Conditional Probability Matrix and the $S^2$-rank

TL;DR

This paper introduces the concept of $S^2$-rank for matrices and demonstrates its application in characterizing the structure of conditional probability matrices, revealing that such matrices typically have $S^2$-rank 1.

Contribution

It defines the $S^2$-rank for specific matrices and establishes its relevance to conditional probability matrices, including a converse result under certain conditions.

Findings

Conditional probability matrices have $S^2$-rank 1.

The $S^2$-rank characterizes the structure of these matrices.

A converse result links $S^2$-rank 1 matrices to conditional probability matrices.

Abstract

Using the $det^{S^2}$ map from [5], we introduce the notion of $S^2$-rank of a matrix of type $d\times \frac{s(s-1)}{2}$. As an application, we show that the conditional probability matrix associated to two random variables has the $S^2$-rank equal to $1$. Under suitable conditions we prove that the converse of this result also holds.