Seminar Details
2023-10-24 (14h) : Tropical toric maximum likelihood estimation
At Euler building (room A.002)
Organized by Mathematical Engineering
Speaker :
Karel Devriendt (max planck institute, Leipzig)
Abstract :
Many common statistical models are parametrized by polynomial maps; some examples are log-linear and graphical models. To study such statistical models,
applied algebraic geometry can be used in an approach which is known as algebraic statistics. One well-studied problem in this setting is maximum likelihood estimation (MLE):
given a model and some data, which points in your model best explain the data? In this talk, we consider the MLE problem when our data depends on a parameter
and we ask what can be said about the convergence rates of the solution as the parameter goes to zero. This problem was solved for linear models by Agostini et al. (2021) and
Ardila-Eur-Penaguiao (2022). Here we consider the problem for log-linear models, also called toric models, where the MLE problem comes down to intersecting a
toric variety with a linear space. Using tools from tropical geometry -- a combinatorical shadow of algebraic geometry -- the problem simplies to intersecting the tropical toric
variety (which is a linear space) with a tropical linear space (which is a polyhedral complex). I will present some preliminary results which show that the tropical MLE points,
i.e. the convergence rates, are given by simple linear transformations of the data, and that the different MLE points are labeled by simplices in a certain triangulation.
This is joint work with Erick Boniface and Serkan Hosten.