Group theoretic limits on the reconstruction of evolutionary processes

Evolutionary processes in deep time such as trait evolution or speciation and extinction are reconstructed from the limited evidence available in the present. Understanding limiting factors on the information content of these reconstructions is crucial to understanding the inferences that can be made from them, and identify the identifiability issues arising from the usage of reconstructed processes in place of evolutionary processes. I show that the identifiability problem decomposes into (1) a global problem about the mode of evolution that can be described by an group action on the space of evolutionary modes, and (2) a local problem about the tempo of evolution that can be described by groups action on the time interval of interest.
Based on weak assumptions on the statistical framework used, these results provide fundamental limits on information that can be inferred from reconstructed evolutionary processes.