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

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 fixed 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 Amendola and Zwiernik. This talk will focus on a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Prufer theorem to complete graphs with weights given by a tree, and on the intersection theory used to compute the ML-degree of the star tree.