Nonparametric Graphical Models

Abstract

This chapter introduces different approaches to graphical modeling for continuous and mixed data, using semiparametric techniques that make weak assumptions compared with the default Gaussian graphical model. It outlines some different ways of making restrictions on the model that lead to computationally tractable models with favorable statistical properties. The chapter describe a family of semiparametric graphical models, called exponential family graphical models. In terms of the potential function representation this family uses linear edge potentials and general vertex potentials. The chapter describes the case where both the edge and vertex potentials are nonparametric, but where tractability is achieved by assuming that the conditional independence graph has no cycles, leading to tree-structured graphical models. It presents an approach based on pairwise tensor products of smoothing splines, a type of log-density ANOVA model where the computational bottleneck of computing the normalizing constant is circumvented by using a surrogate loss function called score matching.