Covariance as a commutator

Abstract

The covariance between real finite variance random variables can be expressed as the commutator of taking expectations and multiplying, both viewed as operators extended to act jointly on pairs of functions. The efficient influence curve of the mean represents a centering operator which we demonstrate to interact with expectations and products through simple commutator identities. These expressions reveal an underlying Lie algebraic structure that endows the calculus of efficient influence curves with a natural differential geometric interpretation.