Hadamard phylogenetic methods and the n-taxon process

The Hadamard transform (Hendy and Penny, Syst. Zool. 38(4):297-309, 1989; Hendy, Syst. Zool. 38(4):310-321, 1989) provides a way to work with stochastic models for sequence evolution without having to deal with the complications of tree space and the graphical structure of trees. Here we demonstrate that the transform can be expressed in terms of the familiar P[tau]=e ( Q[tau]) formula for Markov chains. The key idea is to study the evolution of vectors of states, one vector entry for each taxa; we call this the n-taxon process. We derive transition probabilities for the process. Significantly, the findings show that tree-based models are indeed in the family of (multi-variate) exponential distributions.