Title: Likelihood Geometry of Brownian Motion Tree Models
Abstract: Brownian motion tree models are used to describe the evolution of a continuous trait along a phylogenetic tree under genetic drift. Such a model is obtained by placing linear constraints on a mean-zero multivariate Gaussian distribution according to the topology of the underlying tree. We investigate the enumerative geometry of the standard and dual maximum likelihood estimation problems in these models. In particular, we study the number of complex critical points of the log-likelihood and dual log-likelihood functions, known as the ML-degree and dML-degree, respectively. We use the toric geometry of Brownian motion tree models to give a formula for the dML-degree for all trees. We also prove a formula for the ML-degree of a star tree and show that for general trees, the ML-degree does not depend on the location of the root.