# Foundations of Structural Causal Models with Cycles and Latent Variables

arXiv: https://arxiv.org/abs/1611.06221 · HTML: https://mlsanity.com/paper/1611.06221/

## Key facts

- Cited by 134 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2016.

## Built on by (most cited first, 30 of 134)

- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Causal Abstraction: A Theoretical Foundation for Mechanistic Interpretability](https://mlsanity.com/paper/2301.04709/) (2023)
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- [CLadder: Assessing Causal Reasoning in Language Models](https://mlsanity.com/paper/2312.04350/) (2023)
- [Causal Machine Learning for Healthcare and Precision Medicine](https://mlsanity.com/paper/2205.11402/) (2022)
- [A Survey on Causal Discovery: Theory and Practice](https://mlsanity.com/paper/2305.10032/) (2023)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [From Statistical to Causal Learning](https://mlsanity.com/paper/2204.00607/) (2022)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Compositional Abstraction Error and a Category of Causal Models](https://mlsanity.com/paper/2103.15758/) (2021)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Discovering Agents](https://mlsanity.com/paper/2208.08345/) (2022)
- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [Sharp Bounds for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2305.16988/) (2023)
- [Causal Inference Through the Structural Causal Marginal Problem](https://mlsanity.com/paper/2202.01300/) (2022)
- [Causal Abstraction with Soft Interventions](https://mlsanity.com/paper/2211.12270/) (2022)
- [Self-Compatibility: Evaluating Causal Discovery without Ground Truth](https://mlsanity.com/paper/2307.09552/) (2023)
- [Passive learning of active causal strategies in agents and language models](https://mlsanity.com/paper/2305.16183/) (2023)
- [Improvement-Focused Causal Recourse (ICR)](https://mlsanity.com/paper/2210.15709/) (2022)
- [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023)
- [Orion-MSP: Multi-Scale Sparse Attention for Tabular In-Context Learning](https://mlsanity.com/paper/2511.02818/) (2025)
- [Exploiting Independent Instruments: Identification and Distribution Generalization](https://mlsanity.com/paper/2202.01864/) (2022)
- [What-is and How-to for Fairness in Machine Learning: A Survey, Reflection, and Perspective](https://mlsanity.com/paper/2206.04101/) (2022)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Invariant Policy Learning: A Causal Perspective](https://mlsanity.com/paper/2106.00808/) (2021)
- [Local Constraint-Based Causal Discovery under Selection Bias](https://mlsanity.com/paper/2203.01848/) (2022)
- [Targeted Reduction of Causal Models](https://mlsanity.com/paper/2311.18639/) (2023)
- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [Identifiability of total effects from abstractions of time series causal graphs](https://mlsanity.com/paper/2310.14691/) (2023)

## Abstract (arXiv)

Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a well-studied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist, and if it exists the marginal model does not always respect the latent projection; they do not always satisfy a Markov property; and their graphs are not always consistent with their causal semantics. We prove that for SCMs in general each of these properties does hold under certain solvability conditions. Our work generalizes results for SCMs with cycles that were only known for certain special cases so far. We introduce the class of simple SCMs that extends the class of acyclic SCMs to the cyclic setting, while preserving many of the convenient properties of acyclic SCMs. With this paper we aim to provide the foundations for a general theory of statistical causal modeling with SCMs.