Profile graphical models

This thesis concerns the theory and the inference of a new class of independence models based on a graphical representation that we name profile graphs. Multiple graph models are special cases in this class and the compatibility in terms of independence structure is derived with respect to chain graph models of different types. Inference and model selection based on both Lasso methodology and Bayesian theory are studied and implemented. The thesis is composed of four chapters. In the first chapter, we present a literature review of multiple and chain graphs. Markov properties, parameterization and inference are reviewed for undirected, bidirected, LWF chain and regression graphs. In the second chapter, a class of profile graphs is introduced for modelling the effect of an external factor on the independence structure of a multivariate set of variables. Conditional and marginal independence structures are explored by using profile undirected and bi-directed graphical models, respectively. These two families of graphical models are formally defined with their corresponding Markov properties. Furthermore, necessary conditions are derived to induce, for any profile undirected and bi-directed graph model, a compatible class of chain graph models of different type known as LWF chain graph and regression graph, respectively. In the third chapter, we propose two Bayesian approaches for the selection of Ising models associated to multiple undirected graphs. We devise a Bayesian exact-likelihood inference for low-dimensional binary response data, based on conjugate priors for log- linear parameters. We also propose a quasi-likelihood Bayesian approach for fitting high-dimensional multiple Ising graphs, where the normalization constant results computationally intractable. In both methods, we define a Markov Random Field prior on the graph structures, which encourages the selection of the same edges in related graphs. Finally, in the fourth chapter we present some final remarks on Chapters 2 and 3.