mlsanityData to 7 Oct 2026

Markov Properties for Graphical Models with Cycles and Latent Variables

arXiv 1710.08775 ↗First posted 24 Oct 2017math.ST stat.ME stat.MLcited by 32

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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.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). First posted on arXiv 24 Oct 2017.

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Citing papers per year: 2021: 4; 2022: 8; 2023: 6; 2024: 4; 2025: 7; 2026: 3.

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  1. 2023A Survey on Causal Discovery: Theory and Practicecited by 63
  2. 2022Local Constraint-Based Causal Discovery under Selection Biascited by 15
  3. 2023Causal models in string diagramscited by 14
  4. 2024Dynamic Structural Causal Modelscited by 13
  5. 2022The d-separation criterion in Categorical Probabilitycited by 12
  6. 2023Identifiability of Direct Effects from Summary Causal Graphscited by 12
  7. 2024Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confoundingcited by 8
  8. 2022A Unified Experiment Design Approach for Cyclic and Acyclic Causal Modelscited by 7
  9. 2021Universal Decision Modelscited by 6
  10. 2022Learning Relational Causal Models with Cycles through Relational Acyclificationcited by 6
  11. 2023Establishing Markov Equivalence in Cyclic Directed Graphscited by 5
  12. 2021Causality and independence in perfectly adapted dynamical systemscited by 4
20 more
  1. 2021Causal Homotopycited by 4
  2. 2022Relational Causal Models with Cycles:Representation and Reasoningcited by 4
  3. 2025Differentiable Cyclic Causal Discovery Under Unmeasured Confounderscited by 4
  4. 2025Intuitionistic j-Do-Calculus in Topos Causal Modelscited by 4
  5. 2026Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learningcited by 4
  6. 2021Conditions and Assumptions for Constraint-based Causal Structure Learningcited by 3
  7. 2022Categoroids: Universal Conditional Independencecited by 3
  8. 2025Identifying Macro Causal Effects in a C-DMG over ADMGscited by 3
  9. 2022On The Universality of Diagrams for Causal Inference and The Causal Reproducing Propertycited by 2
  10. 2023On the Foundations of Cycles in Bayesian Networkscited by 2
  11. 2025An Asymmetric Independence Model for Causal Discovery on Path Spacescited by 2
  12. 2025Identifying Macro Causal Effects in C-DMGs over DMGscited by 2
  13. 2023Axiomatization of Interventional Probability Distributionscited by 1
  14. 2024MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Datacited by 1
  15. 2022A Layered Architecture for Universal Causalitycited by 0
  16. 2024Spatio-Temporal Graphical Counterfactuals: An Overviewcited by 0
  17. 2025σ-Maximal Ancestral Graphscited by 0
  18. 2025Topos Causal Modelscited by 0
  19. 2026Data-Driven Covariate Selection for Nonparametric and Cycle-Agnostic Causal Effect Estimationcited by 0
  20. 2026Causal Reasoning with Bipartite Graphical Causal Modelscited by 0