Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915441855692800 |
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| author | Coons, Jane Ivy Cox, Shelby Maraj, Aida Nometa, Ikenna |
| author_facet | Coons, Jane Ivy Cox, Shelby Maraj, Aida Nometa, Ikenna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance Coons, Jane Ivy Cox, Shelby Maraj, Aida Nometa, Ikenna Statistics Theory Algebraic Geometry Populations and Evolution 62R01, 14M25, 62F10 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. |
| title | Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance |
| topic | Statistics Theory Algebraic Geometry Populations and Evolution 62R01, 14M25, 62F10 |
| url | https://arxiv.org/abs/2402.10322 |