A computation of maximum likelihood for 4-states-triplets under Jukes-Cantor and MC

TL;DR

This paper analytically computes the maximum likelihood for a specific 4-state triplet evolutionary model under Jukes-Cantor and molecular clock assumptions, confirming a conjecture about the likelihood's unique maximum.

Contribution

It proves the likelihood function has a unique maximum analytically depending on parameters, under certain inequalities, confirming a conjecture from prior work.

Findings

Likelihood function has a unique maximum depending analytically on parameters.

The proof reduces to an algebraic problem solved with Maple.

The result confirms a conjecture about the model's likelihood.

Abstract

We study the ChorHendySnir2006 evolutionary model, which consists of a rooted phylogenetic tree with three leaves, subject to the Jukes--Cantor (JC69) molecular evolutionary model and molecular clock. We show that the likelihood function associated with this model has a unique maximum which depends analytically of the parameters (as it was conjectured in ChorHendySnir2006), assuming that these parameters verify some very precise inequalities; some of which arise naturally from the model. With a typical argument of differential topology we reduce the proof to answer a question of algebra, very simple, although computationally involved, that we solve using some Maple libraries. We are very indebted to Marta Casanellas, who presented the problem to us and gave us the first insights on it.