# VertiBayes: Learning Bayesian network parameters from vertically partitioned data with missing values

arXiv: https://arxiv.org/abs/2210.17228 · HTML: https://mlsanity.com/paper/2210.17228/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 38 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 3 of 3)

- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Towards Federated Bayesian Network Structure Learning with Continuous Optimization](https://mlsanity.com/paper/2110.09356/) (2021)
- [Bayesian Networks in Healthcare: Distribution by Medical Condition](https://mlsanity.com/paper/2002.00224/) (2020)

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

- [CCBNet: Confidential Collaborative Bayesian Networks Inference](https://mlsanity.com/paper/2405.15055/) (2024)
- [Differentially Private and Federated Structure Learning in Bayesian Networks](https://mlsanity.com/paper/2512.01708/) (2025)
- [Federated Bayesian Network Ensembles](https://mlsanity.com/paper/2402.12142/) (2024)
- [Uncertainty Quantification in Federated Granger Causality Learning](https://mlsanity.com/paper/2602.13004/) (2026)
- [A Survey on Federated Causal Discovery and Inference](https://mlsanity.com/paper/2606.23741/) (2026)

## Abstract (arXiv)

Federated learning makes it possible to train a machine learning model on decentralized data. Bayesian networks are probabilistic graphical models that have been widely used in artificial intelligence applications. Their popularity stems from the fact they can be built by combining existing expert knowledge with data and are highly interpretable, which makes them useful for decision support, e.g. in healthcare. While some research has been published on the federated learning of Bayesian networks, publications on Bayesian networks in a vertically partitioned or heterogeneous data setting (where different variables are located in different datasets) are limited, and suffer from important omissions, such as the handling of missing data. In this article, we propose a novel method called VertiBayes to train Bayesian networks (structure and parameters) on vertically partitioned data, which can handle missing values as well as an arbitrary number of parties. For structure learning we adapted the widely used K2 algorithm with a privacy-preserving scalar product protocol. For parameter learning, we use a two-step approach: first, we learn an intermediate model using maximum likelihood by treating missing values as a special value and then we train a model on synthetic data generated by the intermediate model using the EM algorithm. The privacy guarantees of our approach are equivalent to the ones provided by the privacy preserving scalar product protocol used. We experimentally show our approach produces models comparable to those learnt using traditional algorithms and we estimate the increase in complexity in terms of samples, network size, and complexity. Finally, we propose two alternative approaches to estimate the performance of the model using vertically partitioned data and we show in experiments that they lead to reasonably accurate estimates.