The embedding problem for Markov matrices

TL;DR

This paper investigates the embedding problem for Markov matrices, providing bounds, criteria, and algorithms for determining when a matrix can be expressed as an exponential of a rate matrix, with a complete solution for 4x4 matrices.

Contribution

It introduces a spectrum-based bound, a generic embeddability criterion, and a comprehensive solution for 4x4 matrices in the embedding problem.

Findings

Bound on the number of generators based on spectrum

Criterion for generic embeddability

Complete solution for 4x4 matrices

Abstract

Characterizing whether a Markov process of discrete random variables has an homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old question known in the literature as the embedding problem (Elfving37), which has been only solved for matrices of size $2\times 2$ or $3\times 3$. In this paper, we address this problem and related questions and obtain results in two different lines. First, for matrices of any size, we give a bound on the number of Markov generators in terms of the spectrum of the Markov matrix. Based on this, we establish a criterion for deciding whether a generic Markov matrix (different eigenvalues) is embeddable and propose an algorithm that lists all its Markov generators. Then, motivated and inspired by recent results on substitution models of DNA, we focus in the $4\times 4$ case and completely solve the embedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition.