Equational Axioms for Expected Value Operators

TL;DR

This paper develops an algebraic framework using equational axioms in signed meadows to formalize expected value, variance, and related concepts for probability functions, including conditional values and finite support summation.

Contribution

It introduces a comprehensive algebraic axiomatization of probability concepts within signed meadows, incorporating conditional values and finite support summation.

Findings

Provides equational axioms for expected value, variance, covariance, and correlation squared.

Defines conditional values and relates them to probability mass functions and random variables.

Simplifies probability function requirements using finite support summation.

Abstract

An equational axiomatisation of probability functions for one-dimensional event spaces in the language of signed meadows is expanded with conditional values. Conditional values constitute a so-called signed vector meadow. In the presence of a probability function, equational axioms are provided for expected value, variance, covariance, and correlation squared, each defined for conditional values. Finite support summation is introduced as a binding operator on meadows which simplifies formulating requirements on probability mass functions with finite support. Conditional values are related to probability mass functions and to random variables. The definitions are reconsidered in a finite dimensional setting.