Correlation-Based And-Operations Can Be Copulas: A Proof

TL;DR

This paper proves that correlation-based and-operations, used for estimating joint probabilities from individual event probabilities and correlation, are mathematically copulas, linking two important concepts in probability theory.

Contribution

It provides the first formal proof that correlation-based and-operations are indeed copulas, establishing a theoretical connection between these concepts.

Findings

Proof that correlation-based and-operations are copulas

Clarifies the relationship between correlation and copula functions

Enhances understanding of joint probability estimation methods

Abstract

In many practical situations, we know the probabilities $a$ and $b$ of two events $A$ and $B$, and we want to estimate the joint probability ${\rm Prob}(A\,\&\,B)$. The algorithm that estimates the joint probability based on the known values $a$ and $b$ is called an and-operation. An important case when such a reconstruction is possible is when we know the correlation between $A$ and $B$; we call the resulting and-operation correlation-based. On the other hand, in statistics, there is a widely used class of and-operations known as copulas. Empirical evidence seems to indicate that the correlation-based and-operation derived in https://doi.org/10.1007/978-3-031-08971-8_64 is a copula, but until now, no proof of this statement was available. In this paper, we provide such a proof.