On Sufficient Graphical Models

TL;DR

This paper introduces a nonparametric sufficient graphical model that leverages nonlinear sufficient dimension reduction to evaluate conditional independence, avoiding high-dimensional kernel issues and outperforming existing methods under non-Gaussian conditions.

Contribution

It develops a novel nonparametric graphical model based on sufficient predictors, with proven properties and superior performance in high-dimensional, non-Gaussian scenarios.

Findings

Outperforms existing methods when Gaussian assumptions are violated.

Maintains excellent performance in high-dimensional settings.

Provides theoretical guarantees including convergence and variable selection consistency.

Abstract

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.