Uniformization stable Markov models and their Jordan algebraic structure

We provide a characterisation of the continuous-time Markov models where the Markov matrices from the model can be parameterised 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 analyse two model hierarchies practically important to phylogenetic inference, obtained by assuming (i) time-reversibility and (ii) permutation symmetry, respectively.