2011/09/20 by Emanuel Ben‐David, Tianxi Li, Ben-David, Emanuel +5
Computer Science · Mathematics · #62-09 #62E10 #62J05 #Bayesian Methods and Mixture Models #Bayesian Modeling and Causal Inference #FOS: Computer and information sciences #FOS: Mathematics #Other Statistics (stat.OT) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1109.4371
openalex publication_date 2011/09/20 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28
In this paper, we consider Gaussian models Markov with respect to an\narbitrary DAG. We first construct a family of conjugate priors for the Cholesky\nparametrization of the covariance matrix of such models. This family has as\nmany shape parameters as the DAG has vertices, and naturally extends the work\nof Geiger and Heckerman [8]. From these distributions, we derive prior\ndistributions for the covariance and precision parameters of the Gaussian DAG\nMarkov models. Our works thus extends the work of Dawid and Lauritzen [5] and\nLetac and Massam [16] for Gaussian models Markov with respect to a decomposable\ngraph to arbitrary DAGs. For this reason, we call our distributions DAG-Wishart\ndistributions. An advantage of these distributions is that they possess strong\nhyper Markov properties and thus allow for explicit estimation of the\ncovariance and precision parameters, regardless of the dimension of the\nproblem. They also allow us to develop methodology for model selection and\ncovariance estimation in the space of DAG-Markov models. We demonstrate via\nseveral numerical examples that the proposed method scales well to\nhigh-dimensions.\n