Bayesian DAG Structure Learning with Simultaneous Shrinkage Covariance Estimation under Scale-Mixture Error Distributions in the Proportional High-Dimensional Regime

2026-07-09

AI Summary

This study addresses the challenge of jointly estimating directed acyclic graph (DAG) structures and precision matrices in high-dimensional settings with heavy-tailed or contaminated data. The authors propose R-DACH, a unified Bayesian framework that, for the first time, directly places a global–local horseshoe prior on the strictly lower triangular part of the Cholesky factor, combined with an inverse-Gamma scale mixture error model. This approach enables simultaneous inference of variable ordering, sparse DAG structure, and continuous parameters while inherently achieving robustness to outliers. Theoretical and empirical results demonstrate that R-DACH outperforms graphical horseshoe, DAG-Wishart, and PC algorithms—particularly under contamination—with superior topological consistency and parent selection accuracy, even when the number of variables reaches several hundred. Applied to TCGA RNA-seq data, R-DACH uncovers biologically interpretable gene regulatory relationships missed by competing methods.

Abstract

We propose a unified Bayesian framework namely robust DAG-Cholesky horseshoe (R-DACH) for joint directed acyclic graph (DAG) structure learning and precision matrix estimation in the high-dimensional proportional asymptotic regime $p/n \to c \in (0,\infty)$, under the scale mixture of normal errors. The construction places a global-local horseshoe-type prior directly on the strictly lower-triangular entries of the modified Cholesky factor of the DAG-Markov precision matrix, so that sparsity in the Cholesky parameters induces a coherent parent-set selection consistent with a topological ordering of the variables. A per-observation inverse-gamma scale mixture yields automatic robustness to heavy-tailed and contaminated observations and admits Student-$t$, Laplace, and slash distributions as special cases. We design a partially-collapsed blocked Gibbs sampler that traverses the joint space of orderings, sparsity patterns and continuous parameters. Simulations across $(n,p)$ configurations with $p$ up to several hundreds confirm the theoretical rates and demonstrate substantial gains over graphical-horseshoe, DAG-Wishart, and PC-based competitors under contamination. An application to RNA-seq gene-expression data from \emph{The Cancer Genome Atlas} reveals biologically interpretable regulatory structure that competing methods fail to recover.

Problem

• Bayesian DAG structure learning
• high-dimensional regime
• scale-mixture errors
• precision matrix estimation
• robustness

Innovation

• Bayesian DAG learning
• Cholesky decomposition
• horseshoe prior
• scale-mixture errors
• robust covariance estimation