title: Do tree split probabilities determine the branch lengths?
abstract: The evolution of aligned DNA sequence sites is generally modeled by a Markov process operating along the edges of a phylogenetic tree. It is well known that the probability distribution on the site patterns at the tips of the tree determines the tree topology, and its branch lengths. However, the number of patterns is typically much larger than the number of edges, suggesting considerable redundancy in the branch length estimation. In this paper we ask whether the probabilities of just the 'edge-specific' patterns (the ones that correspond to a change of state on a single edge) suffice to recover the branch lengths of the tree, under a symmetric 2-state Markov process. We first show that this holds provided the branch lengths are sufficiently short, by applying the inverse function theorem. We then consider whether this restriction to short branch lengths is necessary. We show that for trees with up to four leaves it can be lifted. This leaves open the interesting question of whether this holds in general. Our results also extend to certain Markov processes on more than 2-states, such as the Jukes-Cantor model.
authors.affiliation: School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel. Electronic address: benny@cs.tau.ac.il.
authors.forename: Benny
authors.initials: B
authors.lastname: Chor
authors.affiliation: Biomathematics Research Centre, School of Mathematics and Statistics, University of Canterbury, Christchurch 8140, New Zealand. Electronic address: mike.steel@canterbury.ac.nz.
authors.forename: Mike
authors.initials: M
authors.lastname: Steel
mesh_terms: D000465:Algorithms; D019143:Evolution, Molecular; D008390:Markov Chains; D008954:Models, Biological; D010802:Phylogeny; D011336:Probability; D012984:Software
keywords: Evolutionary model; Hadamard transform; Inverse function theorem; Markov process; Phylogenetic tree reconstruction; Systems of polynomial equations