How Does Bayesian Causal Discovery Fail? Characterising Structural Consequences in Linear Gaussian Networks under Latent Confounding
06:00 · July 13, 2026 · arXiv cs.AI RSS
Bayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size -- more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.
Summary
This arXiv paper examines how Bayesian causal discovery behaves when the causal-sufficiency assumption is violated by an unobserved confounder that additively affects exactly two observed variables in a linear Gaussian structural causal model. The authors focus on the posterior distribution over directed acyclic graphs obtained with the Bayesian Gaussian equivalent (BGe) score and show that the latent common cause induces an extra covariance term that no causally sufficient DAG can explain.
They derive an explicit, sample-size-dependent correlation threshold beyond which the marginal likelihood begins to favour graphs containing a spurious edge between the two confounded variables. Because the threshold decreases as the number of observations grows, larger data sets paradoxically make the inclusion of the incorrect edge more likely once the induced correlation exceeds the critical value.
Beyond this threshold the posterior exhibits two qualitatively different failure modes whose character is governed by the local collider structure surrounding the confounded pair. In one regime the posterior silently concentrates on graphs that absorb the spurious dependence without visibly degrading fit on other edges; in the other the posterior becomes noisy, spreading probability mass across multiple incorrect structures. Both regimes are characterised analytically and verified through exact posterior enumeration on small graphs.
The work is carried out by researchers at Utrecht University and Leiden University and supplies a precise structural account of how posterior uncertainty degrades under latent confounding rather than merely noting that identifiability is lost.
Why it matters
Provides novel theoretical characterization of failure modes in Bayesian structure learning directly applicable to Dutch/EU researchers using causal discovery in genomics, epidemiology, or similar domains; high technical depth and novelty for advanced readers.
