A note on probability metrics in a categorical setting

TL;DR

This paper explores how probability metrics can be integrated into a categorical framework, unifying various results and highlighting their structural properties in probability theory.

Contribution

It demonstrates that probability metrics can be embedded in a categorical setting, revealing the categorical nature of certain probability constructions.

Findings

Probability metrics can be naturally incorporated into a categorical framework.

Categorical perspective unifies various results in probability metric theory.

Enhances understanding of the structural properties of probability metrics.

Abstract

Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with the randomness structure. But probability metrics suffer from the same instability under constructions as metrics. In \cite{L15}, as well as in former and related work which can be found in the references of \cite{L15}, a comprehensive setting was developed to deal with this. It is the purpose of this note to point out that these ideas can also be applied to probability metrics thus embedding them in a natural categorical framework, showing that certain constructions performed in the setting of probability theory are in fact categorical in nature. This allows us to deduce various separate results in the literature from a unified approach.