  We introduce a sufficient graphical model by applying the recently developed
nonlinear sufficient dimension reduction techniques to the evaluation of
conditional independence. The graphical model is nonparametric in nature, as it
does not make distributional assumptions such as the Gaussian or copula
Gaussian assumptions. However, unlike a fully nonparametric graphical model,
which relies on the high-dimensional kernel to characterize conditional
independence, our graphical model is based on conditional independence given a
set of sufficient predictors with a substantially reduced dimension. In this
way we avoid the curse of dimensionality that comes with a high-dimensional
kernel. We develop the population-level properties, convergence rate, and
variable selection consistency of our estimate. By simulation comparisons and
an analysis of the DREAM 4 Challenge data set, we demonstrate that our method
outperforms the existing methods when the Gaussian or copula Gaussian
assumptions are violated, and its performance remains excellent in the
high-dimensional setting.
