# The Bayesian Structural EM Algorithm

arXiv: https://arxiv.org/abs/1301.7373 · HTML: https://mlsanity.com/paper/1301.7373/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2013.

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

- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [MissDAG: Causal Discovery in the Presence of Missing Data with Continuous Additive Noise Models](https://mlsanity.com/paper/2205.13869/) (2022)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [Bayesian Structure Scores for Probabilistic Circuits](https://mlsanity.com/paper/2302.12130/) (2023)
- [Learning Latent Structural Causal Models](https://mlsanity.com/paper/2210.13583/) (2022)
- [Temporal-spatial Correlation Attention Network for Clinical Data Analysis in Intensive Care Unit](https://mlsanity.com/paper/2306.01970/) (2023)
- [Estimating Causal Effects from Learned Causal Networks](https://mlsanity.com/paper/2408.14101/) (2024)
- [Flow based approach for Dynamic Temporal Causal models with non-Gaussian or Heteroscedastic Noises](https://mlsanity.com/paper/2506.17065/) (2025)
- [Discovery and density estimation of latent confounders in Bayesian networks with evidence lower bound](https://mlsanity.com/paper/2206.05490/) (2022)
- [Context-Specific Likelihood Weighting](https://mlsanity.com/paper/2101.09791/) (2021)
- [Causal Discovery from Sparse Time-Series Data Using Echo State Network](https://mlsanity.com/paper/2201.02933/) (2022)
- [Learning Multitask Gaussian Bayesian Networks](https://mlsanity.com/paper/2205.05343/) (2022)
- [Learning Cyclic Causal Models from Incomplete Data](https://mlsanity.com/paper/2402.15625/) (2024)
- [Investigating the validity of structure learning algorithms in identifying risk factors for intervention in patients with diabetes](https://mlsanity.com/paper/2403.14327/) (2024)
- [Learning Staged Trees from Incomplete Data](https://mlsanity.com/paper/2405.18306/) (2024)
- [MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data](https://mlsanity.com/paper/2410.18918/) (2024)
- [Stable Blanket with Hidden Variables and Cycles](https://mlsanity.com/paper/2605.01856/) (2026)
- [KG-SoftMAP: Soft Knowledge-Graph Priors for Bayesian Network Structure Learning from Sparse Discrete Data](https://mlsanity.com/paper/2606.10358/) (2026)
- [Identification of Latent Variables From Graphical Model Residuals](https://mlsanity.com/paper/2101.02332/) (2021)
- [Context-specific kernel-based hidden Markov model for time series analysis](https://mlsanity.com/paper/2301.09870/) (2023)
- [Probabilistic Multi-Dimensional Classification](https://mlsanity.com/paper/2306.06517/) (2023)
- [Gaussian Mixture Models for Affordance Learning using Bayesian Networks](https://mlsanity.com/paper/2402.06078/) (2024)
- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)
- [Learning discrete Bayesian networks with hierarchical Dirichlet shrinkage](https://mlsanity.com/paper/2509.13267/) (2025)
- [Shylock: Causal Discovery in Multivariate Time Series based on Hybrid Constraints](https://mlsanity.com/paper/2510.21181/) (2025)
- [RECLAIM: Cyclic Causal Discovery Amid Measurement Noise](https://mlsanity.com/paper/2603.20585/) (2026)

## Abstract (arXiv)

In recent years there has been a flurry of works on learning Bayesian networks from data. One of the hard problems in this area is how to effectively learn the structure of a belief network from incomplete data- that is, in the presence of missing values or hidden variables. In a recent paper, I introduced an algorithm called Structural EM that combines the standard Expectation Maximization (EM) algorithm, which optimizes parameters, with structure search for model selection. That algorithm learns networks based on penalized likelihood scores, which include the BIC/MDL score and various approximations to the Bayesian score. In this paper, I extend Structural EM to deal directly with Bayesian model selection. I prove the convergence of the resulting algorithm and show how to apply it for learning a large class of probabilistic models, including Bayesian networks and some variants thereof.