Distinguishing between convergent evolution and violation of the molecular clock

We give a non-technical introduction to convergence-divergence models, a new modeling approach for phylogenetic data that allows for the usual divergence of species post speciation but also allows for species 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 treelike. We show that there are not always good statistical reasons to prefer the usual class of treelike models over more general convergence-divergence models. Specifically we show many $3$-taxon datasets 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 abundance of evidence that evolution is not strictly treelike, our discussion is an illustration that as phylogeneticists we often need to think clearly about the structural form of the models we use.