The algebraic statistics of sampling, likelihood, and regression

This thesis is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. Featured here are three types of models that have such an algebraic structure. Linear Gaussian covariance models are continuous models which are simple to define but hard to analyze. We compute their maximum likelihood degree in dimension two and find it equal to $2n-3$ generically if the model has $n$ covariates. Discrete models with rational MLE are those discrete models for which likelihood estimation is easiest. We characterize them geometrically by building on the work of Huh and Kapranov on Horn uniformization. Algebraic manifolds are a more general kind of object which is used to encode continuous data. We introduce a new method for computing integrals and sampling from distributions on them, based on intersecting with random linear spaces. A brief report on mathematics in the sciences featuring case studies from soil ecology and nonparametric statistics closes the thesis.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Leipzig

Year

2020

Author dc:creator

• Marigliano, Orlando

• Mathematics
• Statistics
• Algebraic Geometry
• Algebraic Statistics
• Mathematics in the Sciences