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.