On topological and algebraic structures of categorical random variables

TL;DR

This paper introduces a metric for categorical random variables based on entropy and symmetrical uncertainty, and explores its topological and algebraic properties, including a compatible monoid structure.

Contribution

It defines a new metric and algebraic structure on categorical random variables, linking entropy-based measures with topological and monoid frameworks.

Findings

The metric induces a quotient space of categorical variables.

A natural commutative monoid structure exists in this quotient space.

The monoid operation is continuous with respect to the metric topology.

Abstract

Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.