Phylogenetic mixtures and linear invariants for equal input models

TL;DR

This paper studies the equal input model in phylogenetics, showing how linear invariants can restore tree identifiability in mixture models, and characterizes the structure of these invariants for any number of leaves.

Contribution

It extends the understanding of linear invariants in the equal input model, generalizing previous models and providing a detailed algebraic description of the space of mixtures and invariants.

Findings

Characterizes the space of mixtures for any fixed tree and all trees.

Provides a detailed description of invariants for four-leaf trees.

Builds on classic results to connect random processes and linear algebra in phylogenetics.

Abstract

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the `equal input model'. This model generalizes the `Felsenstein 1981' model (and thereby the Jukes--Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a `random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees -- the so called `model invariants'), on any number $n$ of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of $n=4$ leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake in 1987.