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Phylogenetics uses alignments of molecular sequence data to learn aboutevolutionary trees. Substitutions in sequences are modelled through acontinuous-time Markov process, characterised by an instantaneous rate matrix,which standard models assume is time-reversible and stationary. Theseassumptions are biologically questionable and induce a likelihood functionwhich is invariant to a tree's root position. This hampers inference because atree's biological interpretation depends critically on where it is rooted.Relaxing both assumptions, we introduce a model whose likelihood candistinguish between rooted trees. The model is non-stationary, with stepchanges in the instantaneous rate matrix at each speciation event. Exploitingrecent theoretical work, each rate matrix belongs to a non-reversible family ofLie Markov models. These models are closed under matrix multiplication, so ourextension offers the conceptually appealing property that a tree and all itssub-trees could have arisen from the same family of non-stationary models.  We adopt a Bayesian approach, describe an MCMC algorithm for posteriorinference and provide software. The biological insight that our model canprovide is illustrated through an analysis in which non-reversible butstationary, and non-stationary but reversible models cannot identify aplausible root.

Incorporating compositional heterogeneity into Lie Markov models for phylogenetic inference