Numerical algebraic geometry provides numerical descriptions of solution sets of polynomial systems of equations in several unknown. Such sets are called algebraic varieties.
In algebraic statistics, a statistical model is associated to an algebraic variety to study its geometric structure.
This thesis contains my work at UC Berkeley that uses numerical algebraic geometry for the algebraic statistics problem of maximum likelihood estimation.
In Chapter 2 we study the maximum likelihood estimation problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions.
In Chapter 3 we prove a bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of corank r-1.
From the perspective of statistics, we show that maximum likelihood estimation for matrices of rank r is the same problem as minimum likelihood estimation for matrices of corank r-1,
and vice versa.
In Chapter 4, a description of the maximum likelihood estimation problem in terms of dual varieties and conormal varieties is given. With this description, we define the dual likelihood equations. We show how solving these dual likelihood equations give solutions to the maximum likelihood estimation problem without having the defining equations of the model itself.
In Chapter 5,
discrete algebraic statistical models are considered and solutions to the likelihood equations when the data contain zeros are studied.
Focusing on sampling and model zeros, we show that the solutions of the likelihood equations in these cases are contained in a previously studied variety, the likelihood correspondence. The number of solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into computationally easier problems involving sampling and model zeros.
In Chapter 6
the Macaulay2 package Bertini.m2 is introduced.
Macaulay2 is a software system designed to support research in algebraic geometry, and Bertini is a popular software system for numerical algebraic geometry.
The package Bertini.m2 provides an interface to Bertini via Macaulay2.