  Probability modelling for DNA sequence evolution is well established and
provides a rich framework for understanding genetic variation between samples
of individuals from one or more populations. We show that both classical and
more recent models for coalescence (with or without recombination) can be
described in terms of the so-called phase-type theory, where complicated and
tedious calculations are circumvented by the use of matrices. The application
of phase-type theory consists of describing the stochastic model as a Markov
model by appropriately setting up a state space and calculating the
corresponding intensity and reward matrices. Formulae of interest are then
expressed in terms of these aforementioned matrices. We illustrate this by a
few examples calculating the mean, variance and even higher order moments of
the site frequency spectrum in the multiple merger coalescent models, and by
analysing the mean and variance for the number of segregating sites for
multiple samples in the two-locus ancestral recombination graph. We believe
that phase-type theory has great potential as a tool for analysing probability
models in population genetics. The compact matrix notation is useful for
clarification of current models, in particular their formal manipulation
(calculation), but also for further development or extensions.
