Motivation: We consider models useful for learning an evolutionary or phylogenetic tree from data consisting of DNA sequences corresponding to the leaves of the tree. In particular, we consider a general probabilistic model described in Siepel and Haussler that we call the phylogenetic-HMM model which generalizes the classical probabilistic models of Neyman and Felsenstein. Unfortunately, computing the likelihood of phylogenetic-HMM models is intractable. We consider several approximations for computing the likelihood of such models including an approximation introduced in Siepel and Haussler, loopy belief propagation and several variational methods.

Results: We demonstrate that, unlike the other approximations, variational methods are accurate and are guaranteed to lower bound the likelihood. In addition, we identify a particular variational approximation to be best—one in which the posterior distribution is variationally approximated using the classic Neyman–Felsenstein model. The application of our best approximation to data from the cystic fibrosis transmembrane conductance regulator gene region across nine eutherian mammals reveals a CpG effect.

To whom correspondence should be addressed.

Author notes

1Microsoft Research, Redmond, WA 98052, USA, 2Technion—Israel Institute of Technology Computer Science Department, Haifa 32000, Israel, 3Center for Biomolecular Science and Engineering and 4Howard Hughes Medical Institute, University of California Santa Cruz, CA 95064, USA