3:00pm - 5:00pmGraphical Models
Chair(s): Elina Robeva (Massachusetts Institute of Technology, United States of America)
Graphical models are used to express relationships between random variables. They have numerous applications in the natural sciences as well as in machine learning and big data. This minisymposium will feature talks on several different types of graphical models, including latent tree models, max linear models, network models, boltzman machines, and non-Gaussian graphical models, each of which exploits their intrinsic algebraic, geometric, and combinatorial structure.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Brownian motion tree models are toric
Piotr Zwiernik
Universitat Pompeu Fabra
Felsenstein’s classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We give an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation.
Algebra and statistical learning for inferring phylogenetic networks
Elizabeth Gross
University of Hawaii at Manoa
Phylogenetic trees are graphical summaries of the evolutionary history of a set of species. In a phylogenetic tree, the interior nodes represent extinct species, while the leaves represent extant, or living, species. While trees are a natural choice for representing evolution visually, by restricting to the class of trees, it is possible to miss more complicated events such as hybridization and horizontal gene transfer. For more complete descriptions, phylogenetic networks, directed acyclic graphs, are increasingly becoming more common in evolutionary biology. In this talk, we will discuss Markov models on phylogenetic networks and explore how understanding their algebra and geometry can aid in establishing identifiability and model selection. In particular, we will describe a method for network inference that combines computational algebraic geometry and statistical learning. This is joint work with Travis Barton, Colby Long, and Joseph Rusinko.
Geometry of max-linear graphical models
Carlos Amendola
Technical University Munich
Motivated by extreme value theory, max-linear graphical models have been recently introduced and studied as an alternative to the classical Gaussian or discrete distributions often used in graphical modeling. We present max-linear models naturally in the framework of tropical geometry. This perspective allows us to shed light on some known results and to prove others with algebraic techniques, including conditional independence statements and maximum likelihood parameter estimation. This is joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.
Maximum Likelihood Estimation of Toric Fano Varieties motivated by phylogenetics
Dimitra Kosta
University of Glasgow
We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fibre products, and it also follows from a connection to a recent result about staged trees. This is joint work with Carlos Amendola and Kaie Kubjas.