# Incorporating Expert Knowledge into Bayesian Causal Discovery of Mixtures of Directed Acyclic Graphs

arXiv: https://arxiv.org/abs/2510.06735 · HTML: https://mlsanity.com/paper/2510.06735/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 66 references that are in the map.
- First posted on arXiv 8 Oct 2025.

## Builds on (most cited first, 30 of 31)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Modern Bayesian Experimental Design](https://mlsanity.com/paper/2302.14545/) (2023)
- [DiBS: Differentiable Bayesian Structure Learning](https://mlsanity.com/paper/2105.11839/) (2021)
- [A survey of Bayesian Network structure learning](https://mlsanity.com/paper/2109.11415/) (2021)
- [A Tutorial on Learning With Bayesian Networks](https://mlsanity.com/paper/2002.00269/) (2020)
- [Understanding predictive information criteria for Bayesian models](https://mlsanity.com/paper/1307.5928/) (2013)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Why did the distribution change?](https://mlsanity.com/paper/2102.13384/) (2021)
- [The impact of prior knowledge on causal structure learning](https://mlsanity.com/paper/2102.00473/) (2021)
- [Active Bayesian Causal Inference](https://mlsanity.com/paper/2206.02063/) (2022)
- [A Survey on Causal Discovery Methods for I.I.D. and Time Series Data](https://mlsanity.com/paper/2303.15027/) (2023)
- [NTS-NOTEARS: Learning Nonparametric DBNs With Prior Knowledge](https://mlsanity.com/paper/2109.04286/) (2021)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Theory Refinement on Bayesian Networks](https://mlsanity.com/paper/1303.5709/) (2013)
- [Causal Structure Discovery from Distributions Arising from Mixtures of DAGs](https://mlsanity.com/paper/2001.11940/) (2020)
- [Large Language Models are Effective Priors for Causal Graph Discovery](https://mlsanity.com/paper/2405.13551/) (2024)
- [Prior knowledge elicitation: The past, present, and future](https://mlsanity.com/paper/2112.01380/) (2021)
- [Differentiable Causal Discovery Under Latent Interventions](https://mlsanity.com/paper/2203.02336/) (2022)
- [The Landscape of Causal Discovery Data: Grounding Causal Discovery in Real-World Applications](https://mlsanity.com/paper/2412.01953/) (2024)
- [Being Bayesian about Network Structure](https://mlsanity.com/paper/1301.3856/) (2013)
- [Differentiable Multi-Target Causal Bayesian Experimental Design](https://mlsanity.com/paper/2302.10607/) (2023)
- [A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations](https://mlsanity.com/paper/2106.03480/) (2021)
- [Bayesian Optimal Experimental Design for Inferring Causal Structure](https://mlsanity.com/paper/2103.15229/) (2021)
- [Differentiable Bayesian Structure Learning with Acyclicity Assurance](https://mlsanity.com/paper/2309.01392/) (2023)
- [Goal-Oriented Sequential Bayesian Experimental Design for Causal Learning](https://mlsanity.com/paper/2507.07359/) (2025)

## Built on by (most cited first, 2 of 2)

- [Causal Discovery Should Embrace the Wisdom of the Crowd](https://mlsanity.com/paper/2603.02678/) (2026)
- [Coupling Noisy Pairwise Knowledge to the DAG Posterior for Causal Discovery](https://mlsanity.com/paper/2610.04559/) (2026)

## Abstract (arXiv)

Bayesian causal discovery benefits from prior information elicited from domain experts, and in heterogeneous domains any prior knowledge would be badly needed. However, so far prior elicitation approaches have assumed a single causal graph and hence are not suited to heterogeneous domains. We propose a causal elicitation strategy for heterogeneous settings, based on Bayesian experimental design (BED) principles, and a variational mixture structure learning (VaMSL) method -- extending the earlier differentiable Bayesian structure learning (DiBS) method -- to iteratively infer mixtures of causal Bayesian networks (CBNs). We construct an informative graph prior incorporating elicited expert feedback in the inference of mixtures of CBNs. Our proposed method successfully produces a set of alternative causal models (mixture components or clusters), and achieves an improved structure learning performance on heterogeneous synthetic data when informed by a simulated expert. Finally, we demonstrate that our approach is capable of capturing complex distributions in a breast cancer database.