Conditional Independence, Conditional Mean Independence, and Zero Conditional Covariance

TL;DR

This paper explores the relationships among conditional independence, mean independence, and zero covariance, showing implications and conditions under which these notions are equivalent or imply each other in the context of random variables.

Contribution

It clarifies the hierarchy and reversibility of these conditional dependence notions and characterizes when zero covariance implies mean independence, especially for Bernoulli variables.

Findings

Conditional independence implies mean independence and zero covariance, but not vice versa.

For Bernoulli variables, zero covariance implies mean independence due to affine conditional expectation.

The hierarchy of dependence notions is not reversible in general.

Abstract

Investigation of the reversibility of the directional hierarchy in the interdependency among the notions of conditional independence, conditional mean independence, and zero conditional covariance, for two random variables X and Y given a conditioning element Z which is not constrained by any topological restriction on its range, reveals that if the first moments of X, Y, and XY exist, then conditional independence implies conditional mean independence and conditional mean independence implies zero conditional covariance, but the direction of the hierarchy is not reversible in general. If the conditional expectation of Y given X and Z is "affine in X," which happens when X is Bernoulli, then the "intercept" and "slope" of the conditional expectation (that is, the nonparametric regression function) equal the "intercept" and "slope" of the "least-squares linear regression function", as a result of which zero conditional covariance implies conditional mean independence.