Semialgebraic statistics and latent tree models

Semialgebraic statistics and latent tree models

by Zwiernik, Piotr

Type Book
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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…

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Title Statements

  • Full title :Semialgebraic statistics and latent tree models / Piotr Zwiernik.

Summary

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.

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Edition Statements

  • Publication Date:2016.

Publication Statements

  • Publisher(s):Chapman & Hall/CRC
  • Place of Publication:Boca Raton
  • Language:eng
  • Publication Date:2016.

Notes

<P><STRONG>Introduction <BR></STRONG>A statistical model as a geometric object <BR>Algebraic statistics <BR>Toward semialgebraic statistics <BR>Latent tree models <BR>Structure of the book </P> <P></P><B><I> <P>Semialgebraic statistics <BR></I>Algebraic and analytic geometry</B> <BR>Basic concepts <BR>Real algebraic and analytic geometry <BR>Tensors and flattenings <BR>Classical examples <BR>Birational geometry </P> <P></P><B> <P>Algebraic statistical models <BR></B>Discrete measures <BR>Exponential families and their mixtures <BR>Maximum likelihood of algebraic models <BR>Graphical models </P> <P></P><B> <P>Tensors, moments, and combinatorics <BR></B>Posets and Mbius functions <BR>Cumulants and binary <I>L</I>-cumulants <BR>Tensors and discrete measures<BR>Submodularity and log-supermodularity </P> <P></P><B><I> <P>Latent tree graphical models <BR></I>Phylogenetic trees and their models</B> <BR>Trees <BR>Markov process on a tree <BR>The general Markov model <BR>Phylogenetic invariants </P> <P></P><B> <P>The local geometry</B> <BR>Tree cumulant parameterization<BR>Geometry of unidentified subspaces <BR>Examples, special trees, and submodels <BR>Higher number of states </P> <P></P><B> <P>The global geometry <BR></B>Geometry of two-state models <BR>Full semialgebraic description <BR>Examples, special trees, and submodels <BR>Inequalities and estimation </P> <P></P><B> <P>Gaussian latent tree models<BR></B>Gaussian models<BR>Gaussian tree models and Chow-Liu algorithm <BR>Gaussian latent tree models<BR>The tripod tree </P> <P></P> <P><EM>Bibliographical notes appear at the end of each chapter.</EM></P>

The Physical Object

  • Description:1 online resource (xx, 225 pages) : : illustrations (black and white)

Edition Identifiers

  • ISBN:9781466576223, 1466576227, 9781466576216, 1466576219, 9781466576223
  • oclc:on1076627288
  • Dewey Decimal:519.5

Subjects

Latent variables Statistics Variables latentes

Writers

Zwiernik, Piotr

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