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

• Authors: Jan-Christian Hütter, Chandler Squires, Michael Truell, Caroline Uhler and Piotr Zwiernik
• Bernoulli..

We study the problem of maximum likelihood estimation given one 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, for any fixed tree, 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 one-sample MLE over all possible BMTM tree structures 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.