the main formulation; the paper does not extend the method to PAG or FCI-style discovery (Lyu et al., 28 Sep 2025).

Another limitation is computational. Every undirected edge may require multiple LLM calls under multiple answer orderings, so the query budget can become substantial (Lyu et al., 28 Sep 2025). In addition, the method’s notion of confidence remains heuristic rather than probabilistically calibrated: majority consistency under shuffling and ordering by $X-Z-Y$0 provide robustness, but not a formal posterior uncertainty model (Lyu et al., 28 Sep 2025).

A common misconception would be to read MosaCD as an “LLM causal discovery” method in which LLMs replace statistical testing. The paper does not support that interpretation. MosaCD keeps CI-based skeleton learning and uses LLMs only after strict filtering, mainly to supply high-confidence seeds that can be propagated cautiously (Lyu et al., 28 Sep 2025). Another misconception would be to treat it as equivalent to standard PC with extra edge votes; the method’s main novelty lies in the redesigned propagation order and the distinction between stronger non-collider evidence and weaker collider evidence (Lyu et al., 28 Sep 2025).

In the broader literature, this places MosaCD near PC-family methods in terms of statistical backbone, but closer to recent hybrid approaches in its use of external semantic priors. This suggests a broader methodological trend: rather than replacing causal-discovery algorithms with foundation models, stronger results may come from integrating language-model outputs into precisely defined structural subroutines.