Semialgebraic Statistics and Latent Tree Models explains how to analyze statistical models with hidden (latent) variables. It takes a systematic, geometric approach to studying the semialgebraic structure of latent tree models.
The first part of the book gives a general introduction to key concepts in algebraic statistics, focusing on methods that are helpful in the study of models with hidden variables. The author uses tensor geometry as a natural language to deal with multivariate probability distributions, develops new combinatorial tools to study models with hidden data, and describes the semialgebraic structure of statistical models.
The second part illustrates important examples of tree models with hidden variables. The book discusses the underlying models and related combinatorial concepts of phylogenetic trees as well as the local and global geometry of latent tree models. It also extends previous results to Gaussian latent tree models.
This book shows you how both combinatorics and algebraic geometry enable a better understanding of latent tree models. It contains many results on the geometry of the models, including a detailed analysis of identifiability and the defining polynomial constraints.
Table of Contents
Introduction. Semialgebraic statistics: Algebraic and analytic geometry. Algebraic statistical models. Tensors, moments, and combinatorics. Latent tree graphical models: Phylogenetic trees and their models. The local geometry. The global geometry. Gaussian latent tree models.
Piotr Zwiernik is a Marie Skłodowska-Curie International Fellow in the Department of Mathematics at the University of Genoa. His research interests include statistical inference, graphical models with hidden variables, algebraic statistics, singular learning theory, time series analysis, and symbolic methods. He received a PhD in statistics from the University of Warwick.