A semialgebraic description of the general Markov model on phylogenetic trees

TL;DR

This paper provides a complete semialgebraic description of the model space for the k-state general Markov model on phylogenetic trees, using algebraic and semi-algebraic methods to characterize the distributions.

Contribution

It introduces a novel semialgebraic framework for describing the model space of the general Markov model on trees, utilizing hyperdeterminants and quadratic forms.

Findings

Developed semialgebraic descriptions of the model space.

Used hyperdeterminants and quadratic forms to encode parameter constraints.

Explored Sturm sequences for similar characterization methods.

Abstract

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map --- the collection of joint distributions for a model --- forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space, but has been usefully studied with algebraic methods. Of ultimate interest, however, is not the full variety, but only the model space. Here we develop complete semialgebraic descriptions of the model space arising from the k-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently-formulated analogs of Cayley's hyperdeterminant, and the construction of certain quadratic forms from the joint distribution whose positive (semi-)definiteness encodes information about parameter values. We additionally investigate the use of Sturm sequences for obtaining similar results.