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arXiv · 2402.10322

Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance

Jane Ivy Coons·Shelby Cox·Aida Maraj·Ikenna Nometa
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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 Am\'endola 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-Pr\"ufer theorem to complete graphs with weights given by a tree.

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Jane Ivy Coons, Shelby Cox, Aida Maraj, Ikenna Nometa. 2024-02-15. Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance. https://doi.org/10.1016/j.jsc.2025.102482

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