ML degrees of Brownian motion tree models: Star trees and root invariance

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-08-06 DOI:10.1016/j.jsc.2025.102482

Jane Ivy Coons , Shelby Cox , Aida Maraj , Ikenna Nometa

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引用次数: 0

Abstract

A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with

n + 1

leaves is

2^{n + 1} - 2 n - 3

, which was previously conjectured by Améndola and Zwiernik. We also prove that the ML-degree of a BMT model is independent of the choice of the root. The proofs rely on the toric geometry of concentration matrices in a BMT model. Toward this end, we produce a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Prüfer theorem to complete graphs with weights given by a tree.

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布朗运动树模型的ML度：星树和根不变性

布朗运动树（BMT）模型是一种高斯模型，其相关的协方差矩阵集是根据系统发育树的共同祖先线性约束的。我们研究了通过计算最大似然度来推断BMT模型最大似然估计量的复杂性。我们的主要结果是，在有n+1个叶子的星树上，BMT模型的ml度为2n+1−2n−3，这是amsamundola和Zwiernik先前推测的。我们还证明了BMT模型的ml度与根的选择无关。这些证明依赖于BMT模型中浓度矩阵的环几何。为此，我们提出了一个BMT模型浓度矩阵行列式的组合公式，它将cayley - pr fer定理推广到权值由树给出的完全图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.