Search for: All records

Abstract Maximum likelihood estimation is among the most widely-used methods for inferring phylogenetic trees from sequence data. This paper solves the problem of computing solutions to the maximum likelihood problem for 3-leaf trees under the 2-state symmetric mutation model (CFN model). Our main result is a closed-form solution to the maximum likelihood problem for unrooted 3-leaf trees, given generic data; this result characterizes all of the ways that a maximum likelihood estimate can fail to exist for generic data and provides theoretical validation for predictions made in Parks and Goldman (Syst Biol 63(5):798–811, 2014). Our proof makes use of both classical tools for studying group-based phylogenetic models such as Hadamard conjugation and reparameterization in terms of Fourier coordinates, as well as more recent results concerning the semi-algebraic constraints of the CFN model. To be able to put these into practice, we also give a complete characterization to test genericity. 

The Macaulay2 [5] package AlgebraicOptimization implements methods for determining the algebraic degree of an optimization problem. We describe the structure of an algebraic optimization problem and explain how the methods in this package may be used to determine the respective degrees. Special features include determining Euclidean distance degrees and maximum likelihood degrees. To our knowledge, this is the first comprehensive software package combining different methods in algebraic optimization. The package is available at https://github.com/Macaulay2/Workshop-2020-Cleveland/tree/ISSAC-AlgOpt/alg-stat/AlgebraicOptimization.