Overview

SEMdeep provides an integrated framework to train, validate, and explain Structural Equation Models (SEMs) using deep neural network (DNN) and machine learning (ML) algorithms. It combines the strengths of causal inference and predictive modeling to build interpretable systems that remain faithful to a given graph structure.

The package is fully compatible with SEMgraph, which defines and manipulates causal graphs. Together, the two packages form a dual modular system:

• SEMgraph: linear SEM fitting, graph model search, and topological representation.
• SEMdeep: SEM-based DNN and ML fitting, model evaluation, and explainability.

Scientific Rationale and Added Value

While most machine learning models achieve strong predictive performance, they often obscure the causal structure underlying the data. SEMdeep extends the classical SEM paradigm by embedding ML and DNN algorithms directly within the causal graph, thus providing:

• Data-driven estimation aligned with prior causal structure;
• Model interpretability through connection and gradient weights;
• Interoperability with SEMgraph for graph construction and validation.

This integration allows the user to move seamlessly from causal discovery to explainable predictive modeling.

Conceptual Workflow

The conceptual workflow follows a modular and interoperable design:

1. Data preprocessing – sampling, scaling, and feature selection.
2. Graph compilation – conversion of DAGs or SEM specifications to adjacency matrices using SEMgraph.
3. Model training – graph-aligned machine learning or deep neural network fitting via SEMml() and SEMdnn().
4. Parameter optimization – hyperparameter tuning and regularization across models.
5. Model evaluation – predictive accuracy and goodness-of-fit metrics.
6. Model explanation – quantitative and visual interpretation of causal effects.

All preprocessing and graph compilation steps are handled by SEMgraph, while model training, optimization, and explainability modules are managed internally by SEMdeep.

Modeling Framework

Input

SEMdeep operates on two main inputs:

• A data matrix, containing observed or simulated variables.
• A directed graph (adjacency matrix) that encodes the hypothesized or data-driven causal relationships among variables.

These inputs define the structural dependencies that guide model compilation, training, and explanation.

Modeling Strategies

SEMdnn()

SEMdnn() extends this logic to deep neural networks (DNNs), supporting multiple graph-aware fitting strategies:

Each network is trained using torch, enabling efficient backpropagation, and if available, GPU acceleration. This design allows seamless transition from traditional SEMs to fully neural, graph-informed architectures.

SEMml()

SEMml() performs machine learning–based SEM fitting through a nodewise approach. Each node with incoming edges is modeled as a function of its direct parents, resulting in a set of independent predictive models aligned with the directed graph.

The algorithm supports four machine learning modes:

By mapping data onto the input graph, SEMml() creates a set of nodewise models based on directed causal links.

Output and Explainability

SEMdeep provides interpretable outputs through distinct explainability functions for both DNN and ML-based SEMs.

DNN-based Explainability

ML-based Explainability

These measures allow causal-effect visualization through color-coded graphs and node importance plots, bridging predictive and structural interpretability. In addition, if argument boot != 0 in SEMdnn() and SEMml(), Lam’s (2002) cheap bootstrapping is applied to estimate 90% confidence intervals for DNN and ML parameters.

Integration and Applications

Together, SEMgraph and SEMdeep provide a coherent workflow for Causal AI, connecting structural modeling with modern predictive algorithms.

Installation

SEMdeep relies on the native R package torch, which supports CPU, GPU, and Apple Silicon acceleration — no Python required.

# Install dependencies
install.packages("torch")
torch::install_torch()
# Install SEMdeep (CRAN release)
install.packages("SEMdeep")
# or latest development version
# remotes::install_github("BarbaraTarantino/SEMdeep")
install.packages("SEMdeep", repos = c('https://barbaratarantino.r-universe.dev', 'https://cloud.r-project.org'))

Documentation

Comprehensive documentation, tutorials, and examples are available HERE.

Related Package

• SEMgraph — CRAN | GitHub

Future Developments

The next module under development is SEMdgm() function. This extension will introduce deep generative modeling (DGM) architectures, including: Graph Autoencoders (GAE), Variational Autoencoders (VAE), Normalizing Flows (NF), and Generative Adversarial Networks (GAN), to approximate latent structures within complex systems for causal structure discovery. The goal is to enable causally informed representation learning that enhances predictive accuracy while preserving consistency with the underlying causal graph.

Stay tuned for updates to version 1.2.0 and higher.

References

1. Grassi M., Palluzzi F., and Tarantino B. (2022). SEMgraph: An R Package for Causal Network Analysis of High-Throughput Data with Structural Equation Models. Bioinformatics, 38(20), 4829–4830.
   https://doi.org/10.1093/bioinformatics/btac567

2. Breiman L., Friedman J.H., Olshen R.A., and Stone, C.J. (1984). Classification and Regression Trees. Chapman and Hall/CRC.
   https://doi.org/10.1201/9781315139470

3. Breiman L. (2001). Random Forests. Machine Learning, 45(1), 5–32.
   https://doi.org/10.1023/A:1010933404324

4. Chen T., and Guestrin C. (2016). XGBoost: A Scalable Tree Boosting System. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining.
   https://doi.org/10.1145/2939672.2939785

5. Zheng, X., Dan, C., Aragam, B., Ravikumar, P., Xing E. (2020). Learning sparse nonparametric dags. International Conference on Artificial Intelligence and Statistics, PMLR, 3414–3425.
   https://doi.org/10.48550/arXiv.1909.13189

6. Grassi, M., Tarantino, B. (2025). SEMdag: Fast learning of Directed Acyclic Graphs via node or layer ordering. PLoS ONE, 20(1): e0317283.
   https://doi.org/10.1371/journal.pone.0317283

7. Chen A., Shi, R.I., Gao, X., Baptista, R., Krishnan, R.G. (2023). Structured neural networks for density estimation and causal inference. Advances in Neural Information Processing Systems, 36, 66438–66450.
   https://doi.org/10.48550/arXiv.2311.02221

8. Shrivastava, H., Chajewska, U. (2023). Neural graphical models. European Conference on Symbolic and Quantitative Approaches with Uncertainty (pp. 284–307). Springer Nature Switzerland.
   https://doi.org/10.48550/arXiv.2210.00453

9. Lam, H. (2022). Cheap Bootstrap for Input Uncertainty Quantification. Winter Simulation Conference (WSC), Singapore, 2318–2329.
   https://doi.org/10.1109/WSC57314.2022.10015362

Contact

Barbara Tarantino
Email: barbara.tarantino00@gmail.com
GitHub: https://github.com/BarbaraTarantino