Distinguishing between convergent evolution and violation of the molecular clock for three taxa

We give a non-technical introduction to convergence-divergence models, a new modeling approach for phylogenetic data that allows for the usual divergence of lineages after lineage-splitting but also allows for taxa to converge, i.e. become more similar over time. By examining the 3-taxon case in some detail we illustrate that phylogeneticists have been ``spoiled'' in the sense of not having to think about the structural parameters in their models by virtue of the strong assumption that evolution is tree-like. We show that there are not always good statistical reasons to prefer the usual class of tree-like models over more general convergence-divergence models. Specifically we show many 3-taxon data sets can be equally well explained by supposing violation of the molecular clock due to change in the rate of evolution along different edges, or by keeping the assumption of a constant rate of evolution but instead assuming that evolution is not a purely divergent process. Given the abundan...