A simple measure of conditional dependence

TL;DR

This paper introduces a new, model-free coefficient for measuring conditional dependence between variables, which converges without distributional assumptions and underpins a consistent variable selection algorithm called FOCI.

Contribution

The paper proposes a novel conditional dependence coefficient with desirable properties and develops FOCI, a new variable selection method based on this coefficient, with proven consistency.

Findings

Coefficient converges to a limit in [0,1] without distributional assumptions.

FOCI algorithm is model-free and tuning-parameter free.

Applications demonstrate effectiveness on synthetic and real data.

Abstract

We propose a coefficient of conditional dependence between two random variables $Y$ and $Z$ given a set of other variables $X_1,\ldots,X_p$, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in $[0,1]$, where the limit is $0$ if and only if $Y$ and $Z$ are conditionally independent given $X_1,\ldots,X_p$, and is $1$ if and only if $Y$ is equal to a measurable function of $Z$ given $X_1,\ldots,X_p$. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial $R^2$ statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.