Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics

TL;DR

This paper presents methods to approximate high-dimensional finite-state Markov processes with lower-dimensional ones using total variation metrics, optimizing transition probabilities and invariant distributions through water-filling solutions.

Contribution

It introduces two novel approaches for Markov process approximation based on total variation constraints, including transition probability and invariant distribution methods, with explicit water-filling solutions.

Findings

Water-filling solutions effectively approximate Markov processes.

Methods successfully reduce process complexity while controlling approximation error.

Theoretical results are demonstrated through specific examples.

Abstract

The aim of this paper is to approximate a finite-state Markov process by another process with fewer states, called herein the approximating process. The approximation problem is formulated using two different methods. The first method, utilizes the total variation distance to discriminate the transition probabilities of a high dimensional Markov process and a reduced order Markov process. The approximation is obtained by optimizing a linear functional defined in terms of transition probabilities of the reduced order Markov process over a total variation distance constraint. The transition probabilities of the approximated Markov process are given by a water-filling solution. The second method, utilizes total variation distance to discriminate the invariant probability of a Markov process and that of the approximating process. The approximation is obtained via two alternative formulations: (a) maximizing a functional of the occupancy distribution of the Markov process, and (b) maximizing the entropy of the approximating process invariant probability. For both formulations, once the reduced invariant probability is obtained, which does not correspond to a Markov process, a further approximation by a Markov process is proposed which minimizes the Kullback-Leibler divergence. These approximations are given by water-filling solutions. Finally, the theoretical results of both methods are applied to specific examples to illustrate the methodology, and the water-filling behavior of the approximations.