Zwiernik Semialgebraic Statistics and Latent Tree Models

Erscheinungsjahr 2015
ISBN: 978-1-4665-7622-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 245 Seiten

Reihe: Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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.

Zielgruppe

Researchers and graduate students in algebraic statistics and machine learning.

Autoren/Hrsg.

Zwiernik, Piotr

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

Introduction
A statistical model as a geometric object
Algebraic statistics
Toward semialgebraic statistics
Latent tree models
Structure of the book

Semialgebraic statistics
Algebraic and analytic geometry
Basic concepts
Real algebraic and analytic geometry
Tensors and flattenings
Classical examples
Birational geometry

Algebraic statistical models
Discrete measures
Exponential families and their mixtures
Maximum likelihood of algebraic models
Graphical models

Tensors, moments, and combinatorics
Posets and Möbius functions
Cumulants and binary L-cumulants
Tensors and discrete measures
Submodularity and log-supermodularity

Latent tree graphical models
Phylogenetic trees and their models
Trees
Markov process on a tree
The general Markov model
Phylogenetic invariants

The local geometry
Tree cumulant parameterization
Geometry of unidentified subspaces
Examples, special trees, and submodels
Higher number of states

The global geometry
Geometry of two-state models
Full semialgebraic description
Examples, special trees, and submodels
Inequalities and estimation

Gaussian latent tree models
Gaussian models
Gaussian tree models and Chow–Liu algorithm
Gaussian latent tree models
The tripod tree

Bibliographical notes appear at the end of each chapter.

Über Autor(innen)

Piotr Zwiernik is a Marie Sklodowska-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.