Bayesian inference for graphical Gaussian and conditional Gaussian models

Abstract

Graphical Modes are families of distributions satisfying a set of conditional independence relationships, which may be represented by a graph where vertices represent the variables under study and pairwise conditional independences are represented by missing edges.  Much of the work concerning Bayesian inference for these models has dealt with those where all the variables are either all discrete or all continuous.  While the case of purely discrete models has been treated thoroughly, most work on the purely continuous case of graphical Gaussian models has been restricted to decomposable models, while allow analyses to be broken down into sub-analyses of smaller, simple sub-models and graphical models with variables of each type have been given very little attention in a Bayesian context.  The thesis addresses these two issues through the use of Markov chain Monte Carlo methods, a powerful tool for Bayesian inference.  Methodology for inference both for fixed models and under model uncertainty is developed for the entire class of graphical Gaussian models, avoiding the use of conjugate prior distributions.  This approach is then applied to simple mixed graphical models as well as to the larger class of hierarchical interaction models.

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