Covariance as a commutator

TL;DR

This paper reveals a Lie algebraic structure underlying the covariance and influence curves in statistics, interpreting them through the lens of operator commutators and differential geometry.

Contribution

It introduces a novel operator-theoretic framework for understanding covariance and influence functions using commutator identities and Lie algebraic structures.

Findings

Covariance expressed as a commutator of expectation and multiplication operators.

Efficient influence curve acts as a centering operator with commutator interactions.

Reveals a Lie algebraic and differential geometric structure in statistical calculus.

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.