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Maximum Likelihood Estimation of Symmetric Group-Based Models via Numerical Algebraic Geometry.

Dimitra Kosta1, Kaie Kubjas2

  • 1School of Mathematics and Statistics, University of Glasgow, Glasgow, UK.

Bulletin of Mathematical Biology
|October 26, 2018
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Summary

This study uses algebraic geometry to analyze phylogenetic models, revealing challenges in maximum likelihood estimation. Algebraic methods uncover cases where the maximum likelihood estimate for phylogenetic models does not exist.

Keywords:
Algebraic statisticsGroup-based modelsMaximum likelihood estimationNumerical algebraic geometryPhylogeneticsReal algebraic geometry

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Area of Science:

  • Computational Biology
  • Phylogenetics
  • Algebraic Geometry

Background:

  • Phylogenetic models can be parameterized using polynomial maps related to root distribution and edge transition probabilities.
  • Previous work established polynomial inequalities characterizing joint probabilities for symmetric continuous-time group-based models.

Purpose of the Study:

  • To apply numerical algebraic geometry to maximum likelihood estimation in phylogenetic models.
  • To explore scenarios where maximum likelihood estimates may not exist, particularly in complex phylogenetic models.

Main Methods:

  • Utilized polynomial parametrization maps derived from phylogenetic models.
  • Employed numerical algebraic geometry techniques for maximum likelihood estimation.
  • Investigated specific examples to identify limitations of estimation methods.

Main Results:

  • Demonstrated the application of algebraic methods to maximum likelihood estimation in phylogenetics.
  • Identified a specific case where the maximum likelihood estimate for a phylogenetic model does not exist.
  • Highlighted the difficulty of discovering such non-existence issues using traditional methods.

Conclusions:

  • Numerical algebraic geometry provides a powerful framework for analyzing phylogenetic models and their statistical properties.
  • Algebraic methods are crucial for uncovering edge cases and limitations in maximum likelihood estimation for phylogenetics.
  • The study underscores the importance of advanced mathematical tools in computational biology for robust model analysis.