Computing algebraic degrees of phylogenetic varieties

TL;DR

This paper computes algebraic invariants like degrees and maximum likelihood degrees for phylogenetic varieties derived from common nucleotide substitution models on small trees, aiding phylogeny reconstruction.

Contribution

It provides explicit calculations of algebraic degrees for phylogenetic varieties in models where such computations were previously lacking.

Findings

Computed degrees for Cavender-Farris-Neyman, Jukes-Cantor, Kimura 2-parameter, and Kimura 3-parameter models.

Analyzed small phylogenetic trees with up to 5 leaves.

Filled a gap in algebraic statistics literature regarding invariants of phylogenetic varieties.

Abstract

A phylogenetic variety is an algebraic variety parameterized by a statistical model of the evolution of biological sequences along a tree. Understanding this variety is an important problem in the area of algebraic statistics with applications in phylogeny reconstruction. In the broader area of algebra statistics, there have been important theoretical advances in computing certain invariants associated to algebraic varieties arising in applications. Beyond the dimension and degree of a variety, one is interested in computing other algebraic degrees, such as the maximum likelihood degree and the Euclidean distance degree. Despite these efforts, the current literature lacks explicit computations of these invariants for the particular case of phylogenetic varieties. In our work, we fill this gap by computing these invariants for phylogenetic varieties arising from the simplest group-based models of nucleotide substitution Cavender-Farris-Neyman model, Jukes-Cantor model, Kimura 2-parameter model, and the Kimura 3-parameter model on small phylogenetic trees with at most 5 leaves.