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

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.