# Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks

arXiv: https://arxiv.org/abs/2210.00822 · HTML: https://mlsanity.com/paper/2210.00822/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 3 Oct 2022.

## Builds on (most cited first, 11 of 11)

- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [Characterizing and Learning Equivalence Classes of Causal DAGs under Interventions](https://mlsanity.com/paper/1802.06310/) (2018)
- [ABCD-Strategy: Budgeted Experimental Design for Targeted Causal Structure Discovery](https://mlsanity.com/paper/1902.10347/) (2019)
- [GeneDisco: A Benchmark for Experimental Design in Drug Discovery](https://mlsanity.com/paper/2110.11875/) (2021)
- [A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations](https://mlsanity.com/paper/2106.03480/) (2021)
- [Recovering Causal Structures from Low-Order Conditional Independencies](https://mlsanity.com/paper/2010.02675/) (2020)
- [Learning from Pairwise Marginal Independencies](https://mlsanity.com/paper/1508.00280/) (2015)
- [A Transformational Characterization of Unconditionally Equivalent Bayesian Networks](https://mlsanity.com/paper/2203.00521/) (2022)
- [Representation Theory of the Symmetric Group in Voting Theory and Game Theory](https://mlsanity.com/paper/1508.05891/) (2015)
- [A survey of discrete methods in (algebraic) statistics for networks](https://mlsanity.com/paper/1510.02838/) (2015)
- [Marginal Independence Models](https://mlsanity.com/paper/2112.10287/) (2021)

## Built on by (most cited first, 3 of 3)

- [Learning nonparametric latent causal graphs with unknown interventions](https://mlsanity.com/paper/2306.02899/) (2023)
- [Neuro-Causal Factor Analysis](https://mlsanity.com/paper/2305.19802/) (2023)
- [Coarsening Causal DAG Models](https://mlsanity.com/paper/2601.10531/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data, learning an undirected graph we call the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the 20% HPD credible sets include the true structure at a high rate for data-generating graphs with density at least 0.5.