On matrix-analytic Markov modulated models for evolution: application potential and future directions required to...

Markov modulated (MM) models in the theory of matrix-analytic methods (MAMs) are a class of probabilistic models whose key feature is an underlying Markov environment that drives the evolution of some complex and unpredictable system. Our goal in this paper is to emphasise their great potential for understanding evolution and the need for close multidisciplinary collaborations between probabilists and evolutionary biologists to advance the field. We give an overview of the fundamental MM models in MAMs, namely quasi-birth-and-death processes (QBDs) and stochastic fluid models (SFMs), in which the fluctuations in the level variable are modulated by fluctuations in the phase variable. We illustrate their application potential in evolution through an example of phylogenetic inference, state a general tree likelihood formula applicable to a wide range of MM models, and discuss future directions required to advance the theory of MM models as well as in the applications of the existing theory to evolutionary contexts.