Semiparametric Bayesian structure learning of nonparanormal directed acyclic graphs with local--global shrinkage

Abstract

Bayesian structure learning of directed acyclic graphs (DAGs) is central to high-dimensional causal discovery, yet existing methods mostly assume multivariate Gaussian data, which is routinely violated by measurements displaying heavy tails, skewness, or bounded support. We address this by introducing NPN-DAG-HS, a fully Bayesian semiparametric DAG model coupling the nonparanormal family with horseshoe shrinkage on Cholesky off-diagonals via an extended-rank likelihood. This handles arbitrary unknown monotone marginal transformations without estimating them, preserving exact-zero shrinkage and conditionally conjugate posteriors for practical inference in large dimensions. Our method uses a partially collapsed Metropolis-within-Gibbs sampler that augments rank-likelihood Gaussian copies and alternates score-based DAG moves with horseshoe Gibbs updates. Theoretically, in the high-dimensional regime ($p\to \infty$ with $\log p/n\to 0$), we establish posterior contraction at the rate $\sqrt{(s_0+p)\log p/n}$, strong skeleton selection consistency, and a parametric Bernstein-von Mises theorem for smooth total causal-effect functionals, yielding asymptotic frequentist calibration of credible intervals. Simulations confirm our method outperforms Gaussian baselines and standard frequentist learners. Applied to an acute myeloid leukaemia dataset, it recovers a sparse, interpretable network, pruning unsupported edges declared by Gaussian analyses and illustrating the value of our semiparametric relaxation.