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The selection of the most suitable evolutionary model to analyze the givenmolecular data is usually left to biologist's choice. In his famous book, JFelsenstein suggested that certain linear equations satisfied by the expectedprobabilities of patterns observed at the leaves of a phylogenetic tree couldbe used for model selection. It remained open the question regarding whetherthese equations were enough for characterizing the evolutionary model.  Here we prove that, for equivariant models of evolution, the space ofdistributions satisfying these linear equations coincides with the space ofdistributions arising from mixtures of trees on a set of taxa. In other words,we prove that an alignment is produced from a mixture of phylogenetic treesunder an equivariant evolutionary model if and only if its distribution ofcolumn patterns satisfies the linear equations mentioned above. Moreover, foreach equivariant model and for any number of taxa, we provide a set of linearlyindependent equations defining this space of phylogenetic mixtures. This is apowerful tool that has already been successfully used in model selection. Wealso use the results obtained to study identifiability issues for phylogeneticmixtures.

The space of phylogenetic mixtures for equivariant models