  We introduce a nonparametric graphical model for discrete node variables
based on additive conditional independence. Additive conditional independence
is a three way statistical relation that shares similar properties with
conditional independence by satisfying the semi-graphoid axioms. Based on this
relation we build an additive graphical model for discrete variables that does
not suffer from the restriction of a parametric model such as the Ising model.
We develop an estimator of the new graphical model via the penalized estimation
of the discrete version of the additive precision operator and establish the
consistency of the estimator under the ultrahigh-dimensional setting. Along
with these methodological developments, we also exploit the properties of
discrete random variables to uncover a deeper relation between additive
conditional independence and conditional independence than previously known.
The new graphical model reduces to a conditional independence graphical model
under certain sparsity conditions. We conduct simulation experiments and
analysis of an HIV antiretroviral therapy data set to compare the new method
with existing ones.
