[Paper Review] A computation of maximum likelihood for 4-states-triplets under Jukes-Cantor and MC

Maria Emilíia Alonso, Ernesto Álvarez|arXiv (Cornell University)|Jan 21, 2026

Genome Rearrangement Algorithms0 citations

TL;DR

The paper proves that the likelihood for a JC69 4-states triplet with molecular clock has a unique interior maximum, which depends analytically on the parameters within a restricted admissible region, using Morse-theoretic arguments and Maple-based algebraic elimination.

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.

Motivation & Objective

• Motivate and study the likelihood landscape for a rooted tripod under JC69 with molecular clock.
• Establish the existence and uniqueness of the maximum likelihood estimate within a restricted parameter region.
• Show that the ML value depends analytically on model parameters.
• Characterize the admissible region of parameters ensuring well-behaved ML optimization.

Proposed method

• Define the JC69 triplet likelihood L(a,f) with a-parameters a0,a1,a2,a3,a4 and pattern frequencies f0,f12,f3,f4.
• Reduce to a two-variable optimization problem in x2 and x3 under molecular clock constraints (x_i = e^{-4 q_i}).
• Apply a Morse-function framework (Christensen 2017) to guarantee a unique global maximum by checking gradient and Hessian conditions.
• Reformulate the problem into real-algebraic conditions and perform elimination using Maple libraries (SolveTools) to relate f-parameters to x-parameters.
• Prove that the Hessian is positive at interior critical points within the restricted admissible region, ensuring a unique ML solution.

Experimental results

Research questions

• RQ1 Is there a unique global maximum of the log-likelihood h_f for the JC69 MC triplet model in the admissible region?
• RQ2 Does the maximum likelihood solution depend analytically on the observed data parameters within the restricted region?
• RQ3 What are the necessary parameter constraints (admissible region) that guarantee a well-posed ML optimization under molecular clock?
• RQ4 Can algebraic elimination via Maple provide explicit relationships linking pattern frequencies to model parameters?

Key findings

• There exists a unique critical point in the interior of the restricted domain A' and it is the global maximum for the log-likelihood.
• The Hessian at critical points is positive within the restricted admissible region, yielding a local (hence global) maximum by Morse theory arguments.
• The log-likelihood varies analytically with the model parameters in the restricted admissible region.
• A restricted admissible region is identified by linear and inequality constraints on the pattern frequencies f0,f3,f12,f4, ensuring well-posedness of the ML optimization.
• The authors translate the optimization into algebraic conditions and perform real-elimination to relate f-parameters to x-parameters, enabling explicit ML characterization.