Maximum likelihood estimation of phylogenetic tree and substitution   rates via generalized neighbor-joining and the EM algorithm
  A central task in the study of molecular sequence data from present-day species is the reconstruction of the ancestral relationships. The most established approach to tree reconstruction is the maximum likelihood (ML) method. In this method, evolution is described in terms of a discrete-state continuous-time Markov process on a phylogenetic tree. The substitution rate matrix, that determines the Markov process, can be estimated using the expectation maximization (EM) algorithm. Unfortunately, an exhaustive search for the ML phylogenetic tree is computationally prohibitive for large data sets. In such situations, the neighbor-joining (NJ) method is frequently used because of its computational speed. The NJ method reconstructs trees by clustering neighboring sequences recursively, based on pairwise comparisons between the sequences. The NJ method can be generalized such that reconstruction is based on comparisons of subtrees rather than pairwise distances. In this paper, we present an algorithm for simultaneous substitution rate estimation and phylogenetic tree reconstruction. The algorithm iterates between the EM algorithm for estimating substitution rates and the generalized NJ method for tree reconstruction. Preliminary results of the approach are encouraging. 
Asger Hobolth and Ruriko Yoshida