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

TL;DR

The paper tackles causal discovery under model misspecification due to heterogeneity by enabling expert-guided, Bayesian inference over mixtures of causal DAGs (CBNs). It introduces an informative latent graph prior, derives a BED-based elicitation strategy to query experts efficiently, and extends DiBS to VaMSL for joint inference over graph structure and parameters across multiple components, including nonlinear BNs. Empirical results on synthetic and breast cancer data show that incorporating expert information improves structure learning (lower ESHD) and downstream classification in heterogeneous settings, achieving performance close to supervised baselines in some cases. The approach is practical for domains with limited data but rich domain expertise, enabling robust, interpretable causal discovery with explicit handling of heterogeneity and expert input.

Abstract

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.

Figures (18)

• Figure 1: Generative model for expert edge beliefs ($\psi_{ij}$), with hyperparameters $\alpha_0,\beta_0$, and trials observed by the expert $(\mathcal{K}_{ij})$.
• Figure 2: Generative model for VaMSL using DiBS.
• Figure 3: Boxplot (and values) of average ESHD between learned graphs and true graphs (left) and classification accuracy (right) by method, in data generated from a mixture of two linear (top) and non-linear (bottom) ER BNs with Gaussian errors. Learned graphs not available for Causal k-means and GMMs.
• Figure 4: Comparison between BED and random approaches for query selection by mean and bootstrapped 95% confidence interval for homogeneous experiments with varying number of queries (top) and varying expert reliability (bottom) when inferring Gaussian ER network in the linear (left) and non-linear case (right).
• Figure 5: Boxplot and values of classification accuracy by method in out-of-sample data for the breast cancer dataset, in ten splits of the data.

Theorems & Definitions (1)

• Proposition 2: Latent mixture posterior expectation