High dimensional Bayesian inference for Gaussian directed acyclic graph models
arXiv:1109.4371
Abstract
We study centered Gaussian models Markov with respect to a directed acyclic graph (DAG) whose vertices have a fixed parent ordering. We construct a conjugate family on the modified Cholesky parameters, with one shape parameter per vertex, and derive its induced distributions on incomplete covariance and precision coordinates. The distribution is proper exactly when \(α_i>|\mathrm{pa}(i)|+2\) for every vertex, and its nodewise conditional-variance and regression parameters are independent across vertices. This factorization gives a closed-form normalizing constant, conjugate updating, marginal likelihoods, and explicit full-matrix posterior means. We distinguish these posterior means from nonlinear completions of incomplete-coordinate means. We also distinguish the transformed Cholesky-coordinate mode from modes defined using covariance or precision coordinates. The model-selection procedure searches only over DAGs compatible with the specified ordering. Historical simulation and data examples illustrate the method; their evidentiary limitations and reproducibility requirements are stated explicitly.
arXiv comment55 pages, 8 figures, 12 table
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