Title: Conditional Independence, Conditional Mean Independence, and Zero Conditional Covariance,
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.