A deep learning framework for causal estimation using the cascade additive noise model: An optimization perspective

Sohail Ahmad; Mona F. ElWakeel; Moiz Qureshi; Hasnain Iftikhar; Paulo Canas Rodrigues; Sohail Ahmad; Mona F. ElWakeel; Moiz Qureshi; Hasnain Iftikhar; Paulo Canas Rodrigues

doi:10.3934/math.20261031

AIMS Mathematics

2026, Volume 11, Issue 8: 25744-25767. doi: 10.3934/math.20261031

Previous Article Next Article

Research article

A deep learning framework for causal estimation using the cascade additive noise model: An optimization perspective

1.
Discipline of Statistics, The University of Agriculture Swat, Mingora 19120, Pakistan
2.
Department of Finance and Banking, College of Business Administration, Dar Al Uloom University, Riyadh 13314, Saudi Arabia
3.
Department of Statistics, University of Sindh, Jamshoro 76080, Pakistan
4.
Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
5.
Department of Statistics, University of Peshawar, Peshawar 25120, Pakistan
6.
Department of Statistics, Federal University of Bahia, Salvador 40170-110, Brazil
7.
Department of Business Management, University of Pretoria, Pretoria 0002, South Africa

Received: 30 June 2026 Revised: 21 July 2026 Accepted: 06 August 2026 Published: 19 August 2026
MSC : 62H22, 68T07, 68T09, 90C26

Causal inference in nonlinear, high-dimensional systems remains challenging due to complex dependencies and unobserved confounders. To address latent variables and indirect effects, a variational autoencoder (VAE) was incorporated into the cascade additive noise model (CANM). However, the CANM-VAE and its extension suffer from limitations in modeling long-range dependencies in high-dimensional settings because confounding variables are handled only at the latent-variable stage. To overcome these limitations, we developed the transformer-based variational CANM (TV-CANM), which leverages self-attention mechanisms to effectively capture complex nonlinear relationships and long-range dependencies. We trained the TV-CANM on three arrhythmia benchmark datasets, applied the test likelihood to the causal direction, and used the mean squared error (MSE) and root mean squared error (RMSE) to measure performance. Findings indicate that TV-CANM is significantly more effective than the original CANM at identifying causal directions consistent with domain knowledge and substantially reducing error rates. The proposed framework provides reliable causal knowledge that can support optimization-based decision-making and operational planning in complex data-driven systems. Moreover, the TV-CANM is found to be sound in nonlinear, high-dimensional causal discovery, compared with five traditional algorithms: additive noise model (ANM), information-geometric causal inference (IGCI), causal additive model (CAM), post-nonlinear model (PNL), and a loss-based algorithm.
- causal inference,
- cascade additive noise model,
- variational autoencoder,
- transformer,
- self-attention mechanism,
- high-dimensional data,
- optimization,
- operations research applications
Citation: Sohail Ahmad, Mona F. ElWakeel, Moiz Qureshi, Hasnain Iftikhar, Paulo Canas Rodrigues. A deep learning framework for causal estimation using the cascade additive noise model: An optimization perspective[J]. AIMS Mathematics, 2026, 11(8): 25744-25767. doi: 10.3934/math.20261031

Related Papers:

Abstract

Causal inference in nonlinear, high-dimensional systems remains challenging due to complex dependencies and unobserved confounders. To address latent variables and indirect effects, a variational autoencoder (VAE) was incorporated into the cascade additive noise model (CANM). However, the CANM-VAE and its extension suffer from limitations in modeling long-range dependencies in high-dimensional settings because confounding variables are handled only at the latent-variable stage. To overcome these limitations, we developed the transformer-based variational CANM (TV-CANM), which leverages self-attention mechanisms to effectively capture complex nonlinear relationships and long-range dependencies. We trained the TV-CANM on three arrhythmia benchmark datasets, applied the test likelihood to the causal direction, and used the mean squared error (MSE) and root mean squared error (RMSE) to measure performance. Findings indicate that TV-CANM is significantly more effective than the original CANM at identifying causal directions consistent with domain knowledge and substantially reducing error rates. The proposed framework provides reliable causal knowledge that can support optimization-based decision-making and operational planning in complex data-driven systems. Moreover, the TV-CANM is found to be sound in nonlinear, high-dimensional causal discovery, compared with five traditional algorithms: additive noise model (ANM), information-geometric causal inference (IGCI), causal additive model (CAM), post-nonlinear model (PNL), and a loss-based algorithm.

References

[1]	Y. Li, O. Deng, A. Ogihara, S. Nishimura, Q. Jin, Causal discovery of health features from wearable device and traditional Chinese medicine diagnosis data, In: HCI international 2023 – Late breaking papers, 2023,556–569. https://doi.org/10.1007/978-3-031-48041-6_37
[2]	C. Glymour, K. Zhang, P. Spirtes, Review of causal discovery methods based on graphical models, Front. Genet., 10 (2019), 524. https://doi.org/10.3389/fgene.2019.00524 doi: 10.3389/fgene.2019.00524
[3]	M. Huber, An introduction to causal discovery, Swiss J. Econ. Stat., 160 (2024), 14. https://doi.org/10.1186/s41937-024-00131-4 doi: 10.1186/s41937-024-00131-4
[4]	L. Jin, Y. Zhang, Y. Li, S. Wang, H. H. Yang, J. Wu, M. Zhang, MoE$^{2}$: Optimizing collaborative inference for edge large language models, IEEE/ACM Trans. Netw., 34 (2026), 4637–4651. https://doi.org/10.1109/TON.2026.3677371 doi: 10.1109/TON.2026.3677371
[5]	H. Zhang, Y. Ren, Y. Xia, S. Zhou, J. Guan, Towards effective causal partitioning by edge cutting of adjoint graph, IEEE Trans. Pattern Anal. Mach. Intell., 46 (2024), 10259–10271. https://doi.org/10.1109/TPAMI.2024.3435503 doi: 10.1109/TPAMI.2024.3435503
[6]	J. Pearl, Causality, 2Eds., Cambridge: Cambridge University Press, 2013. https://doi.org/10.1017/CBO9780511803161
[7]	M. Caliendo, S. Kopeinig, Some practical guidance for the implementation of propensity score matching, J. Econ. Surv., 22 (2008), 31–72. https://doi.org/10.1111/j.1467-6419.2007.00527.x doi: 10.1111/j.1467-6419.2007.00527.x
[8]	M. Baiocchi, J. Cheng, D. S. Small, Instrumental variable methods for causal inference, Stat. Med., 33 (2014), 2297–2340. https://doi.org/10.1002/sim.6128 doi: 10.1002/sim.6128
[9]	J. Bor, E. Moscoe, T. Bärnighausen, Three approaches to causal inference in regression discontinuity designs, Epidemiology, 26 (2015), e28–e30. https://doi.org/10.1097/EDE.0000000000000256 doi: 10.1097/EDE.0000000000000256
[10]	F. F. Stanaway, A. Diaz, R. Maddox, Causal inference, mediation analysis and racial inequities, Int. J. Epidemiol., 53 (2024), dyae038. https://doi.org/10.1093/ije/dyae038 doi: 10.1093/ije/dyae038
[11]	Q. Liu, P. Chen, K. Lin, K. Zhao, J. Ding, Y. Li, Sample-efficient backtrack temporal difference deep reinforcement learning, Knowl.-Based Syst., 330 (2025), 114613. https://doi.org/10.1016/j.knosys.2025.114613 doi: 10.1016/j.knosys.2025.114613
[12]	R. Visontay, L. Mewton, T. Slade, I. M. Aris, M. Sunderland, Moderate alcohol consumption and depression: A marginal structural model approach promoting causal inference, Am. J. Psychiatry, 180 (2023), 209–217. https://doi.org/10.1176/appi.ajp.22010043 doi: 10.1176/appi.ajp.22010043
[13]	Y. Lu, Q. Zheng, D. Quinn, Introducing causal inference using Bayesian networks and do-calculus, J. Stat. Data Sci. Educ., 31 (2023), 3–17. https://doi.org/10.1080/26939169.2022.2128118 doi: 10.1080/26939169.2022.2128118
[14]	F. Podestà, Combining process tracing and synthetic control method: Bridging two ways of making causal inference in evaluation research, Evaluation, 29 (2023), 50–66. https://doi.org/10.1177/13563890221139511 doi: 10.1177/13563890221139511
[15]	B. Vakulenko-Lagun, C. Magdamo, M. L. Charpignon, B. Zheng, M. W. Albers, S. Das, causalCmprsk: An R package for nonparametric and Cox-based estimation of average treatment effects in competing risks data, Comput. Methods Programs Biomed., 242 (2023), 107819. https://doi.org/10.1016/j.cmpb.2023.107819 doi: 10.1016/j.cmpb.2023.107819
[16]	Z. Yan, Z. Pan, G. Hu, H. Yan, Finite-time annular domain guaranteed cost control for uncertain mean-field stochastic systems with Wiener and Poisson noises, IEEE Trans. Syst. Man Cybern. Syst., 55 (2025), 7981–7993. https://doi.org/10.1109/TSMC.2025.3598993 doi: 10.1109/TSMC.2025.3598993
[17]	P. Spirtes, C. Glymour, R. Scheines, Causation, prediction, and search, MIT Press, 2001. https://doi.org/10.7551/mitpress/1754.001.0001
[18]	B. Huang, K. Zhang, J. Zhang, J. Ramsey, C. Glymour, B. Schölkopf, Causal discovery from heterogeneous/nonstationary data, J. Mach. Learn. Res., 21 (2020), 1–53.
[19]	H. Huang, W. Zhang, Y. Xu, H. Zhang, J. Zou, A method for PWM frequency harmonic suppression using zero-vector switching SVPWM strategy in three-level inverter drive systems, IEEE J. Emerg. Sel. Top. Power Electron., 13 (2025), 1482–1491. https://doi.org/10.1109/JESTPE.2024.3485962 doi: 10.1109/JESTPE.2024.3485962
[20]	K. Zhang, Z. Wang, J. Zhang, B. Schölkopf, On estimation of functional causal models: General results and application to the post-nonlinear causal model, ACM Trans. Intell. Syst. Technol., 7 (2015), 13. https://doi.org/10.1145/2700476 doi: 10.1145/2700476
[21]	P. Hoyer, D. Janzing, J. M. Mooij, J. Peters, B. Schölkopf, Nonlinear causal discovery with additive noise models, Adv. Neural Inf. Process. Syst., 2008,689–696.
[22]	S. Shimizu, P. O. Hoyer, A. Hyvärinen, A. Kerminen, A linear non-Gaussian acyclic model for causal discovery, J. Mach. Learn. Res., 7 (2006), 2003–2030.
[23]	K. Zhang, A. Hyvarinen, On the identifiability of the post-nonlinear causal model, arXiv preprint, 2012, arXiv: 1205.2599. https://doi.org/10.48550/arXiv.1205.2599
[24]	Z. Liang, L. Liu, J. Yang, J. Yu, S. Zhang, X. Zhang, et al., AUV cooperative localization method based on M–H resampling and state covariance matrix optimized particle filter, IEEE Trans. Instrum. Meas., 74 (2025), 8513314. https://doi.org/10.1109/TIM.2025.3609326 doi: 10.1109/TIM.2025.3609326
[25]	X. Zheng, B. Aragam, P. K. Ravikumar, E. P. Xing, DAGs with no tears: Continuous optimization for structure learning, Adv. Neural Inf. Process. Syst., 31 (2018).
[26]	Y. Yu, J. Chen, T. Gao, M. Yu, DAG-GNN: DAG structure learning with graph neural networks, Proc. 36th Int. Conf. Mach. Learn., 97 (2019), 7154–7163.
[27]	P. Lippe, T. Cohen, E. Gavves, Efficient neural causal discovery without acyclicity constraints, arXiv preprint, 2021, arXiv: 2107.10483. https://doi.org/10.48550/arXiv.2107.10483
[28]	I. Tsamardinos, L. E. Brown, C. F. Aliferis, The Max-min hill-climbing Bayesian network structure learning algorithm, Mach. Learn., 65 (2006), 31–78. https://doi.org/10.1007/s10994-006-6889-7 doi: 10.1007/s10994-006-6889-7
[29]	J. M. Mooij, S. Magliacane, T. Claassen, Joint causal inference from multiple contexts, J. Mach. Learn. Res., 21 (2020), 1–108.
[30]	Z. Zhou, X. Zhang, L. Zhang, J. Liu, E. Cambria, C. Li, FineFake: A knowledge-enriched dataset for fine-grained multi-domain fake news detection, Inf. Fusion, 132 (2026), 104253. https://doi.org/10.1016/j.inffus.2026.104253 doi: 10.1016/j.inffus.2026.104253
[31]	D. P. Kingma, M. Welling, Auto-encoding variational Bayes, arXiv preprint, 2013, arXiv: 1312.6114. https://doi.org/10.48550/arXiv.1312.6114
[32]	A. Almodóvar, J. Parras, S. Zazo, Propensity weighted federated learning for treatment effect estimation in distributed imbalanced environments, Comput. Biol. Med., 178 (2024), 108779. https://doi.org/10.1016/j.compbiomed.2024.108779 doi: 10.1016/j.compbiomed.2024.108779
[33]	L. Wu, K. Wang, K. Nie, S. Guo, C. Gao, Z. Wang, S. Li, TFGIN: Tight-fitting graph inference network for table-based fact verification, ACM Trans. Inf. Syst., 43 (2025), 130. https://doi.org/10.1145/3734520 doi: 10.1145/3734520
[34]	S. Yin, Z. Xiang, Adaptive operator selection with dueling deep Q-network for evolutionary multi-objective optimization, Neurocomputing, 581 (2024), 127491. https://doi.org/10.1016/j.neucom.2024.127491 doi: 10.1016/j.neucom.2024.127491
[35]	J. Qiao, R. Cai, K. Zhang, Z. Zhang, Z. Hao, Causal discovery with confounding cascade nonlinear additive noise models, ACM Trans. Intell. Syst. Technol., 12 (2021), 1–28. https://doi.org/10.1145/3482879 doi: 10.1145/3482879
[36]	S. Ahmad, H. Wang, Transformer-variational autoencoder for estimating individual treatment effect using causal inference framework, Appl. Intell., 55 (2025), 855. https://doi.org/10.1007/s10489-025-06738-1 doi: 10.1007/s10489-025-06738-1
[37]	S. Ahmad, K. Shah, A. Debbouche, Structural equation modeling for causal effect estimation with machine learning, J. Comput. Appl. Math., 475 (2026), 117020. https://doi.org/10.1016/j.cam.2025.117020 doi: 10.1016/j.cam.2025.117020
[38]	S. Ahmad, H. Wang, TV-CCANM: A transformer variational inference in confounding cascade additive noise model for causal effect estimation, J. Stat. Comput. Simul., 95 (2025), 3048–3076. https://doi.org/10.1080/00949655.2025.2516793 doi: 10.1080/00949655.2025.2516793
[39]	A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, et al., Attention is all you need, Adv. Neural Inf. Process. Syst., 30 (2017). https://doi.org/10.48550/arXiv.1706.03762 doi: 10.48550/arXiv.1706.03762
[40]	H. Qiu, J. Zhang, J. Zhou, G. Shi, G. Yao, X. Cai, A novel delta-type transformerless unified power flow controller with multiple power regulation ports, IEEE Trans. Power Electron., 40 (2025), 15820–15834. https://doi.org/10.1109/TPEL.2025.3575074 doi: 10.1109/TPEL.2025.3575074
[41]	X. Glorot, A. Bordes, Y. Bengio, Deep sparse rectifier neural networks, Proc. Fourteenth Int. Conf. Artif. Intell. Stat., 15 (2011), 315–323.
[42]	J. M. Mooij, J. Peters, D. Janzing, J. Zscheischler, B. Schölkopf, Distinguishing cause from effect using observational data: Methods and benchmarks, J. Mach. Learn. Res., 17 (2016), 1–102.
[43]	F. Quispe, E. Salcedo, H. Iftikhar, A. Zafar, M. Khan, J. E. Turpo-Chaparro, et al., Multi-step ahead ozone level forecasting using a component-based technique: A case study in Lima, Peru, AIMS Environ. Sci., 11 (2024), 401–425. https://doi.org/10.3934/environsci.2024020 doi: 10.3934/environsci.2024020
[44]	H. Iftikhar, M. Qureshi, J. Zywiołek, J. L. López-Gonzales, O. Albalawi, Short-term PM 2.5 forecasting using a unique ensemble technique for proactive environmental management initiatives, Front. Environ. Sci., 12 (2024), 1442644. https://doi.org/10.3389/fenvs.2024.1442644 doi: 10.3389/fenvs.2024.1442644
[45]	A. Gretton, K. Fukumizu, C. Teo, L. Song, B. Schölkopf, A. Smola, A kernel statistical test of independence, Adv. Neural Inf. Process. Syst., 20 (2007).
[46]	T. Chen, C. Guestrin, XGBoost: A scalable tree boosting system, Proc. 22nd ACM SIGKDD Int. Conf. Knowl. Discov. Data Min., 2016,785–794. https://doi.org/10.1145/2939672.2939785 doi: 10.1145/2939672.2939785
[47]	D. Janzing, J. Mooij, K. Zhang, J. Lemeire, J. Zscheischler, P. Daniušis, B. Steudel, B. Schölkopf, Information-geometric approach to inferring causal directions, Artif. Intell., 182–183 (2012), 1–31. https://doi.org/10.1016/j.artint.2012.01.002 doi: 10.1016/j.artint.2012.01.002
[48]	P. Buhlmann, J. Peters, J. Ernest, CAM: Causal additive models, high-dimensional order search and penalized regression, Ann. Stat., 42 (2014), 2526–2556. https://doi.org/10.1214/14-AOS1260 doi: 10.1214/14-AOS1260
[49]	J. Mooij, D. Janzing, J. Peters, B. Schölkopf, Regression by dependence minimization and its application to causal inference in additive noise models, Proc. 26th Annu. Int. Conf. Mach. Learn., 2009,745–752. https://doi.org/10.1145/1553374.1553470 doi: 10.1145/1553374.1553470
[50]	Y. Zhang, J. Xiang, J. Li, Denoising self-supervised learning for disease-gene association prediction, BMC Bioinform., 26 (2025), 260. https://doi.org/10.1186/s12859-025-06281-3 doi: 10.1186/s12859-025-06281-3
[51]	H. Iftikhar, M. Khan, J. E. Turpo-Chaparro, P. C. Rodrigues, J. L. López-Gonzales, Forecasting stock prices using a novel filtering-combination technique: Application to the Pakistan stock exchange, AIMS Mathematics, 9 (2024), 3264–3288. https://doi.org/10.3934/math.2024159 doi: 10.3934/math.2024159
[52]	Y. Tian, H. Tian, A multi-layer dynamic model of information propagation considering individual three-phase linear modulated behaviors, Physica A, 689 (2026), 131427. https://doi.org/10.1016/j.physa.2026.131427 doi: 10.1016/j.physa.2026.131427

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)