ML MC Forks: Closed Form Analytic Solutions∗

Maximum likelihood (ML) is increasingly used as an optimality criterion for se-lecting evolutionary trees (Felsenstein, 1981), but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM), are used in order to find op-timal parameters for a given tree. So far, analytic solutions were derived only for the simplest model- three taxa, two state characters, under a molecular clock (MC). Quoting Ziheng Yang (2000), who initiated the analytic approach, “this seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogenetic estimation”. In this work, we give general analytic solutions for a family of trees with four taxa, two state characters, under a molecular clock. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. We start by presenting the general maximum likelihood problem on phylogenetic trees as a constrained optimization problem, an