Estimating Causal Bounds from Observational Data With Neural Generative Models

Max Zhu; Srivathsan Amruth; Sina Akbari; Qingyuan Zhao

stat.MEOct 18, 2026

Estimating Causal Bounds from Observational Data With Neural Generative Models

Authors: Max Zhu, Srivathsan Amruth, Sina Akbari, Qingyuan Zhao

Organizations: Department of Pure Mathematics and Mathematical Statistics University of Cambridge

Abstract

Identifying causal bounds from discrete observational data is a fundamental problem in causal inference. While exact analytical bounds exist for certain graphs, this is not generally the case. In this paper, we introduce a framework for estimating such bounds using neural generative models together with a new extrapolation procedure to estimate these bounds more accurately. Across multiple causal inference settings, it produces bounds that are consistently closer to the true bounds than those obtained with existing general neural approaches. We validate the framework on a range of benchmark problems for which exact bounds are available.

Figures & tables

Panel of Figure 1 — Figure 1: Causal graphs used in the simulation studies. Left: instrumental variable graph (IV). Centre: instrumental variable graph with a proxy variable (Proxy). Right: mediated instrumental variable graph (Mediated). $U$ is exogenous and the other nodes are endogenous categorical variables.

Figure 2: Model-predicted ATE and Energy Distance as

\lambda

varies. Left: values in the extrapolation regime, where fitted ATE intercepts recover the predicted bounds and Energy Distance follows a quadratic trend. The linear fit and

R^{2}

goodness-of-fit values are shown for the plots. Right: values across a wider range of

\lambda

beyond the linear and quadratic regimes.

Dataset	Autobounds		BCCE		NeuralPartialID		Ours
	Upper	Lower	Upper	Lower	Upper	Lower	Upper	Lower
IV-Binary	0	0	$0.135\text{\scriptsize$ ,\pm,0.012 $}$	$0.087\text{\scriptsize$ ,\pm,0.021 $}$	$0.105\text{\scriptsize$ ,\pm,0.026 $}$	$0.159\text{\scriptsize$ ,\pm,0.041 $}$	$0.020\text{\scriptsize$ ,\pm,0.005 $}$	$0.011\text{\scriptsize$ ,\pm,0.004 $}$
IV-Ternary	0	0	$0.154\text{\scriptsize$ ,\pm,0.043 $}$	$0.125\text{\scriptsize$ ,\pm,0.018 $}$	$0.191\text{\scriptsize$ ,\pm,0.065 $}$	$0.162\text{\scriptsize$ ,\pm,0.047 $}$	$0.018\text{\scriptsize$ ,\pm,0.011 $}$	$0.010\text{\scriptsize$ ,\pm,0.002 $}$
Proxy	$0.033\text{\scriptsize$ ,\pm,0.011 $}$ *	$0.020\text{\scriptsize$ ,\pm,0.006 $}$ *	$0.093\text{\scriptsize$ ,\pm,0.027 $}$	$0.107\text{\scriptsize$ ,\pm,0.029 $}$	$0.067\text{\scriptsize$ ,\pm,0.020 $}$	$0.077\text{\scriptsize$ ,\pm,0.032 $}$	$0.032\text{\scriptsize$ ,\pm,0.012 $}$	$0.026\text{\scriptsize$ ,\pm,0.009 $}$
Mediated	$0.136\text{\scriptsize$ ,\pm,0.038 $}$ *	$0.171\text{\scriptsize$ ,\pm,0.041 $}$ *	$0.056\text{\scriptsize$ ,\pm,0.020 $}$	$0.037\text{\scriptsize$ ,\pm,0.016 $}$	$0.128\text{\scriptsize$ ,\pm,0.053 $}$	$0.099\text{\scriptsize$ ,\pm,0.036 $}$	$0.020\text{\scriptsize$ ,\pm,0.009 $}$	$0.021\text{\scriptsize$ ,\pm,0.010 $}$

Table 1: Absolute error

\downarrow

of the predicted upper and lower bounds for the ATE, compared with the true bounds. Standard errors are recorded over 10 datasets. (*) Autobounds does not fully converge and gives looser bounds than the true bound.

Figure 3: Absolute error between the analytical and predicted ATE bounds for our model, BCCE, and NeuralPartialID, averaged over the upper and lower bounds, over different neural network widths. Solid lines indicate a constant learning rate, while dashed lines indicate

\mu

P scaling.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Width	Lower error	Upper error
16	0.008	0.010
32	0.010	0.011
64	0.009	0.014
128	0.062	0.151
256	0.371	0.399
512	0.572	0.780

Appendix

Table 2: Error of the GAN generator from BCCE for different NN widths for the upper bound model and lower bound model. Lower error indicates better fit. The learning rate follows

\mu

P scaling.

Width	$\alpha$	$\chi_{\min}$	$\chi_{\max}$
16	0.131	12.48	10.55
32	0.132	22.54	24.51
64	0.136	14.18	27.61
128	0.136	0.21	36.56
256	0.128	7.62	20.97
512	0.137	8.07	29.98

Appendix

Table 3: Final parameters of the NeuralPartialID models for different width NNs.

\chi_{\min}

, and

\chi_{\max}

are the final Lagrange multipliers, for the lower and upper bounds and

\alpha

is the fit quality threshold.

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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.

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