Nonparanormal Bayesian Learning of Directed Acyclic Graphs under Gamma and Inverse-Gamma Innovation Priors: Closed-Form Scores and Informed Sampling

Abstract

Bayesian structure learning for directed acyclic graphs (DAGs) is a central tool for reconstructing biological networks, yet it often assumes the data are jointly Gaussian. In motivating proteomic applications, this assumption is routinely violated by skewed and heavy-tailed data, causing Gaussian DAGs to recover spurious or misdirected edges. We develop a fully Bayesian framework for DAG learning in the nonparanormal family, replacing Gaussianity with the weaker requirement that unknown strictly increasing marginal transformations are jointly Gaussian. Working on the modified Cholesky parameterization of the latent precision matrix, we introduce two innovation-variance priors: a non-conjugate Normal-Gamma prior, which decouples coefficient shrinkage from variance regularization, and a conjugate Normal-Inverse-Gamma prior. For both, we obtain the node-wise marginal likelihood in closed form -- through a modified Bessel function of the third kind for the Gamma prior and a Student-$t$ form for the Inverse-Gamma prior -- allowing MCMC sampler moves to be scored without numerical integration. Exploiting these, we build a locally-balanced informed sampler and a Bessel-free score that scales the sampler to hundreds of nodes. On simulated data, our nonparanormal samplers match Gaussian methods when data are Gaussian and dominate them sharply when margins are skewed. On human T-cell protein-signalling data, they recover well-established interactions at high posterior probability, clearly outperforming constraint-based competitors and performing comparably to a Gaussian Bayesian model.