A distance function for stochastic matrices

2024-10-16

arXiv.org

AI Summary

This paper addresses the lack of a rigorous distance metric for comparing Markov chains on stochastic matrix spaces—particularly in applications such as clinical pathway analysis. We propose an information-geometric distance based on the Bhattacharyya angle, the first to simultaneously satisfy all metric axioms (non-negativity, symmetry, triangle inequality), admit a closed-form expression, and enable efficient O(n²) numerical computation. Under ergodicity, it is rigorously equivalent to the sequence-level Bhattacharyya distance. Theoretically, we derive explicit bounds linking its convergence rate to the Markov chain’s mixing time. Empirically, experiments on real-world healthcare process data demonstrate its strong discriminability, robustness to noise, and interpretability—enabling principled, computationally tractable comparison of dynamical system behaviors. This work provides a provably sound and practical tool for quantitative analysis of stochastic processes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Define distance function for stochastic matrices
• Compare short and long term Markov chains
• Evaluate convergence bounds and mixing times

Innovation

• Advocates Bhattacharyya angle for Markov chains
• Introduces new distance function for stochastic matrices
• Derives bounds on distance and mixing times