Measures of distinguishability between stochastic processes

TL;DR

This paper introduces a family of divergence rate measures for quantifying the distinguishability of stochastic processes, emphasizing a specific measure that is computationally efficient and robust across various scenarios.

Contribution

The paper proposes a set of criteria for process distinguishability measures and introduces divergence rates, especially the co-emission divergence rate, that meet these criteria.

Findings

Co-emission divergence rate can be computed efficiently.

It behaves similarly to existing measures in applicable regimes.

It remains well-behaved when other measures fail.

Abstract

Quantifying how distinguishable two stochastic processes are lies at the heart of many fields, such as machine learning and quantitative finance. While several measures have been proposed for this task, none have universal applicability and ease of use. In this Letter, we suggest a set of requirements for a well-behaved measure of process distinguishability. Moreover, we propose a family of measures, called divergence rates, that satisfy all of these requirements. Focussing on a particular member of this family -- the co-emission divergence rate -- we show that it can be computed efficiently, behaves qualitatively similar to other commonly-used measures in their regimes of applicability, and remains well-behaved in scenarios where other measures break down.