Analytic Solutions for Three-Taxon MLMC Trees with Variable Rates Across Sites

We consider the problem of finding the maximum likelihood rooted tree under a molecular clock (MLMC), with three species and 2-state characters under a symmetric model of substitution. For identically distributed rates per site this is probably the simplest phylogenetic estimation problem, and it is readily solved numerically. Analytic solutions, on the other hand, were obtained only recently (Yang, 2000). In this work we provide analytic solutions for any distribution of rates across sites (provided the moment generating function of the distribution is strictly increasing over the negative real numbers). This class of distributions includes, among others, identical rates across sites, as well as the Gamma, the uniform, and the inverse Gaussian distributions. Therefore, our work generalizes Yang's solution. In addition, our derivation of the analytic solution is substantially simpler. We employ the Hadamard conjugation (Hendy and Penny, 1993) and convexity of an entropy-like function.