Structural Causal Model Essentials

• Structural Causal Models are formal frameworks that use directed acyclic graphs and structural equations to define interventional and counterfactual distributions.
• Recent extensions integrate latent variables, temporal dynamics, and deep generative mechanisms, broadening applications in fields like medicine and computer vision.
• Causal structure learning leverages constraint-based, score-based, and invariance methods to identify model parameters and address challenges such as non-acyclicity.

A Structural Causal Model (SCM) is a formal framework for encoding and reasoning about the causal relationships among a set of variables via (usually) a directed acyclic graph (DAG) and a set of structural equations that specify how each variable is generated from its direct causes plus exogenous noise. SCMs support the precise definition and computation of interventional and counterfactual distributions, are the backbone of modern causal inference, and are central to research in fields as varied as economics, medicine, computer vision, and language modeling. In recent work, SCMs have been extended to accommodate latent variables, temporal dynamics, functional constraints, deep generative mechanisms, and systematic encoding of expert and data-driven priors.

1. Formal Definition and Core Structure

SCMs are typically defined as a tuple $(V, U, F, P_U)$, where:

• $V$ is the set of endogenous (observed) random variables, indexed $j = 1,\dots,p$;
• $U$ is a set of exogenous noise variables, often mutually independent but sometimes correlated to model latent confounding;
• $F$ is a collection of deterministic functions $f_j$ specifying $X_j \leftarrow f_j(X_{\text{pa}(j)}, U_j)$, with $\text{pa}(j)$ the parent set of $j$;
• $P_U$ is a joint distribution over the exogenous variables.

Associated to each SCM is a DAG $G = (V, E)$ where an edge $i \to j$ exists if $i \in \text{pa}(j)$. The Markov factorization is $P(V) = \prod_{j=1}^p P(X_j | X_{\text{pa}(j)})$; in linear models, $X = B X + \epsilon$ with $B$ the weighted adjacency and $\epsilon$ the noise vector (Heinze-Deml et al., 2017).

SCMs encode Pearl's "do"-calculus semantics: an intervention $\text{do}(X_j = x_j)$ replaces the $j$th equation by $X_j = x_j$ and updates the joint distribution accordingly. Counterfactuals and potential outcomes are computed by the abduction-action-prediction pathway.

2. Extensions to Latent Variables and Deep Causal Models

SCMs are generalized to accommodate latent causal factors (unobserved confounders or mechanisms) and deep generative structures:

• Latent SCMs: In image or high-dimensional tasks, the causal variables $Z = (Z_1, \dots, Z_d)$ are unobserved, and the entire structure, parameters, and latent states must be inferred from low-level observations $X \in \mathbb{R}^D$ (Subramanian et al., 2022). Linear additive-noise SCMs are written $Z = W^T Z + \epsilon$, $\epsilon \sim \mathcal{N}(0, \sigma^2 I)$, with $W$ parameterizing edges. Observed $X$ are decoded from $Z$, e.g., via a neural network likelihood $p_\psi(X|Z)$.
• Deep Structural Causal Models for Meshes: Instead of standard Euclidean variables, SCMs can model complex objects like 3D anatomical meshes. In "Deep Structural Causal Shape Models" (Rasal et al., 2022), nodes $A$ (age), $S$ (sex), $B$ (brain volume), $V$ (stem volume), and $X$ (mesh) are linked by flows and conditional deep VAEs, with explicit formulas for interventions and counterfactual mesh generation.

3. Temporal and Interference-Driven SCMs

Time-series and systems under external interference are modeled by temporal SCMs with explicit latent processes:

• TLV-SCM (Temporal Latent Variable SCM) (Cai et al., 13 Nov 2025): Observed nodes $X_1, ..., X_m$ and latent "interference" variables $Z_1, ..., Z_n$ are linked across time by equations:

$$\begin{aligned} \mathbf{X}(t) &= (\mathbf{A}^{XX} \odot \mathbf{W}^{XX}) \mathbf{X}(t-1) + (\mathbf{A}^{XZ} \odot \mathbf{W}^{XZ}) \mathbf{Z}(t-1) + \mathbf{N}^{X}(t) \ \mathbf{Z}(t) &= (\mathbf{A}^{ZZ} \odot \mathbf{W}^{ZZ}) \mathbf{Z}(t-1) + \mathbf{N}^{Z}(t) \end{aligned}$$

Here, $\mathbf{A}$ encodes adjacency, $\mathbf{W}$ strength, and estimation employs ELBO-based variational inference with explicit sparsity priors and expert knowledge integration.

4. Causal Structure Learning Algorithms and Priors

Identifying the DAG and mechanisms of an SCM from data is a central challenge, with multiple algorithmic paradigms:

• Constraint-based (PC, FCI, rankPC): Exploit conditional independence to infer skeletons and orient v-structures. FCI allows for latent confounders and outputs a partial ancestral graph (PAG) (Heinze-Deml et al., 2017).
• Score-based (GES, GIES, MMHC): Search for DAG maximizing penalized likelihood (BIC, etc.). GIES exploits known interventions; hybrid methods combine constraint and likelihood.
• LiNGAM: Assumes linearity, acyclicity, and non-Gaussian noise; exploits ICA for identifiability.
• Environmental or invariance-based (BACKSHIFT): Recovers structure in multi-environmental data with unknown interventions exploiting joint diagonalization.

In CKH (Adib et al., 2022), causal knowledge is tiered by confidence into levels $\{L_1, L_2, L_3\}$ (expert, data-driven, literature), weighted convexly and encoded as soft/ hard constraints or regularization terms during structure learning.

5. Generalizations: Causal Constraints Models and Non-Standard SCMs

SCMs are insufficient for dynamical equilibria, functional laws, or physical systems with conservation constraints. CCMs generalize SCMs by encoding arbitrary algebraic or differential causal constraints with activation sets for each intervention regime (Blom et al., 2018):

• A CCM is $(V, U, \mathcal{C}, P_U)$, with each constraint $(f_k, c_k, A_k)$ active only under specified interventions.
• E.g., the ideal gas law $PV = Nk_B T$ is a CCM constraint holding under any intervention unless $P$ or $T$ is set independently.

6. Applications: Recommendation, LLMs, and Text

• Recommendation systems: SCMs encode both user decision mechanisms and RS-induced interventions as competing pathways, optimizing a mixture log-likelihood subject to DAG constraints. Augmented Lagrangian solvers with Gumbel-Softmax reparameterization enforce acyclicity (Xu et al., 2022).
• SD-SCMs for LLMs: LLMs can serve as causal mechanism generators, with prompts substituted for explicit equations. A DAG over covariates or treatments is mapped to a template set of possible outputs, and interventions are implemented by prompt engineering and forced assignment (Bynum et al., 12 Nov 2024).
• Text summarization: SCMs can distinguish causal content/style and non-causal factors for abstractive summarization, enabling identifiability of latent structure via VAE reformulations under exponential-family priors and additive noise (Chen et al., 2023).

7. Identifiability, Counterfactuals, and Confidence

Identifiability is a central property: in linear Gaussian SCMs with equal error variances, the DAG and weights are uniquely determined by the covariance (Strieder et al., 2023). For latent and partially-specified SCMs, causal EM algorithms approximate posteriors over exogenous distributions and yield credible intervals for causal effects and counterfactuals, with NP-hardness for general polytree graphs (Zaffalon et al., 2020).

Counterfactuals are computed via abduction-action-prediction in either full or sampled models. When structure is uncertain, confidence sets for causal effects are formed by test inversion across all DAGs consistent with the data, integrating both parameteric and structural sources of uncertainty.

8. Limitations, Open Problems, and Theoretical Guarantees

• SCMs assume acyclicity, causal sufficiency, faithfulness, and invariance; violations result in partial identification or require generalization (e.g., CCMs, latent variable models).
• For time series and high-dimensional domains, computational complexity and scalability are active research challenges (Cai et al., 13 Nov 2025).
• Empirical robustness to prior misspecification is attained via hierarchical or convex weighting (CKH), and performance metrics center on SHD, AUROC, F1, recall, precision, and MCC.
• SCMs are strictly less expressive than CCMs in systems governed by multiple constraints, functional laws, or equilibrium selection (Blom et al., 2018).

In summary, Structural Causal Models provide a rigorous, extensible foundation for causal graphical reasoning, interventional and counterfactual analysis, and structure learning. Extensions encompassing latent variables, temporality, expert knowledge, deep generative components, and constraint-based generalizations have advanced the scope and practical applicability of SCMs in domains spanning the sciences, recommendation, high-dimensional generative modeling, and language processing (Subramanian et al., 2022, Cai et al., 13 Nov 2025, Blom et al., 2018, Adib et al., 2022, Heinze-Deml et al., 2017, Strieder et al., 2023, Rasal et al., 2022, Bynum et al., 12 Nov 2024, Chen et al., 2023).