# Hard and Soft EM in Bayesian Network Learning from Incomplete Data

arXiv: https://arxiv.org/abs/2012.05269 · HTML: https://mlsanity.com/paper/2012.05269/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2020.

## Built on by (most cited first, 5 of 5)

- [Greedy structure learning from data that contain systematic missing values](https://mlsanity.com/paper/2107.04184/) (2021)
- [Learning Staged Trees from Incomplete Data](https://mlsanity.com/paper/2405.18306/) (2024)
- [High-Dimensional Inference in Bayesian Networks](https://mlsanity.com/paper/2112.09217/) (2021)
- [LUME-DBN: Full Bayesian Learning of DBNs from Incomplete data in Intensive Care](https://mlsanity.com/paper/2511.04333/) (2025)
- [EM-NeSy: Expectation Maximization for Neurosymbolic Learning](https://mlsanity.com/paper/2606.14463/) (2026)

## Abstract (arXiv)

Incomplete data are a common feature in many domains, from clinical trials to industrial applications. Bayesian networks (BNs) are often used in these domains because of their graphical and causal interpretations. BN parameter learning from incomplete data is usually implemented with the Expectation-Maximisation algorithm (EM), which computes the relevant sufficient statistics ("soft EM") using belief propagation. Similarly, the Structural Expectation-Maximisation algorithm (Structural EM) learns the network structure of the BN from those sufficient statistics using algorithms designed for complete data. However, practical implementations of parameter and structure learning often impute missing data ("hard EM") to compute sufficient statistics instead of using belief propagation, for both ease of implementation and computational speed. In this paper, we investigate the question: what is the impact of using imputation instead of belief propagation on the quality of the resulting BNs? From a simulation study using synthetic data and reference BNs, we find that it is possible to recommend one approach over the other in several scenarios based on the characteristics of the data. We then use this information to build a simple decision tree to guide practitioners in choosing the EM algorithm best suited to their problem.