# Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample

arXiv: https://arxiv.org/abs/2112.00816 · HTML: https://mlsanity.com/paper/2112.00816/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2021.

## Built on by (most cited first, 4 of 4)

- [Learning and Testing Latent-Tree Ising Models Efficiently](https://mlsanity.com/paper/2211.13291/) (2022)
- [Network Topology Inference with Sparsity and Laplacian Constraints](https://mlsanity.com/paper/2309.00960/) (2023)
- [Adaptive Estimation of Graphical Models under Total Positivity](https://mlsanity.com/paper/2210.15471/) (2022)
- [EM's Convergence in Gaussian Latent Tree Models](https://mlsanity.com/paper/2211.11904/) (2022)

## Abstract (arXiv)

We study the problem of maximum likelihood estimation given one data sample (n=1) over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on trees. BMTMs are often used as a null model in phylogenetics, where the one-sample regime is common. Specifically, we show that, almost surely, the one-sample BMTM maximum likelihood estimator (MLE) exists, is unique, and corresponds to a fully observed tree. Moreover, we provide a polynomial time algorithm for its exact computation. We also consider the MLE over all possible BMTM tree structures in the one-sample case and show that it exists almost surely, that it coincides with the MLE over diagonally dominant M-matrices, and that it admits a unique closed-form solution that corresponds to a path graph. Finally, we explore statistical properties of the one-sample BMTM MLE through numerical experiments.