# Being Bayesian about Network Structure

arXiv: https://arxiv.org/abs/1301.3856 · HTML: https://mlsanity.com/paper/1301.3856/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2013.

## Built on by (most cited first, 17 of 17)

- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Bayesian learning of Causal Structure and Mechanisms with GFlowNets and Variational Bayes](https://mlsanity.com/paper/2211.02763/) (2022)
- [A Meta-Learning Approach to Bayesian Causal Discovery](https://mlsanity.com/paper/2412.16577/) (2024)
- [Active causal structure learning with advice](https://mlsanity.com/paper/2305.19588/) (2023)
- [Interactive Causal Structure Discovery in Earth System Sciences](https://mlsanity.com/paper/2107.01126/) (2021)
- [Learning Latent Structural Causal Models](https://mlsanity.com/paper/2210.13583/) (2022)
- [A Circuit Complexity Formulation of Algorithmic Information Theory](https://mlsanity.com/paper/2306.14087/) (2023)
- [Causal structure learning with momentum: Sampling distributions over Markov Equivalence Classes of DAGs](https://mlsanity.com/paper/2310.05655/) (2023)
- [Estimation of partially known Gaussian graphical models with score-based structural priors](https://mlsanity.com/paper/2401.14340/) (2024)
- [Causal Discovery under Off-Target Interventions](https://mlsanity.com/paper/2402.08229/) (2024)
- [Incorporating Expert Knowledge into Bayesian Causal Discovery of Mixtures of Directed Acyclic Graphs](https://mlsanity.com/paper/2510.06735/) (2025)
- [Causal Discovery on Higher-Order Interactions](https://mlsanity.com/paper/2511.14206/) (2025)
- [Identification of Latent Variables From Graphical Model Residuals](https://mlsanity.com/paper/2101.02332/) (2021)
- [A general framework for adaptive two-index fusion attribute weighted naive Bayes](https://mlsanity.com/paper/2202.11963/) (2022)
- [Learning Dynamic Bayesian Networks from Data: Foundations, First Principles and Numerical Comparisons](https://mlsanity.com/paper/2406.17585/) (2024)
- [Empirical Bayes for Dynamic Bayesian Networks Using Generalized Variational Inference](https://mlsanity.com/paper/2406.17831/) (2024)

## Abstract (arXiv)

In many domains, we are interested in analyzing the structure of the underlying distribution, e.g., whether one variable is a direct parent of the other. Bayesian model-selection attempts to find the MAP model and use its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible posterior. Thus, we want compute the Bayesian posterior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to efficiently compute a sum over the exponential number of networks that are consistent with a fixed ordering over network variables. This allows us to compute, for a given ordering, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approximates the Bayesian posterior of a feature. Our approach uses a Markov Chain Monte Carlo (MCMC) method, but over orderings rather than over network structures. The space of orderings is much smaller and more regular than the space of structures, and has a smoother posterior `landscape'. We present empirical results on synthetic and real-life datasets that compare our approach to full model averaging (when possible), to MCMC over network structures, and to a non-Bayesian bootstrap approach.