Learning Higher-Order Graph Structure with Features by Structure Penalty

In discrete undirected graphical models, the conditional independence of node labels Y is specified by the graph structure. We study the case where there is another input random vector X (e.g. The main contribution of this paper is to learn the graph structure and the functions conditioned on X at the same time. We prove that discrete undirected graphical models with feature X are equivalent to mul- tivariate discrete models. The reparameterization of the potential functions in graphical models by conditional log odds ratios of the latter offers advantages in representation of the conditional independence structure.