A construction of a graphical model

TL;DR

This paper introduces a nonparametric graphical model that uses conditional dependence coefficients to represent conditional independence among variables, with a new structure learning method based on sample estimates and thresholding.

Contribution

It proposes a novel nonparametric graphical model framework and a two-step procedure for structure learning using conditional dependence coefficients.

Findings

Effective graph recovery on artificial datasets

Successful application to real datasets

Adaptation for elliptical distribution partial correlation graphs

Abstract

We present a nonparametric graphical model. Our model uses an undirected graph that represents conditional independence for general random variables defined by the conditional dependence coefficient (Azadkia and Chatterjee (2021)). The set of edges of the graph are defined as $E=\{(i,j):R_{i,j}\neq 0\}$, where $R_{i,j}$ is the conditional dependence coefficient for $X_i$ and $X_j$ given $(X_1,\ldots,X_p) \backslash \{X_{i},X_{j}\}$. We propose a graph structure learning by two steps selection procedure: first, we compute the matrix of sample version of the conditional dependence coefficient $\widehat{R_{i,j}}$; next, for some prespecificated threshold $\lambda>0$ we choose an edge $\{i,j\}$ if $ \left|\widehat{R_{i,j}} \right| \geq \lambda.$ The graph recovery structure has been evaluated on artificial and real datasets. We also applied a slight modification of our graph recovery procedure for learning partial correlation graphs for the elliptical distribution.