Mixed and Covariate Dependent Graphical Models.

Abstract

Graphical models have proven to be a useful tool in understanding the conditional depen- dency structure of multivariate distributions. In this thesis, we consider two types of undirected graphical models that are motivated by particular types of applications. The First model we consider is a mixed graphical model, linking both continuous and discrete variables. The pro- posed model is simple enough to be suitable for high-dimensional data, yet exible enough to represent all possible graph structures. We develop a computationally efficient regression- based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by applying group lasso penalty which can be approximated by a weighted L1 penalty. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation data set (CAL500), obtaining a sparse and interpretable graphical model. The second model we consider is a covariate dependent Ising model which studies the conditional dependency within the binary data and how they are dependent on the additional covariates. This results in subject-specific Ising models, where the subjects' covariates in uence the strength of association between the binary variables. We add L1 penalty to induce sparsity in the fitted graphs and in the number of selected covariates. Two algorithms to fit the model are proposed and compared on a set of simulated data, and asymptotic results are established. The results on a study of genes involved in breast cancer and their biological significance are discussed. 1