University of Washington - Department of Statistics
We develop a framework for the modeling of joint distributions of high-dimensional data that is robust to a variety of data types and modeling paradigms. Central to our considerations are the issues of structural learning and posterior parameter estimation. By building our framework from Gaussian Graphical Models (GGMs) we are able to separate the learning and estimation problems, thereby proposing a methodology possessing desirable computational features. We further show that restrictions in the space of GGMs yield a variety of models, including regression, multivariate regression and directed acyclic graphs. After developing these settings we extend the framework beyond variables that are jointly Gaussian by introducing Copula Gaussian Graphical Models (CGGMs), based on the semiparametric Gaussian copula. This allows data of mixed type to be modeled in a convenient framework that still accommodates conditional independence assumptions provided by GGMs. Throughout, we are motivated by emerging modeling questions. We conduct a battery of simulation studies as well as short data analyses that show the utility of various components of this framework. We conclude with a discussion of future work, including incorporation of an instrumental variable (IV) estimation procedure, matrix variate graphical models and the extension to Bayesian nonparametrics via Dirichlet processes.