Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance
arxiv(2024)
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+1-2n-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.
MoreTranslated text
AI Read Science
Must-Reading Tree
Example
![](https://originalfileserver.aminer.cn/sys/aminer/pubs/mrt_preview.jpeg)
Generate MRT to find the research sequence of this paper
Chat Paper
Summary is being generated by the instructions you defined