mlsanityData to 7 Oct 2026

Margins of discrete Bayesian networks

arXiv 1501.02103 ↗First posted 9 Jan 2015math.ST stat.ML stat.THcited by 26

Not on the map. Why?

Abstract (arXiv)

Bayesian network models with latent variables are widely used in statistics and machine learning. In this paper we provide a complete algebraic characterization of Bayesian network models with latent variables when the observed variables are discrete and no assumption is made about the state-space of the latent variables. We show that it is algebraically equivalent to the so-called nested Markov model, meaning that the two are the same up to inequality constraints on the joint probabilities. In particular these two models have the same dimension. The nested Markov model is therefore the best possible description of the latent variable model that avoids consideration of inequalities, which are extremely complicated in general. A consequence of this is that the constraint finding algorithm of Tian and Pearl (UAI 2002, pp519-527) is complete for finding equality constraints. Latent variable models suffer from difficulties of unidentifiable parameters and non-regular asymptotics; in contrast the nested Markov model is fully identifiable, represents a curved exponential family of known dimension, and can easily be fitted using an explicit parameterization.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). First posted on arXiv 9 Jan 2015.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 4; 2022: 9; 2023: 3; 2024: 3; 2025: 4; 2026: 3.

26 papers.

Sort:

  1. 2021Partial Counterfactual Identification from Observational and Experimental Datacited by 64
  2. 2021Learning latent causal graphs via mixture oraclescited by 55
  3. 2022Causal Structure Learning: a Combinatorial Perspectivecited by 46
  4. 2021An Automated Approach to Causal Inference in Discrete Settingscited by 39
  5. 2025Towards Interpretable Deep Generative Models via Causal Representation Learningcited by 12
  6. 2023Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractionscited by 11
  7. 2022On Testability of the Front-Door Model via Verma Constraintscited by 8
  8. 2024The Fragility of Fairness: Causal Sensitivity Analysis for Fair Machine Learningcited by 8
  9. 2022Causal Discovery in Linear Latent Variable Models Subject to Measurement Errorcited by 7
  10. 2022On the Complexity of Counterfactual Reasoningcited by 7
  11. 2024Deriving Causal Order from Single-Variable Interventions: Guarantees & Algorithmcited by 5
  12. 2021Causal Homotopycited by 4
14 more
  1. 2022Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theorycited by 4
  2. 2023When does the ID algorithm fail?cited by 4
  3. 2026Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learningcited by 4
  4. 2022The Causal Marginal Polytope for Bounding Treatment Effectscited by 3
  5. 2022Categoroids: Universal Conditional Independencecited by 3
  6. 2022On The Universality of Diagrams for Causal Inference and The Causal Reproducing Propertycited by 2
  7. 2024Efficiently Deciding Algebraic Equivalence of Bow-Free Acyclic Path Diagramscited by 1
  8. 2025Recursive Equations For Imputation Of Missing Not At Random Data With Sparse Pattern Supportcited by 1
  9. 2022A Layered Architecture for Universal Causalitycited by 0
  10. 2023Supermodular Rank: Set Function Decomposition and Optimizationcited by 0
  11. 2025On Theoretical Identifiability of Discrete Latent Causal Graphical Modelscited by 0
  12. 2025A fine-grained look at causal effects in causal spacescited by 0
  13. 2026Verifying formulas for interventional distributionscited by 0
  14. 2026A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Modelscited by 0