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

Maria Emilíia Alonso, Ernesto Álvarez

TL;DR

This work analyzes maximum-likelihood estimation for 4-state triplets under the Jukes-Cantor JC69 model with a molecular clock on a rooted tripod. By parameterizing pattern probabilities via $x_i=e^{-4q_i}$ and forming $h_f=\ln L$ on a restricted semi-algebraic region, the authors apply a Morse-theoretic criterion and Maple-based real elimination to study the Hessian. They prove that within the restricted admissible region there is a single interior critical point, with $\det(\mathrm{Hess}\, h_f)>0$, hence a unique global maximum that depends analytically on the observed frequencies $f$. The results provide a rigorous foundation for ML inference in this phylogenetic setting and clarify the parameter region where ML estimates are well-defined and stable with respect to data.

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.

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

TL;DR

This work analyzes maximum-likelihood estimation for 4-state triplets under the Jukes-Cantor JC69 model with a molecular clock on a rooted tripod. By parameterizing pattern probabilities via and forming on a restricted semi-algebraic region, the authors apply a Morse-theoretic criterion and Maple-based real elimination to study the Hessian. They prove that within the restricted admissible region there is a single interior critical point, with , hence a unique global maximum that depends analytically on the observed frequencies . The results provide a rigorous foundation for ML inference in this phylogenetic setting and clarify the parameter region where ML estimates are well-defined and stable with respect to data.

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.
Paper Structure (4 sections, 8 theorems, 15 equations, 8 figures)

This paper contains 4 sections, 8 theorems, 15 equations, 8 figures.

Table of Contents

  1. The 4-states Triplets under Jukes-Cantor model of evolution, following the approach of Chor-Hendy-Snir (2006)
  2. Uniqueness of the ML and analytical dependency on the parameters
  3. Description of a suitable region of the parameters
  4. Study of the sign of the determinant of the Hessian of $h_f$ in the region of interest and admissible parameters

Key Result

Theorem 2.1

(Christensen2017, here for $n=2$) Let $h:M\subset \mathbb{R}^2\rightarrow \mathbb{R}$ be a Morse function defined on a smooth, compact, contractible manifold. If $M$ has a boundary, suppose that the gradient of $h$, $\nabla h$ is well defined and points inward from its boundary. Then $h$ has a uniqu

Figures (8)

  • Figure 1: Phylogenetic tree model by ChorHendySnir2006.
  • Figure 2: $a_0=0$ blue(+), $a_1=0$ black(-), $a_0+a_1=0$ orange(+), $b_0+b_1=0$ yellow(-), $-a_0c_1+c_0a_1=0$ green(-).
  • Figure 3: $a_0^2 \alpha_2 - a_0 a_1\alpha_1 + a_1^2 \alpha_0=0$ purple(+), $a_1=0$ black(-), $a_0+a_1=0$ orange(+), $-a_0c_1+c_0a_1=0$ green(-).
  • Figure 4: $b_0^2 \ \alpha_2 - b_0 \ b_1 \ \alpha_1 + b_1^2 \ \alpha_0 =0$ purple(+), $a_0 + a_1=0$ orange(+), $a_1 \ = \ 0$ black(-), $a_0 \ = 0$ blue (+), $\alpha_2 \ = 0$ "sienna"brown (+).
  • Figure 5: $c_0^2 \ \alpha_2 - c_0 \ c_1 \ \alpha_1 + c_1^2 \ \alpha_0=0$ purple(+), $a_0+a_1=0$ orange(+),$a_1 \ = 0$ black(-), $-a_0 \ c_1 \ + \ c_0 \ a_1 \ =0$ green(+).
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Remark 2.10