Uniformization Stable Markov Models and Their Jordan Algebraic Structure

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1822-1851, December 2023.
Abstract. We provide a characterization of the continuous-time Markov models where the Markov matrices from the model can be parameterized directly in terms of the associated rate matrices (generators). That is, each Markov matrix can be expressed as the sum of the identity matrix and a rate matrix from the model. We show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (so-called) linear models. We connect this property to the well-known uniformization procedure for continuous-time Markov chains by demonstrating that the property is equivalent to all Markov matrices from the model taking the same form as the corresponding discrete-time Markov matrices in the uniformized process. We apply our results to analyze two model hierarchies practically important to phylogenetic inference, obtained by assuming (i) time reversibility and (ii) permutation symmetry, respectively.