# Margins of discrete Bayesian networks

arXiv: https://arxiv.org/abs/1501.02103 · HTML: https://mlsanity.com/paper/1501.02103/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2015.

## Built on by (most cited first, 26 of 26)

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- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [An Automated Approach to Causal Inference in Discrete Settings](https://mlsanity.com/paper/2109.13471/) (2021)
- [Towards Interpretable Deep Generative Models via Causal Representation Learning](https://mlsanity.com/paper/2504.11609/) (2025)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [On Testability of the Front-Door Model via Verma Constraints](https://mlsanity.com/paper/2203.00161/) (2022)
- [The Fragility of Fairness: Causal Sensitivity Analysis for Fair Machine Learning](https://mlsanity.com/paper/2410.09600/) (2024)
- [Causal Discovery in Linear Latent Variable Models Subject to Measurement Error](https://mlsanity.com/paper/2211.03984/) (2022)
- [On the Complexity of Counterfactual Reasoning](https://mlsanity.com/paper/2211.13447/) (2022)
- [When does the ID algorithm fail?](https://mlsanity.com/paper/2307.03750/) (2023)
- [Deriving Causal Order from Single-Variable Interventions: Guarantees & Algorithm](https://mlsanity.com/paper/2405.18314/) (2024)
- [Causal Homotopy](https://mlsanity.com/paper/2112.01847/) (2021)
- [Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theory](https://mlsanity.com/paper/2209.06262/) (2022)
- [Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning](https://mlsanity.com/paper/2603.04780/) (2026)
- [The Causal Marginal Polytope for Bounding Treatment Effects](https://mlsanity.com/paper/2202.13851/) (2022)
- [Categoroids: Universal Conditional Independence](https://mlsanity.com/paper/2208.11077/) (2022)
- [On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property](https://mlsanity.com/paper/2207.02917/) (2022)
- [Efficiently Deciding Algebraic Equivalence of Bow-Free Acyclic Path Diagrams](https://mlsanity.com/paper/2406.09049/) (2024)
- [Recursive Equations For Imputation Of Missing Not At Random Data With Sparse Pattern Support](https://mlsanity.com/paper/2507.16107/) (2025)
- [A Layered Architecture for Universal Causality](https://mlsanity.com/paper/2212.08981/) (2022)
- [Supermodular Rank: Set Function Decomposition and Optimization](https://mlsanity.com/paper/2305.14632/) (2023)
- [On Theoretical Identifiability of Discrete Latent Causal Graphical Models](https://mlsanity.com/paper/2505.18410/) (2025)
- [A fine-grained look at causal effects in causal spaces](https://mlsanity.com/paper/2512.11919/) (2025)
- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)
- [A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Models](https://mlsanity.com/paper/2607.23439/) (2026)

## 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.