DeDiCa: A Scalable Deconfounding Discrete Causal Generative Model

Kunchangtai Liang; Adrián Javaloy; Antonio Vergari; Isabel Valera

cs.LGOct 16, 2026

DeDiCa: A Scalable Deconfounding Discrete Causal Generative Model

Authors: Kunchangtai Liang, Adrián Javaloy, Antonio Vergari, Isabel Valera

Organizations: Department of Computer Science, Saarland University, Saarbrücken, Germany · Department of Computer Science, University of Bath, Bath, United Kingdom · School of Informatics, University of Edinburgh, Edinburgh, United Kingdom

Abstract

We introduce DeDiCa, a practical framework for efficient causal inference with discrete variables and hidden confounding that revisits classical probabilistic modeling through a tensorized parameterization of causal Bayesian networks, enabling tractable inference. Given a causal graph, DeDiCa is trained once on observational data by maximum likelihood, exactly marginalizing over hidden confounders, and can then estimate any (proxy-)identifiable interventional query. Despite this deliberately simple parameterization, DeDiCa matches state-of-the-art neural causal generative models in ATE estimation, while supporting memory-efficient CPU training and exact interventional inference in milliseconds. Compared with causal foundation models, DeDiCa scales more effectively with additional observational samples and achieves competitive accuracy at substantially lower inference time and memory. DeDiCa provides a tractable, accurate, and scalable approach to causal inference in discrete settings with hidden confounding.

Figures & tables

Panel of Figure 1 — Figure 1: Our proposed pipeline to build, learn and use DeDiCa . First, we define and build a tensorized confounded CBN (T-CCBN) guided by a known causal graph and the observational training data. Then, we specialize the T-CCBN with a join tree to optimize the computation of causal queries. Next, we learn the parameters using the observational data via maximum likelihood, leading to the proposed DeDiCa which can then tractably and accurately estimate (proxy-)identifiable queries without retraining.

Table 1: Examples of tensors and tensor operations over binary variables

X

Y

and

Z

. Note that we flattened

(X,Y)

in (f) on top of the product to ease its representation.

Panel of Figure 2 — Figure 2: Prior proxy-identifiability works [ 24 , 2 , 20 , 47 ] have mainly focused on traditional proxies (a). Yet, we can have proxies through mechanisms such as instrumental variables (IVs, b) or colliders (c). Dashed edges are optional and do not invalidate identifiability.

Panel of Figure 3 — Figure 3: The same join tree for the causal graph in Fig. 1 used to compute an observational (a) and interventional query (b). The labels on the edges indicate the size of intermediate tensors sent to the next node, and $\varnothing$ denotes a no-operation used to maintain a binary tree structure in the join tree.

		DeDiCa baseline with $\bm{Z}$ observed	DeDiCa with $\mathcal{C}_{Z_{i}}$ equal to…
		DeDiCa baseline with $\bm{Z}$ observed	$\nicefrac{{2}}{{3}}\,\text{GT}$	GT	$\nicefrac{{4}}{{3}}\,\text{GT}$	$2\,\text{GT}$
Sachs (GT $=3$ , est. $=2$ )	Val NLL	$6.322_{\pm 0.000}$	$6.333_{\pm 0.012}$	$6.323_{\pm 0.001}$	$6.322_{\pm 0.000}$	$6.322_{\pm 0.000}$
Sachs (GT $=3$ , est. $=2$ )	Test ATE MAE	$0.001_{\pm 0.000}$	$0.043_{\pm 0.031}$	$0.071_{\pm 0.023}$	$0.059_{\pm 0.021}$	$0.071_{\pm 0.029}$
Child (GT $=6$ , est. $=6$ )	Val NLL	$11.563_{\pm 0.000}$	$11.600_{\pm 0.011}$	$11.565_{\pm 0.003}$	$11.563_{\pm 0.000}$	$11.563_{\pm 0.000}$
Child (GT $=6$ , est. $=6$ )	Test ATE MAE	$0.001_{\pm 0.000}$	$0.013_{\pm 0.002}$	$0.007_{\pm 0.013}$	$0.016_{\pm 0.020}$	$0.009_{\pm 0.006}$

Table 2: DeDiCa obtains the best performance at the estimated value of hidden categories. Results show validation log-likelihood and test ATE MAE for different values of

\mathcal{C}_{\bm{Z}}

, with 95%-CI shown as error range. Here, GT denotes the ground-truth value of

\mathcal{C}_{Z_{i}}

, and est. the estimation using Section 4.2 . We set all

Z_{i}\in\bm{Z}

to the same

\mathcal{C}_{Z_{i}}

		Asia			Sachs (binarized)			Child (binarized)			Ecoli70
$\bm{Z}$	Model	ATE	Train	Eval	ATE	Train	Eval	ATE	Train	Eval	ATE	Train	Eval
observed	CausalPC	$0.289$	$667.44\text{\,}\mathrm{s}$	$0.81\text{\,}\mathrm{s}$	$0.193$	$1094.09\text{\,}\mathrm{s}$	$0.45\text{\,}\mathrm{s}$	$0.312$	$1841.70\text{\,}\mathrm{s}$	$0.44\text{\,}\mathrm{s}$	$0.188$	$4847.21\text{\,}\mathrm{s}$	$1.06\text{\,}\mathrm{s}$
	CNF	$0.009$	$2312.28\text{\,}\mathrm{s}$	$38.82\text{\,}\mathrm{s}$	$0.023$	$2585.43\text{\,}\mathrm{s}$	$160.84\text{\,}\mathrm{s}$	$0.031$	$2947.71\text{\,}\mathrm{s}$	$234.59\text{\,}\mathrm{s}$	$0.050$	$4821.50\text{\,}\mathrm{s}$	$563.58\text{\,}\mathrm{s}$
	DeDiCa	$0.000$	$0.30\text{\,}\mathrm{s}$	$1.90\text{\,}\mathrm{ms}$	$0.024$	$0.57\text{\,}\mathrm{s}$	$2.62\text{\,}\mathrm{ms}$	$0.011$	$1.45\text{\,}\mathrm{s}$	$4.19\text{\,}\mathrm{ms}$	$0.001$	$9.33\text{\,}\mathrm{s}$	$11.53\text{\,}\mathrm{ms}$
hidden	NCM †	$0.002$	$7240.24\text{\,}\mathrm{s}$	$3.87\text{\,}\mathrm{s}$	$0.047$	$106\,962.04\text{\,}\mathrm{s}$	$20.60\text{\,}\mathrm{s}$	$0.207$ ∗	$111\,661.80\text{\,}\mathrm{s}$ ∗	$22\,306.46\text{\,}\mathrm{s}$	OOM	OOM	OOM
	GAN-NCM †	$0.026$	$6069.06\text{\,}\mathrm{s}$	$0.38\text{\,}\mathrm{s}$	$0.097$	$4035.75\text{\,}\mathrm{s}$	$0.24\text{\,}\mathrm{s}$	$0.084$	$18\,801.49\text{\,}\mathrm{s}$	$0.69\text{\,}\mathrm{s}$	$0.103$	$81\,119.67\text{\,}\mathrm{s}$	$2.17\text{\,}\mathrm{s}$

Table 3: DeDiCa scales well while providing accurate causal estimates under hidden confounding. Results show ATE MAE, training time and evaluation time for one query, averaged over 10 runs. Aside from CausalPC which requires per-query training, the rest train only once. The NCMs run on GPU (denoted † ), and others on CPU. Here, OOM denotes out-of-memory (

>

64\text{,}\mathrm{GiB}

CPU) and an asterisk ∗ a timeout ( $>$120\,000\text{\,}\mathrm{s}

). Bold indicates best non-overlapping confidence intervals per dataset, see Tab. 6 for full details.

	Model	ATE	Train	Eval	ATE	Train	Eval	ATE	Train	Eval	ATE	Train	Eval
		Asia			Sachs (binarized)			Child (binarized)			Ecoli70
$\bm{Z}$ is observed	PFN †	$0.016$	—	$331.73\text{\,}\mathrm{s}$	$0.020$	—	$368.31\text{\,}\mathrm{s}$	$0.014$	—	$539.50\text{\,}\mathrm{s}$	$0.055$	—	$746.75\text{\,}\mathrm{s}$
$\bm{Z}$ is observed	DeDiCa	$0.001$	$77.66\text{\,}\mathrm{ms}$	$2.15\text{\,}\mathrm{ms}$	$0.024$	$0.19\text{\,}\mathrm{s}$	$2.18\text{\,}\mathrm{ms}$	$0.012$	$0.53\text{\,}\mathrm{s}$	$3.42\text{\,}\mathrm{ms}$	$0.001$	$2.93\text{\,}\mathrm{s}$	$8.74\text{\,}\mathrm{ms}$
$\bm{Z}$ is hidden	PFN †	$0.052$	—	$345.26\text{\,}\mathrm{s}$	$0.039$	—	$371.03\text{\,}\mathrm{s}$	$0.026$	—	$541.86\text{\,}\mathrm{s}$	$0.055$	—	$737.20\text{\,}\mathrm{s}$
$\bm{Z}$ is hidden	DeDiCa	$0.002$	$683.03\text{\,}\mathrm{s}$	$2.37\text{\,}\mathrm{ms}$	$0.040$	$1255.73\text{\,}\mathrm{s}$	$20.78\text{\,}\mathrm{ms}$	$0.035$	$893.53\text{\,}\mathrm{s}$	$3.44\text{\,}\mathrm{ms}$	$0.002$	$2553.24\text{\,}\mathrm{s}$	$8.71\text{\,}\mathrm{ms}$

Table 4: DeDiCa is much more efficient than CausalPFN while providing competitive causal estimates. Results show ATE MAE, training time and time for evaluating one query, averaged over 10 subsampled datasets (complete version in Tab. 7 ). CausalPFN needs one evaluation time for each query, while DeDiCa can amortize the training time and evaluate extremely fast. CausalPFN runs on GPU (denoted † ), while DeDiCa runs on CPU.

Panel of Figure 4 — Figure 4: DeDiCa performs consistently well on similar graphs , while CausalPFN fails to capture subtle differences in causal structure. Numbers shown are ATE MAE of an identifiable query $\mathrm{Pr}(y|{\mathrm{do}(t)})$ .

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Another example (b) of proxies ( $N,W$ ) enabled by taking other proxies ( $B_{1},B_{2}$ ) as colliders, as compared to the previous collider example (a, same as Fig. 2(c) ) where $B_{1},B_{2}$ are essential endogenous backdoor.

Panel of Figure 6 — Figure 6: The synthetic causal graphs used in the experiments that allows for varying number of variables. The chain (a) and dense (c) represent the lowest ( $\mathcal{W}=1$ ) and highest ( $\mathcal{W}=D-1$ ) treewidth, respectively, while conf (b) is a middle ground ( $\mathcal{W}=\frac{D}{2}$ ) as well as a representative proxy case.

Panel of Figure 7 — Figure 7: Time cost of different queries on different graphs, with mean and 95%-CI over 5 runs. The top row (a) shows the time cost for $\mathcal{C}=2$ , while the bottom row (b) shows $\mathcal{C}=4$ . The treewidth increases from left to right, in the order of chain , conf , dense .

Graph	Query	$\mathcal{C}$	$D$	$\mathcal{W}$	Memory (MiB)
chain	cond	2	20	1	$3.8_{\pm 0.6}$
chain		4	10	1	$4.5_{\pm 1.3}$
conf		2	20	10	$19.7_{\pm 3.5}$
conf		4	10	5	$37.2_{\pm 4.0}$
dense		2	20	19	$10234.6_{\pm 15.9}$
dense		4	10	9	$4606.5_{\pm 12.8}$

Appendix

Table 5: Memory cost for different graphs and different queries at

\mathcal{C}^{D}=2^{20}

, with mean and 95%-CI over 5 runs. The upper half is the query cond on different graphs, and the lower half is the graph conf with different queries.

Panel of Figure 8 — Figure 8: Ablation study on rank identification with different sizes of hidden confounders and different sizes of training data. $S$ means the number of proxy pairs (cf. Fig. 6(b) ). The colored band is the 95%-CI over 10 runs. Note that the y -axis is non-linear, zooming in around 0.

Panel of Figure 9 — Figure 9: Comparison of rank identification with different relative tolerance. Note that the right (c) has a y -axis scaled differently.

Panel of Figure 10 — Figure 10: Estimation quality on identifiable queries with different training set sizes, aggregating queries with different $\mathcal{C}_{Z}$ and $S$ into one line.

Panel of Figure 11 — Figure 11: Estimation quality with different training set sizes on an unidentifiable query, aggregating queries with different $\mathcal{C}_{Z}$ and $S$ into one line.

Panel of Figure 12 — Figure 12: Comparison among DeDiCa (Ours) and [ 46 ] (CausalPC) and [ 50 ] (NCM) and [ 51 ] (GAN-NCM) with the conf graph and varying $S$ . We also include our T-CCBN trained using closed-form MLE with $Z$ observed.

		$\bm{Z}$ is observed			$\bm{Z}$ is hidden
Model		CausalPC	CNF	DeDiCa	NCM †	NCM v2 †	GAN-NCM †	DeCaFlow	DeDiCa
Asia	ATE	$$0.289 $_{\pm$ 0.042 $}$	$$0.009 $_{\pm$ 0.002 $}$	$$0.000 $_{\pm$ 0.000 $}$	$$0.002 $_{\pm$ 0.000 $}$	$$0.002 $_{\pm$ 0.001 $}$	$$0.026 $_{\pm$ 0.025 $}$	$$0.012 $_{\pm$ 0.005 $}$	$$0.001 $_{\pm$ 0.001 $}$
	Train	$$667.44\text{,}\mathrm{s} $_{\pm$ 157.89\text{,}\mathrm{s} $}$	$$2312.28\text{,}\mathrm{s} $_{\pm$ 271.69\text{,}\mathrm{s} $}$	$$0.30\text{,}\mathrm{s} $_{\pm$ 0.02\text{,}\mathrm{s} $}$	$$7240.24\text{,}\mathrm{s} $_{\pm$ 1208.48\text{,}\mathrm{s} $}$	$$81,739.75\text{,}\mathrm{s} $_{\pm$ 12,905.15\text{,}\mathrm{s} $}$	$$6069.06\text{,}\mathrm{s} $_{\pm$ 1763.37\text{,}\mathrm{s} $}$	$$3000.45\text{,}\mathrm{s} $_{\pm$ 225.75\text{,}\mathrm{s} $}$	$$2569.04\text{,}\mathrm{s} $_{\pm$ 422.89\text{,}\mathrm{s} $}$
	Eval	$$0.81\text{,}\mathrm{s} $_{\pm$ 0.18\text{,}\mathrm{s} $}$	$$38.82\text{,}\mathrm{s} $_{\pm$ 5.19\text{,}\mathrm{s} $}$	$$1.90\text{,}\mathrm{ms} $_{\pm$ 0.45\text{,}\mathrm{ms} $}$	$$3.87\text{,}\mathrm{s} $_{\pm$ 0.01\text{,}\mathrm{s} $}$	$$0.10\text{,}\mathrm{s} $_{\pm$ 0.01\text{,}\mathrm{s} $}$	$$0.38\text{,}\mathrm{s} $_{\pm$ 0.00\text{,}\mathrm{s} $}$	$$37.66\text{,}\mathrm{s} $_{\pm$ 4.95\text{,}\mathrm{s} $}$	$$2.26\text{,}\mathrm{ms} $_{\pm$ 0.51\text{,}\mathrm{ms} $}$
Sachs (binarized)	ATE	$$0.193 $_{\pm$ 0.005 $}$	$$0.023 $_{\pm$ 0.002 $}$	$$0.024 $_{\pm$ 0.000 $}$	$$0.047 $_{\pm$ 0.005 $}$	$$0.033 $_{\pm$ 0.008 $}$	$$0.097 $_{\pm$ 0.037 $}$	$$0.078 $_{\pm$ 0.048 $}$	$$0.039 $_{\pm$ 0.010 $}$
	Train	$$1094.09\text{,}\mathrm{s} $_{\pm$ 324.80\text{,}\mathrm{s} $}$	$$2585.43\text{,}\mathrm{s} $_{\pm$ 151.11\text{,}\mathrm{s} $}$	$$0.57\text{,}\mathrm{s} $_{\pm$ 0.06\text{,}\mathrm{s} $}$	$$106,962.04\text{,}\mathrm{s} $_{\pm$ 23,632.31\text{,}\mathrm{s} $}$	$$129,259.27\text{,}\mathrm{s} $_{\pm$ 25,891.05\text{,}\mathrm{s} $}$	$$4035.75\text{,}\mathrm{s} $_{\pm$ 54.94\text{,}\mathrm{s} $}$	$$3679.98\text{,}\mathrm{s} $_{\pm$ 439.64\text{,}\mathrm{s} $}$	$$4908.72\text{,}\mathrm{s} $_{\pm$ 1284.22\text{,}\mathrm{s} $}$

Appendix

Table 6: Complete version of Tab. 3 . DeDiCa scales well while providing accurate causal estimates under hidden confounding. Results show ATE MAE, training time and time for evaluating one query, averaged over 10 runs with 95%-CI. Aside from CausalPC which requires per-query training, the rest train only once. The NCMs run on GPU (denoted † ), and others on CPU. NCM v2 is the reimplemented version in [ 51 ] , which behaves differently from the original version [ 50 ] . Here, OOM denotes out-of-memory (

>

64\text{,}\mathrm{GiB}

CPU or $>$140\text{\,}\mathrm{GiB}

GPU) and an asterisk ∗ a timeout (

>

120,000\text{,}\mathrm{s}$$ ).

		$\bm{Z}$ is observed		$\bm{Z}$ is hidden
Model		PFN †	DeDiCa	PFN †	DeDiCa
Asia	ATE	$$0.016 $_{\pm$ 0.004 $}$	$$0.001 $_{\pm$ 0.000 $}$	$$0.052 $_{\pm$ 0.008 $}$	$$0.002 $_{\pm$ 0.001 $}$
	Train	—	$$77.66\text{,}\mathrm{ms} $_{\pm$ 30.70\text{,}\mathrm{ms} $}$	—	$$683.03\text{,}\mathrm{s} $_{\pm$ 155.92\text{,}\mathrm{s} $}$
	Eval	$$331.73\text{,}\mathrm{s} $_{\pm$ 7.92\text{,}\mathrm{s} $}$	$$2.15\text{,}\mathrm{ms} $_{\pm$ 0.82\text{,}\mathrm{ms} $}$	$$345.26\text{,}\mathrm{s} $_{\pm$ 17.72\text{,}\mathrm{s} $}$	$$2.37\text{,}\mathrm{ms} $_{\pm$ 0.84\text{,}\mathrm{ms} $}$
Sachs (binarized)	ATE	$$0.020 $_{\pm$ 0.001 $}$	$$0.024 $_{\pm$ 0.000 $}$	$$0.039 $_{\pm$ 0.002 $}$	$$0.040 $_{\pm$ 0.010 $}$
	Train	—	$$0.19\text{,}\mathrm{s} $_{\pm$ 0.04\text{,}\mathrm{s} $}$	—	$$1255.73\text{,}\mathrm{s} $_{\pm$ 484.20\text{,}\mathrm{s} $}$

Appendix

Table 7: Complete version of Tab. 4 . DeDiCa is much more efficient than CausalPFN while providing competitive causal estimates. Results show ATE MAE, training time and time for evaluating one query, averaged over 10 subsampled datasets with 95%-CI. CausalPFN needs one evaluation time for each query, while DeDiCa can amortize the training time and evaluate extremely fast. Moreover, CausalPFN runs on GPU (denoted † ), while DeDiCa on CPU.

Panel of Figure 13 — Figure 13: ATE MAE, CPU memory usage for evaluation (GPU memory usage for CausalPFN not included), and time cost (training time for DeDiCa, evaluation time for CausalPFN) across datasets and varying sample sizes. DeDiCa uses $<$ 20\text{,}\mathrm{MiB}$$ in terms of peak resident size change under all configurations and thus negligible in the second column of figures.

Panel of Figure 14 — Figure 14: Complete version of Fig. 4 . Numbers shown are ATE MAE of an identifiable query $\mathrm{Pr}(Y|{\mathrm{do}(T)})$ averaged over 10 runs with 95%-CI. DeDiCa performs consistently well on similar graphs , while CausalPFN fails to capture subtle differences in causal structure.

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