Demistifying Data and Simulator Assumptions in Supervised Causal Discovery

Pingchuan Ma; Rui Ding; Bojun Huang; Shuai Wang

cs.AISep 29, 2026

Demistifying Data and Simulator Assumptions in Supervised Causal Discovery

Authors: Pingchuan Ma, Rui Ding, Bojun Huang, Shuai Wang

Organizations: Zhejiang University of Technology, China · Microsoft Research Asia · Sony Research · HKUST, Hong Kong SAR

Abstract

Supervised causal discovery learns to infer causal structure for a new dataset from training datasets paired with structural labels. These training pairs are typically simulated, making the simulator both a source of supervision and a carrier of assumptions about causal graphs, mechanisms, and noise. Understanding the resulting predictions therefore requires examining how these assumptions supplement the information available in observational data, which may be compatible with multiple causal graphs. This paper examines that relationship across representative methods available through June 2026. We organize these methods by prediction target, prediction granularity, encoder, structural decoder, and training regime to relate what each method predicts to how it uses data and simulator-based supervision. Using this framework, we distinguish two questions: whether the target is identifiable under the assumed model class, and whether a trained predictor generalizes beyond its training distribution. Restrictions on mechanisms and noise can make otherwise ambiguous causal directions identifiable, but predictive accuracy under those restrictions does not establish transfer when they change. This distinction motivates evaluation that matches metrics to the identifiable graph target and tests changes in graphs, mechanisms, and noise between training and deployment. Extending such evaluation to real data also requires documenting the external causal evidence and uncertainty behind benchmark reference graphs. Together, these analyses guide method comparison and identify open questions in transfer, test-time adaptation, and uncertainty assessment.

Figures & tables

Method	Target / Assumption	Granularity	Encoder	Decoder	Training Regime
Local and global supervised predictors
RCC [ 13 ]	Pairwise relation (sim. prior)	Bivariate pair	Kernel mean embedding	Classifier	Synthetic pairs; transductive tuning on Tübingen inputs
ML4C [ 16 ]	CPDAG given skeleton (A1–A3)	Unshielded triple	Data + skeleton; vicinity statistics	Classifier + orientation rules	Synthetic triple labels
DAG-EQ [ 14 ]	DAG (sim. prior)	DAG	Correlation + EQ network	Edge scores + cycle-avoiding assembly	Synthetic DAG tasks
CSIvA [ 17 ]	DAG labels (sim. prior)	Graph	Transformer (alt. attn.)	Sequential; no DAG constraint	Observational / interventional tasks
SiCL [ 11 ]	CPDAG (A1–A3)	CPDAG	Transformer (feature attn.)	CPDAG-aware decoder	Skeleton / collider labels

TABLE I: Representative methods compared by prediction target, granularity, model components, and training regime. The assumptions qualify the interpretation of the target. A simulator prior defines a prediction problem but does not by itself guarantee identification. ADAG is included as a related reconstruction approach without supervised graph labels.

Method	Simulation Target	Instantiation
RCC [ 13 ]	Synthetic Bivariate ANM	$\begin{gathered}\eta=(P_{X},f,P_{N},n)\sim\Pi_{\mathrm{ANM}},\hskip 8.19447ptX\sim\sum_{k}\pi_{k}\mathcal{N}(\mu_{k},\sigma_{k}^{2}),\hskip 8.19447ptY=f(X)+N,\\ N\perp\!\!\!\perp X,\hskip 8.19447ptN\sim\mathcal{N}(0,\sigma_{N}^{2}),\hskip 8.19447ptf\sim\Pi_{\mathrm{spline}},\hskip 8.19447pt\tau(\eta)=X\!\to\!Y,\hskip 8.19447pt(Y,X)\mapsto Y\!\to\!X.\end{gathered}$
	Transductive Tübingen Setting	Generator parameters are fitted against unlabeled test-pair embeddings. Training labels come from synthetic pairs; Tübingen directions are evaluation labels. A separate ChaLearn experiment uses labeled pairs.
ML4S [ 15 ]	Dataset-Specific Discrete Skeleton	$\begin{gathered}(G_{0},\theta_{0})=\operatorname{FitBN}(D_{\mathrm{test}}),\hskip 8.19447ptG^{\prime}\sim q_{\mathrm{vic}}(\cdot\mid G_{0}),\\ \theta^{\prime}=\operatorname{AdjustCPT}(G^{\prime},G_{0},\theta_{0}),\hskip 8.19447ptD^{\prime}\sim P_{G^{\prime},\theta^{\prime}}^{(n)},\\ \eta=(G^{\prime},\theta^{\prime},n),\hskip 16.38895pt\tau(\eta)=\operatorname{skel}(G^{\prime}).\end{gathered}$
SPOT [ 24 ]	Linear Gaussian MAG Skeleton	$\begin{gathered}\eta=(G,H,\theta,n)\sim\Pi_{\mathrm{ADMG}},\hskip 8.19447ptM=\operatorname{MAG}(G,H),\\ D\sim P_{\eta}^{(n)},\hskip 8.19447pt\tau(\eta)=\operatorname{skel}(M).\end{gathered}$
ML4C [ 16 ]	Discrete V-Structure	$\begin{gathered}\eta=(G,\theta,n)\sim\Pi_{\mathrm{ML4C}},\hskip 8.19447ptG\sim\Pi_{\mathrm{ER/SF}},\hskip 8.19447ptD\sim\prod_{i}P_{\theta_{i}}(X_{i}\mid X_{\operatorname{Pa}_{G}(i)}),\\ (i,k,j)\in\operatorname{UT}(G),\hskip 8.19447pt\tau(\eta)=\operatorname{vstr}(G).\end{gathered}$
DAG-EQ [ 14 ]	Equal-Variance Gaussian DAG	$\begin{gathered}\eta=(G,B_{G},n)\sim\Pi_{\mathrm{LG}},\hskip 8.19447ptG\sim\Pi_{\mathrm{ER/SF}},\hskip 8.19447ptX=B_{G}^{\top}X+\varepsilon,\hskip 8.19447pt\varepsilon\sim\mathcal{N}(0,I),\\ D=\{x^{(r)}\}_{r=1}^{n},\hskip 16.38895pt\tau(\eta)=A(G).\end{gathered}$

TABLE II: Simulators used by representative supervised causal discovery methods. Here

\eta

denotes one simulated data-generating instance;

\tau(\eta)

denotes the structural target. For ADAG, the structure is an evaluation target, not a supervised training label.

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