# Markov Properties for Graphical Models with Cycles and Latent Variables

arXiv: https://arxiv.org/abs/1710.08775 · HTML: https://mlsanity.com/paper/1710.08775/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2017.

## Built on by (most cited first, 30 of 32)

- [A Survey on Causal Discovery: Theory and Practice](https://mlsanity.com/paper/2305.10032/) (2023)
- [Local Constraint-Based Causal Discovery under Selection Bias](https://mlsanity.com/paper/2203.01848/) (2022)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Dynamic Structural Causal Models](https://mlsanity.com/paper/2406.01161/) (2024)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Identifiability of Direct Effects from Summary Causal Graphs](https://mlsanity.com/paper/2306.16958/) (2023)
- [Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding](https://mlsanity.com/paper/2407.07934/) (2024)
- [Universal Decision Models](https://mlsanity.com/paper/2110.15431/) (2021)
- [A Unified Experiment Design Approach for Cyclic and Acyclic Causal Models](https://mlsanity.com/paper/2205.10083/) (2022)
- [Learning Relational Causal Models with Cycles through Relational Acyclification](https://mlsanity.com/paper/2208.12210/) (2022)
- [Establishing Markov Equivalence in Cyclic Directed Graphs](https://mlsanity.com/paper/2309.03092/) (2023)
- [Intuitionistic j-Do-Calculus in Topos Causal Models](https://mlsanity.com/paper/2510.17944/) (2025)
- [Causality and independence in perfectly adapted dynamical systems](https://mlsanity.com/paper/2101.11885/) (2021)
- [Causal Homotopy](https://mlsanity.com/paper/2112.01847/) (2021)
- [Relational Causal Models with Cycles:Representation and Reasoning](https://mlsanity.com/paper/2202.10706/) (2022)
- [Identifying Macro Causal Effects in a C-DMG over ADMGs](https://mlsanity.com/paper/2504.01551/) (2025)
- [Differentiable Cyclic Causal Discovery Under Unmeasured Confounders](https://mlsanity.com/paper/2508.08450/) (2025)
- [Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning](https://mlsanity.com/paper/2603.04780/) (2026)
- [Conditions and Assumptions for Constraint-based Causal Structure Learning](https://mlsanity.com/paper/2103.13521/) (2021)
- [Categoroids: Universal Conditional Independence](https://mlsanity.com/paper/2208.11077/) (2022)
- [Identifying Macro Causal Effects in C-DMGs over DMGs](https://mlsanity.com/paper/2506.19650/) (2025)
- [On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property](https://mlsanity.com/paper/2207.02917/) (2022)
- [On the Foundations of Cycles in Bayesian Networks](https://mlsanity.com/paper/2301.08608/) (2023)
- [An Asymmetric Independence Model for Causal Discovery on Path Spaces](https://mlsanity.com/paper/2503.09859/) (2025)
- [Axiomatization of Interventional Probability Distributions](https://mlsanity.com/paper/2305.04479/) (2023)
- [MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data](https://mlsanity.com/paper/2410.18918/) (2024)
- [A Layered Architecture for Universal Causality](https://mlsanity.com/paper/2212.08981/) (2022)
- [Spatio-Temporal Graphical Counterfactuals: An Overview](https://mlsanity.com/paper/2407.01875/) (2024)
- [σ-Maximal Ancestral Graphs](https://mlsanity.com/paper/2507.00093/) (2025)
- [Topos Causal Models](https://mlsanity.com/paper/2508.08295/) (2025)

## Abstract (arXiv)

We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that can model cycles. We define and analyse several different Markov properties that relate the graphical structure of a HEDG with a probability distribution on a corresponding product space over the set of nodes, for example factorization properties, structural equations properties, ordered/local/global Markov properties, and marginal versions of these. The various Markov properties for HEDGes are in general not equivalent to each other when cycles or hyperedges are present, in contrast with the simpler case of directed acyclic graphical (DAG) models (also known as Bayesian networks). We show how the Markov properties for HEDGes - and thus the corresponding graphical Markov models - are logically related to each other.