A distance function for stochastic matrices

TL;DR

This paper introduces a new, efficient distance function for stochastic matrices inspired by information geometry, enabling better comparison of Markov chain models, especially in healthcare applications.

Contribution

A novel true distance measure for stochastic matrices with a closed form, facilitating model comparison and analysis of Markov chains.

Findings

The Bhattacharyya angle effectively compares Markov chain runs.

The new distance aligns with the Bhattacharyya angle for ergodic chains.

Bounds on convergence and mixing times are established.

Abstract

Motivated by information geometry, a distance function on the space of stochastic matrices is advocated. Starting with sequences of Markov chains the Bhattacharyya angle is advocated as the natural tool for comparing both short and long term Markov chain runs. Bounds on the convergence of the distance and mixing times are derived. Guided by the desire to compare different Markov chain models, especially in the setting of healthcare processes, a new distance function on the space of stochastic matrices is presented. It is a true distance measure which has a closed form and is efficient to implement for numerical evaluation. In the case of ergodic Markov chains, it is shown that considering either the Bhattacharyya angle on Markov sequences or the new stochastic matrix distance leads to the same distance between models.