High dimensional Bayesian inference for Gaussian directed acyclic graph models

Emanuel Ben-David; Tianxi Li; Helene Massam; Bala Rajaratnam

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arXiv · 1109.4371

High dimensional Bayesian inference for Gaussian directed acyclic graph models

Emanuel Ben-David·Tianxi Li·Helene Massam·Bala Rajaratnam

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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 $\alpha_i>|\mathrm{pa}(i)|+2$ 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.

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Emanuel Ben-David, Tianxi Li, Helene Massam, Bala Rajaratnam. 2011-09-20. High dimensional Bayesian inference for Gaussian directed acyclic graph models. https://arxiv.org/abs/1109.4371

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