Scalable Bayesian structure learning of directed acyclic graphs via Laplace approximation, with an application to breast cancer gene expression networks

2026-07-11

AI Summary

This work addresses the computational challenges in Bayesian DAG structure learning arising from the super-exponential growth of the DAG space and the intractability of marginal likelihoods under non-conjugate priors. The authors propose a novel approach based on a modified Cholesky parameterization of the precision matrix, deriving—for the first time—the node-wise marginal likelihood under a Normal–Gamma non-conjugate prior in the form of a generalized inverse Gaussian distribution. Leveraging its asymptotic properties for large parameters, they construct an efficient Laplace approximation-based scoring function embedded within a Metropolis–Hastings sampler for DAG search. Additionally, a probit link couples latent variables with binary clinical outcomes. This framework overcomes the limitations of conjugate priors and enables exact posterior sampling of conditional variances. Experiments demonstrate superior performance over PC, GES, NOTEARS, DAGMA, and conjugate baselines on synthetic data, successful recovery of known structures in the Sachs signaling pathway and Wisconsin breast cancer datasets, and accurate prediction of malignancy with a cross-validated ROC-AUC of 0.94.

Abstract

Structure learning of directed acyclic graphs (DAGs) from observational data is a foundational task in causal discovery and is widely used to infer regulatory networks from medical and genomic measurements. The Bayesian formulation quantifies model uncertainty and admits prior biological knowledge, but its practical use has been hampered by the super-exponential growth of the DAG space and by the intractability of the node-marginal likelihood under flexible, non-conjugate priors. Existing closed-form solutions are largely confined to the conjugate Normal--Inverse-Gamma prior. We develop a Laplace-approximated Bayesian scoring function for the non-conjugate Normal--Gamma prior on the modified Cholesky parameterisation of the precision matrix, embed it in a Metropolis--Hastings sampler over DAGs, and couple the latent Gaussian network to a binary clinical outcome through a probit link. We show that the node-marginal integral is of generalised inverse-Gaussian form, so that its exact value is a modified Bessel function of the second kind and the proposed scoring function is its leading large-argument asymptotic; the posterior of each conditional variance is likewise generalised inverse-Gaussian and is sampled exactly. In simulation, the proposed prior improves on the conjugate baseline and on the PC, greedy-equivalence-search, NOTEARS, and DAGMA benchmarks at sample sizes typical of clinical cohorts. On two real datasets, the Sachs protein-signalling network, scored against its validated consensus graph, and the Wisconsin Diagnostic Breast Cancer data, the method recovers known structure and, through the DAG-probit extension, predicts malignancy from nuclear morphometry with a cross-validated ROC-AUC of $0.94$ using a sparse, interpretable set of direct predictors.

Problem

• Bayesian structure learning
• directed acyclic graphs
• causal discovery
• non-conjugate priors
• gene expression networks

Innovation

• Bayesian structure learning
• Laplace approximation
• non-conjugate priors
• directed acyclic graphs
• probit regression