Test-Time Learning of Causal Structure from Interventional Data

Wei Chen; Rui Ding; Bojun Huang; Yang Zhang; Qiang Fu; Yuxuan Liang; Han Shi; Dongmei Zhang

cs.LGFeb 22, 2026

Test-Time Learning of Causal Structure from Interventional Data

Authors: Wei Chen, Rui Ding, Bojun Huang, Yang Zhang, Qiang Fu, Yuxuan Liang, Han Shi, Dongmei Zhang

Organizations: HKUST · HKUST(GZ) · Microsoft Research · Sony Research ·

Abstract

Supervised causal learning has shown promise in causal discovery, yet it often struggles with generalization across diverse interventional settings, particularly when intervention targets are unknown. To address this, we propose TICL (Test-time Interventional Causal Learning), a novel method that synergizes Test-Time Training with Joint Causal Inference. Specifically, we design a self-augmentation strategy to generate instance-specific training data at test time, effectively avoiding distribution shifts. Furthermore, by integrating joint causal inference, we developed a PC-inspired two-phase supervised learning scheme, which effectively leverages self-augmented training data while ensuring theoretical identifiability. Extensive experiments on bnlearn benchmarks demonstrate TICL's superiority in multiple aspects of causal discovery and intervention target detection.

Figures & tables

Figure 1: (Empirical dominance across interventional SCL tasks). TICL consistently outperforms all SoTA methods on both

\mathcal{I}

-CPDAG discovery and intervention targets detection, across diverse intervention families and limited-sample regimes.

Figure 2: Left: The workflow of test-time learning of causal structure. Right: An identifiability example.

Figure 3: The overall workflow of TICL for test-time learning of causal structure from interventional data.

Methods	TICL			JCI-PC			JCI-BLIP			JCI-HC			JCI-GOLEM			AVICI ∗			ENCO ∗			IGSP ∗			UT-IGSP			GIES
Methods	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1	SHD	SID	F1
Earthquake	0	0	1.00	0	0	1.00	0	0	1.00	5	14	0.40	6	19	0.00	3	6	0.40	3	12	0.67	0	0	1.00	0	0	1.00	1	0	0.89
Survey	0	0	1.00	5	18	0.40	1	4	0.91	0	0	1.00	7	23	0.40	8	18	0.22	5	21	0.29	1	4	0.91	1	4	0.91	1	4	0.91
Asia	2	13	0.86	4	19	0.75	3	18	0.77	6	28	0.46	15	36	0.00	7	28	0.31	6	13	0.74	4	17	0.62	5	24	0.73	4	14	0.77
Sachs	3	6	0.90	19	56	0.52	7	41	0.67	5	29	0.74	18	61	0.11	37	46	0.22	41	42	0.26	10	16	0.70	13	21	0.61	15	38	0.48
Child	7	67	0.81	40	299	0.62	10	188	0.78	16	211	0.65	41	330	0.21	/	/	/	110	251	0.06	29	124	0.63	33	214	0.45	46	197	0.29

Table 1: Performance comparison of

\mathcal{I}\textit{-CPDAG}

(SHD

\downarrow

/ SID

\downarrow

/ F1

\uparrow

). : Best, : Next. / : Crashes, – : Timeout.

Datasets	Earthquake	Survey	Asia	Sachs	Child	Insurance	Water	Mildew	Alarm	Barley	Hailfinder	Hepar2	Win95pts	Pathfinder	Rank (F1)
UT-IGSP	0.50	1.00	0.44	0.33	0.35	0.17	/	0.22	0.15	/	0.27	0.23	/	0.04	5.14 $\pm$ 3.27
CITE	1.00	1.00	0.67	0.36	0.67	0.32	/	0.38	0.56	/	0.67	0.61	/	0.60	3.07 $\pm$ 2.21
PreDITEr	1.00	1.00	0.67	0.50	–	–	/	–	–	/	–	–	/	/	5.21 $\pm$ 5.17
JCI-GOLEM	0.67	0.40	0.40	0.22	0.18	0.11	0.23	–	–	–	–	–	–	–	6.29 $\pm$ 2.20
JCI-HC	1.00	0.50	0.67	0.57	0.40	0.29	0.33	0.34	0.29	0.26	0.23	0.23	–	–	4.14 $\pm$ 1.98
JCI-BLIP	1.00	1.00	0.80	0.67	0.57	0.45	0.44	0.35	0.38	0.34	0.34	0.82	0.29	0.23	2.43 $\pm$ 0.67

Table 2: Performance comparison of Intervention Targets Detection (F1

\uparrow

). : Best, : Next. / : Crashes, – : Timeout.

Figure 4: The impact of different synthetic data strategies on the performance of causal structure learning.

Figure 5: The impact of different initialization strategies on the performance of IS-MCMC convergence.

Figure 6: Effect of training data instances on the performance of causal structure learning.

Figure 7: Effect of training sample size on the performance of causal structure learning.

Figure 8: Efficiency study of our method.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Figure B.1 : The detailed framework of TICL .

Figure B.2 : Feature extraction example of edge

X_{1}-X_{2}

in 1-order skeleton graph.

Skeleton Feature
Type	Name	Calculate	Example Raw Results	Dimension
CI Test Information	$k$ -order Conditional Dependence	$\begin{Bmatrix}X_{i}\sim X_{j}\big\|\mathcal{S}_{2}\end{Bmatrix},\penalty\ \penalty\ \|\mathcal{S}_{2}\|=2$	[0.6, 0.3, 0.7, 0.4, 0.6, 0.1]	Not Fixed $\rightarrow$ 15
CI Test Information	Residual Conditional Dependence	$\begin{Bmatrix}X_{i}\sim X_{j}\big\|\mathcal{S}_{1}\end{Bmatrix}-\min_{X_{q}}\left\{X_{i}\sim X_{j}\big\|\mathcal{S}_{2}\cup\{X_{q}\}\right\}$ , $\|\mathcal{S}_{1}\|=1,\penalty\ \|\mathcal{S}_{2}\|=2$	[0.4, 0.2, 0.4 ,0.7]	Not Fixed $\rightarrow$ 15
Structural Information	Competitiveness	$\frac{\left\|\left\{X_{q}\|X_{q}\in nbd(X_{i}),{C}_{k-1}(X_{i},X_{j})>{C}_{k-1}(X_{i},X_{q})\right\}\right\|}{\left\|nbd_{\mathcal{G}_{k-1}}(X_{i})\right\|-1},$ $\frac{\left\|\left\{X_{q}\|X_{q}\in nbd(X_{j}),{C}_{k-1}(X_{j},X_{i})>{C}_{k-1}(X_{j},X_{q})\right\}\right\|}{\left\|nbd_{\mathcal{G}_{k-1}}(X_{j})\right\|-1},$ ${C}_{k-1}(X_{i},X_{j})$	[0.5, 0.75, 0.8]	3
	Degree	$deg(X_{i}),deg(X_{j})$	[4, 4]	2
	Density	$\frac{\left\|nbd(X_{i})\cap nbd(X_{j})\right\|}{\min{(\|nbd(X_{i})\|,\|nbd(X_{j})\|)}}$	[1]	1

Appendix

Table B.1 : The features extracted for skeletal learning. As shown in Figure B.2 ,

k=2

, is used to illustrate the design intuition and computational examples.

Figure B.3 : Orientation feature extraction example of unshielded triple

X_{a}-X_{c}-X_{b}

Orientation Feature
Type	Name	Calculate	Example Raw Results	Dimension
CI Test Information	Vicinity Conditional Dependence	$\{\text{Variable}\mid\text{Condition}\},$ $\text{Variable: }\{X_{a}\!\sim\!X_{b},X_{a}\!\sim\!PC_{X_{b}},$ $PC_{X_{a}}\!\sim\!X_{b},PC_{X_{a}}\!\sim\!PC_{X_{b}}\}$ $\text{Condition: }\{\emptyset,SS,X_{c},X_{c}\vee SS,$ $PC_{X_{c}},PC_{X_{c}}\vee SS\}$	[0.9, 0.8, 0.8, 0.6, 0.8, 0.7, 0.7, 0.8, 0.8, 0.7, 0.5, 0.4, 0.8, 0.4, 0.4, 0.3, 0.6, 0.2, 0.9, 0.8, 0.9, 0.7, 0.6, 0.4]	24
Structural Information	Overlap	$\frac{\|set_{1}\cap set_{2}\|}{\min(\|set_{1}\|,\|set_{2}\|)},$ $\text{where }set\in\{PC_{X_{a}},PC_{X_{b}},PC_{X_{c}},\mathbf{SS}\},$ $\text{and }set_{1}=X_{c},\;set_{2}=\mathbf{SS}$	[0.5, 0, 0.5, 0.5, 0, 0.5, 1]	7
Structural Information	Scaling	$\#PC_{X_{a}},\#PC_{X_{b}},\#PC_{X_{c}},$ $\#\mathbf{SS},$ $\frac{1}{\#\mathbf{SS}}\sum_{SS_{i}\in\mathbf{SS}}\#SS_{i}$	$[2,3,2,3,\frac{4}{3}]$	5

Appendix

Table B.2 : The features extracted for orientation learning, as shown in Figure B.3 , are used to illustrate the design intuition and computational examples.

Figure C.1 : Convergence Visualization Analysis on different datasets.

Network	#Nodes	#Edges	#Parameters	Max in-degree	Avg. degree
Earthquake	5	4	10	2	1.60
Survey	6	6	21	2	2.00
Asia	8	8	18	2	2.00
Sachs	11	17	178	3	3.09
Child	20	25	230	2	2.50
Insurance	27	52	1,008	3	3.85

Appendix

Table E.1 : Statistics and description of bayesian networks we used.

Panel of Figure 18 — Figure E.1 : Graph Visualizations.

Methods	Groun-Truth DAG		Testing Data			Intervention Experiment
Methods	Type	#Variable	Type	#Obs.	#Int. / per exp.	Type	#Int. Family	#Int. Target
GIES	Random Graph ( 2007 ) (Linear Gaussian SEM)	10 $\sim$ 50	Continue	50 $\sim$ 10k	50 $\sim$ 10k	Single / Multiple Know Hard	#Var. * { .0 / .2 / .4 / .6 / .8 / 1.0 }	1 $\sim$ 4
IGSP	ER (Linear Gaussian SEM)	10 $\sim$ 20	Continue	1k $\sim$ 100k	1k $\sim$ 100k	Single / Multiple Know Hard / Soft	1 $\sim$ 2	#Var. * { 0.4 / 0.5 }
UT-IGSP	ER (Linear Gaussian SEM)	20	Continue	1k $\sim$ 5k	1 $\sim$ 5k	Single / Multiple Know / Unknow Hard / Soft	Know: (5) + Unknow: (5)	Know: (1) + Unknow: (0 $\sim$ 3)
DCDI	ER (LGM / ANM / NN)	10-20	Continue	10k	10k	Single / Multiple Know / Unknow Hard / Soft	#Variable	#Var. * 0.1
SDI	Random Graph (NN) bnlearn	3 $\sim$ 13	Discrete	2560 (from code)	1111 (from code)	Single Know Soft	#Variable	1

Appendix

Table E.2 : Summary of basic settings for various methods of causal discovery from intervention data

Methods	Causal Insufficiency	Interv. Frequency		Interv. Mechanism		Data Type		Domain
Methods	Causal Insufficiency	Single	Multiple	Hard	Soft	Continue	Discrete	Single	Multiple
UT-IGSP	✗	✓	✓	✓	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✓
CITE	✗	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✗
PreDITEr	✗	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✗
LIT	✓	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✓	$\color[rgb]{0.3555,0.6094,0.8359}{\bm{\sim}}$	✓	✓

Appendix

Table E.3 : Comparison of situations (more suitable) supported by intervention target detection methods.

in Predict→ in Ground-Truth↓	identifiable (directed)		unidentifiable (undirected)	missing in skeleton
	right	wrong
identifiable	➀ ✓	➁ ✗	➂ ✗	➃ ✗
unidentifiable	➄ ✗		➅ ✓	➆ ✗
nonexist	➇ ✗		➈ ✗	➉ ✓

Appendix

Table E.4 : SHD calculation details.

Category	Detail	Hyperparameters or Settings
Training Data	#Training Instances	400 (default)
		100 $\sim$ 1600 (others, see 4.2 )
	#Simples / per Instance	10k (default)
		1k $\sim$ 20k (others, see 4.2 )
Testing Data	#Observation Sample	10k (default)
		2k $\sim$ 10k (others, see F.2.2 )

Appendix

Table E.5 : Basic configuration of our experiments and methods

Datasets #nodes / #edges	Metric	TICL	JCI-GOLEM	JCI-PC	JCI-BLIP	JCI-HC	AVICI ∗	ENCO ∗	IGSP ∗	GIES	UT-IGSP
Earthquake 5 / 4	F1 Score	1.00	0.00	1.00	1.00	0.00	0.29	0.67	1.00	1.00	1.00
	SHD	0	6	0	0	4	4	2	0	0	0
	SID	0	13	0	0	20	10	2	0	0	0
Survey 6 / 6	F1 Score	1.00	0.40	0.40	0.73	0.73	0.00	0.44	0.91	0.91	0.91
	SHD	0	7	4	2	2	6	4	1	1	1
	SID	0	23	18	9	9	17	16	4	4	4

Appendix

Table F.1 : Performance comparison of

\mathcal{I}\textit{-CPDAG}

under perfect intervention (F1 Score

\uparrow

/ SHD

\downarrow

/ SID

\downarrow

Datasets	Earthquake	Survey	Asia	Sachs	Child	Insurance	Water
Datasets	1	1	2	2	4	5	6
UT-IGSP	0.40	0.50	0.44	0.40	0.36	0.22	/
CITE	1.00	1.00	1.00	0.57	0.40	0.55	/
PreDITEr	0.67	/	1.00	0.67	–	–	/
JCI-GOLEM	0.67	1.00	0.00	0.18	0.22	0.28	0.21
JCI-HC	0.50	1.00	0.36	0.67	0.47	0.34	0.32

Appendix

Table F.2 : Performance comparison of Intervention Targets Detection under perfect intervention (F1 Score

\uparrow

Datasets #nodes / #edges	Metric	TICL	JCI-GOLEM	JCI-PC	JCI-BLIP	JCI-HC	IGSP ∗	GIES	UT-IGSP
Earthquake 5 / 4	F1 Score	0.60	0.00	0.55	1.00	0.25	1.00	0.40	1.00
	SHD	4	5	4	0	6	0	7	0
	SID	9	18	3	0	15	0	6	0
Survey 6 / 6	F1 Score	0.83	0.00	0.40	0.91	0.83	0.91	0.00	0.91
	SHD	2	9	4	1	1	1	6	1
	SID	10	30	18	4	6	4	30	4

Appendix

Table F.3 : Performance comparison of

\mathcal{I}\textit{-CPDAG}

under multiple interventions (F1 Score

\uparrow

/ SHD

\downarrow

/ SID

\downarrow

Datasets	Earthquake	Survey	Asia	Sachs	Child	Insurance	Water
Datasets	3	3	6	6	4	5	12
UT-IGSP	0.86	0.86	0.67	0.57	0.80	0.24	/
CITE	0.80	1.00	0.67	0.43	0.33	0.71	/
PreDITEr	0.80	0.50	/	0.29	–	–	/
JCI-GOLEM	0.33	0.80	0.62	0.47	0.21	0.14	0.23
JCI-HC	0.57	1.00	0.67	0.67	0.50	0.33	0.38

Appendix

Table F.4 : Performance comparison of Intervention Targets Detection under multiple interventions (F1 Score

\uparrow

Datasets #nodes / #edges	Metric	TICL	JCI-GOLEM	JCI-PC	JCI-BLIP	JCI-HC	AVICI ∗	ENCO ∗	IGSP ∗	GIES	UT-IGSP
Earthquake 5 / 4	F1 Score	1.00	0.00	0.89	1.00	1.00	0.00	0.67	1.00	1.00	1.00
	SHD	0	7	1	0	1	4	2	0	0	0
	SID	0	12	0	0	0	10	2	0	0	0
Survey 6 / 6	F1 Score	0.73	0.00	0.33	0.73	0.00	0.00	0.60	0.91	0.91	0.91
	SHD	2	11	6	2	7	5	3	1	1	1
	SID	9	30	22	9	30	17	14	4	4	4

Appendix

Table F.5 : Performance comparison of

\mathcal{I}\textit{-CPDAG}

under limited interventional data sample sizes (F1

\uparrow

/ SHD

\downarrow

/ SID

\downarrow

Datasets	Earthquake	Survey	Asia	Sachs	Child	Insurance	Water
Datasets	1	1	2	2	4	5	6
UT-IGSP	0.33	1.00	0.57	1.00	0.36	0.27	/
CITE	1.00	0.67	0.67	1.00	0.40	0.22	/
PreDITEr	0.67	1.00	/	0.40	0.33	–	/
JCI-GOLEM	0.00	1.00	0.00	0.27	0.15	0.11	0.00
JCI-HC	1.00	0.67	0.50	0.80	0.40	0.37	0.36

Appendix

Table F.6 : Performance comparison of Intervention Targets Detection under limited interventional data sample sizes (F1

\uparrow

Datasets #nodes / #edges	Metric	TICL	JCI-GOLEM	JCI-PC	JCI-BLIP	JCI-HC	AVICI ∗	ENCO ∗	IGSP ∗	GIES	UT-IGSP
Earthquake 5 / 4	F1 Score	0.86	0.00	0.86	1.00	1.00	0.17	0.57	1.00	0.00	1.00
	SHD	1	6	1	0	0	8	3	0	4	0
	SID	1	16	1	0	0	13	2	0	16	0
Survey 6 / 6	F1 Score	0.73	0.00	0.40	0.73	0.29	0.00	0.25	0.50	0.36	0.50
	SHD	2	12	4	2	6	7	5	4	4	4
	SID	11	30	18	9	29	22	18	11	20	11

Appendix

Table F.7 : Performance comparison of

\mathcal{I}\textit{-CPDAG}

under limited observational data sample sizes (F1

\uparrow

/ SHD

\downarrow

/ SID

\downarrow

Datasets	Earthquake	Survey	Asia	Sachs	Child	Insurance	Water
Datasets	1	1	2	2	4	5	6
UT-IGSP	0.33	0.67	0.40	0.67	0.81	0.24	/
CITE	/	1.00	/	1.00	0.33	0.38	/
PreDITEr	/	1.00	/	/	0.40	–	/
JCI-GOLEM	0.00	0.00	0.00	0.27	0.15	0.27	0.20
JCI-HC	1.00	0.67	0.44	0.50	0.57	0.37	0.30

Appendix

Table F.8 : Performance comparison of Intervention Targets Detection under limited observational data sample sizes (F1

\uparrow

Figure F.1 : Exploration of the impact of different size of intervention experiments on performance

Methods	Learning Paradigm	Problem Space	Intervention Targets						Test-time Training
Methods	Learning Paradigm	Problem Space	Unknown	Known	Hard	Soft	Single	Multiple	Test-time Training
DAG-GFlowNet ( Deleu et al., 2022 )	Unsupervised Generative Learning	Interventional (Passive)	✗	✓	✓	✗	✓	✓	✗
GFlowCausal ( Li et al., 2022 )	Unsupervised Generative Learning	Observational	✗	✗	✗	✗	✗	✗	✗
CORE ( Sauter et al., 2024 )	Reinforcement Learning	Interventional (Active)	✗	✓	✓	✗	✓	✓	✗
meta-CGNN ( Ton et al., 2021 )	Meta-Learning	Observational (Bivariate)	✗	✗	✗	✗	✗	✗	✗
DAG-GNN ( Yu et al., 2019 )	Unsupervised Learning	Observational	✗	✗	✗	✗	✗	✗	✗

Appendix

Table G.1 : Comparison of different methods on causal discovery.

Feature	Other MCMC sampling methods ( Ellis and Wong, 2008 ; Choi et al., 2020 )	IS-MCMC (Our)
Sampling Space	Standard graph space (for posterior estimation)	Intervention-augmented graph space (JCI framework)
Intervention Handling	Cannot handle unknown intervention targets	Handles unknown intervention targets
Usage	Posterior inference	Generates diverse $\langle G_{i},D_{i}\rangle$ training pairs for TTT
Constraints	Standard graph constraints	Enforces intervention constraints
Objective	Posterior estimation	Data generation for adaptation

Appendix

Table G.2 : Comparison of IS-MCMC and Other MCMC Methods

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