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Embedding of reversible Markov matrices

Ellen Baake, Michael Baake, Jeremy Sumner

TL;DR

This work analyzes the embeddability problem for reversible Markov matrices beyond irreducible cases, allowing weak reversibility and negative eigenvalues. It develops an algebraic framework using p-balanced pairs and reversibility, clarifies when a real matrix logarithm yields a Markov generator, and provides concrete, testable criteria for reversible embeddability based on the minimal polynomial and spectrum. The paper shows that, even with positive spectrum, embeddability can be unique, but degeneracies can yield multiple, sometimes non-commuting, reversible and non-reversible embeddings, illustrated with explicit constructions. It also outlines practical extensions to time-inhomogeneous Markov flows and contexts where reversible models serve as useful approximations or starting points for inference in applications such as phylogenetics.

Abstract

The embeddability of reversible Markov matrices into time-homogeneous Markov semigroups is revisited, with some focus on simplifications and extensions. In particular, we do not demand irreducibility and consider weakly reversible matrices as well as reversible matrices with negative eigenvalues.

Embedding of reversible Markov matrices

TL;DR

This work analyzes the embeddability problem for reversible Markov matrices beyond irreducible cases, allowing weak reversibility and negative eigenvalues. It develops an algebraic framework using p-balanced pairs and reversibility, clarifies when a real matrix logarithm yields a Markov generator, and provides concrete, testable criteria for reversible embeddability based on the minimal polynomial and spectrum. The paper shows that, even with positive spectrum, embeddability can be unique, but degeneracies can yield multiple, sometimes non-commuting, reversible and non-reversible embeddings, illustrated with explicit constructions. It also outlines practical extensions to time-inhomogeneous Markov flows and contexts where reversible models serve as useful approximations or starting points for inference in applications such as phylogenetics.

Abstract

The embeddability of reversible Markov matrices into time-homogeneous Markov semigroups is revisited, with some focus on simplifications and extensions. In particular, we do not demand irreducibility and consider weakly reversible matrices as well as reversible matrices with negative eigenvalues.

Paper Structure

This paper contains 9 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and preparatory results
  3. Reversible Markov matrices
  4. General structure
  5. Embedding
  6. Concrete conditions
  7. Multiple embeddings
  8. Outlook
  9. Poisson process approach to Example \ref{['ex:strange']}

Key Result

Lemma 2.6

Let $B\in\mathrm{Mat}(d,\mathbb{R}\space)$ satisfy $B=\space\mathrm{e}^Q = \space\mathrm{e}^R$ with $Q,R \in \mathrm{Mat}(d,\mathbb{R}\space)$, and consider $\varTheta \mathrel{\mathop:}= \{ t \in \mathbb{R}\space : \space\mathrm{e}^{t \space Q} = \space\mathrm{e}^{t R} \}$. Then, either $\varTheta

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.6
  • proof
  • proof
  • Lemma 2.9
  • proof
  • Definition 3.1
  • Example 3.2
  • ...and 22 more