Scalable Bayesian structure learning of directed acyclic graphs via Laplace approximation, with an application to breast cancer gene expression networks
A Laplace-approximated Bayesian scoring function for the non-conjugate Normal--Gamma prior is developed 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.
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.
We develop a network-structured Bayesian hierarchical model for sparse association mapping between genomic alterations and quantitative treatment-response phenotypes. The framework combines a Gaussian Markov random field prior that borrows strength across pathway-connected genes, a global-local horseshoe prior inducing sparsity, and a conjugate Gibbs sampler requiring no Metropolis-Hastings steps. Though broadly applicable to high-dimensional settings with known predictor networks, we validate it using cancer cell-line drug-sensitivity data. Applied to GDSC2 ($N=951$ cell lines, $G=219$ driver genes, $D=295$ drugs), the model identifies 126 gene-drug associations (0.195\% of 64{,}605 pairs), concentrated in EZH2 (45 drugs, all sensitivity-direction, mean effect $-0.911$ $\ln$IC50) and KMT2D (36 drugs, all sensitivity-direction, mean effect $-0.496$ $\ln$IC50). These markers show external support in an independent PRISM screen (1{,}518 compounds), with KMT2D achieving complete directional replication (36/36) and EZH2 partial replication (8/12). Five-fold cross-validated predictive log-likelihood confirms each prior layer's value: the full model outperforms the no-network ablation by $+3{,}109$ log-units per fold and the no-horseshoe ablation by $+14{,}039$ log-units, consistently across folds. Simulations under three scenarios show the full model achieves the highest precision and lowest false-discovery rate throughout, while the network prior improves sensitivity recovery under network-structured signal. A tissue-stratified extension identifies coherent subgroup refinements, including lung-specific EGFR-inhibitor sensitivity and skin-specific BRAF-Dabrafenib sensitivity. These results show the framework identifies sparse, interpretable, externally supported drug-sensitivity markers while enabling principled investigation of tissue-specific departures from shared effects.
Hammed A. Olayinka, Saheed O. Olayemi· 0 citations
SVI-DAG is proposed, a structured variational inference approach to Bayesian causal discovery using observational data and prior beliefs that uses normalizing flows to model dependencies between edges, supporting expressive and multimodal posterior learning over DAGs.
This work proposes SURE-Ridge, a non-iterative, closed-form estimator for equal variance linear Gaussian SEM, which achieves the lowest structural Hamming distance in the small-sample regime and the lowest run time across all sample sizes tested, compared with NOTEARS, DAGMA, and GBNSL baselines.
Exponential random graph models (ERGMs) describe dependence among network ties, but inference becomes difficult when the likelihood is intractable and candidate network statistics are strongly correlated. We introduce BERGM Elastic Net, an adaptive empirical-Bayes approach that combines lasso shrinkage with ridge stabilization in a Bayesian ERGM. A latent-variable formulation supports approximate exchange sampling, while empirical-Bayes updates adapt the amount of regularization to the observed network. We connect the proposed prior to elastic-net penalized likelihood and clarify the interpretation of thresholded reporting and coefficient grouping. The method is developed for over-specified network models containing many related structural and covariate effects.
Dan Han, Vicki Modisette, Tinghan Li et al.· 0 citations
The results show that representing both sources as probabilistic uncertainty over edge existence and orientation is a practical and effective way to improve causal graph accuracy.
N. K. Kitson, Anthony C. Constantinou· 0 citations
Bayesian networks (BNs) are widely used tools for the modeling of complex dependencies among variables. However, their application to multilevel or clustered data remains constrained, especially in scenarios involving multiple random effects, due to a lack of methods and more pressing, efficient computational frameworks. In this article, we propose an innovative framework, INLA-MBN, that integrates multilevel BNs (MBNs) with the integrated nested Laplace approximation (INLA), thereby facilitating efficient structure and parameter learning in both longitudinal and cross-sectional multilevel data contexts. To the best of our knowledge, this is the first implementation of INLA for BNs that involves more than one correlated random effect. More precisely, we advance existing research by modeling longitudinal data with two correlated random effects and multilevel cross-sectional data with more than two correlated random effects. Our approach encompasses both Gaussian and hybrid MBNs that incorporate Gaussian and categorical nodes. Through an exhaustive simulation study, we demonstrate that INLA-MBN exhibit superior performance compared to networks based on maximum likelihood estimation, particularly, in scenarios characterized by small group sizes or limited repeated measurements per subject. INLA-MBN achieves a fully inferred MBN, while negating the cost associated with MBNs based on Markov Chain Monte Carlo. The proposed method was also applied to real-life longitudinal child morbidity data to demonstrate its practical applicability.
B. E. Yirdaw, L. K. Debusho, J. van Niekerk et al.· Statistical Methods in Medic...· 0 citations
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