Algebraic statistics

• Verfasst von: Sullivant, Seth [VerfasserIn]
• Titel: Algebraic statistics
• Verf.angabe: Seth Sullivant
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2018
• Jahr: [2018]
• Umfang: 1 Online-Ressource (xiii, 490 Seiten)
• Gesamttitel/Reihe: Graduate studies in mathematics ; 194
• ISBN: 978-1-4704-4980-3
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Sullivant, Seth: Algebraic statistics. - Providence, Rhode Island : American Mathematical Society, 2018. - xiii, 490 Seiten
• Sach-SW: Electronic books
• K10plus-PPN: 1043848592
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

Algebraic statistics uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. The starting point for this connection is the observation that many statistical models are semialgebraic sets. The algebra/statistics connection is now over twenty years old, and this book presents the first broad introductory treatment of the subject. Along with background material in probability, algebra, and statistics, this book covers a range of topics in algebraic statistics including algebraic exponential families, likelihood inference, Fisher's exact test, bounds on entries of contingency tables, design of experiments, identifiability of hidden variable models, phylogenetic models, and model selection. With numerous examples, references, and over 150 exercises, this book is suitable for both classroom use and independent study

Abstract

Cover -- Title page -- Preface -- Chapter 1. Introduction -- 1.1. Discrete Markov Chain -- 1.2. Exercises -- Chapter 2. Probability Primer -- 2.1. Probability -- 2.2. Random Variables and their Distributions -- 2.3. Expectation, Variance, and Covariance -- 2.4. Multivariate Normal Distribution -- 2.5. Limit Theorems -- 2.6. Exercises -- Chapter 3. Algebra Primer -- 3.1. Varieties -- 3.2. Ideals -- 3.3. Gröbner Bases -- 3.4. First Applications of Gröbner Bases -- 3.5. Computational Algebra Vignettes -- 3.6. Projective Space and Projective Varieties -- 3.7. Exercises -- Chapter 4. Conditional Independence -- 4.1. Conditional Independence Models -- 4.2. Primary Decomposition -- 4.3. Primary Decomposition of CI Ideals -- 4.4. Exercises -- Chapter 5. Statistics Primer -- 5.1. Statistical Models -- 5.2. Types of Data -- 5.3. Parameter Estimation -- 5.4. Hypothesis Testing -- 5.5. Bayesian Statistics -- 5.6. Exercises -- Chapter 6. Exponential Families -- 6.1. Regular Exponential Families -- 6.2. Discrete Regular Exponential Families -- 6.3. Gaussian Regular Exponential Families -- 6.4. Real Algebraic Geometry -- 6.5. Algebraic Exponential Families -- 6.6. Exercises -- Chapter 7. Likelihood Inference -- 7.1. Algebraic Solution of the Score Equations -- 7.2. Likelihood Geometry -- 7.3. Concave Likelihood Functions -- 7.4. Likelihood Ratio Tests -- 7.5. Exercises -- Chapter 8. The Cone of Sufficient Statistics -- 8.1. Polyhedral Geometry -- 8.2. Discrete Exponential Families -- 8.3. Gaussian Exponential Families -- 8.4. Exercises -- Chapter 9. Fisher's Exact Test -- 9.1. Conditional Inference -- 9.2. Markov Bases -- 9.3. Markov Bases for Hierarchical Models -- 9.4. Graver Bases and Applications -- 9.5. Lattice Walks and Primary Decompositions -- 9.6. Other Sampling Strategies -- 9.7. Exercises -- Chapter 10. Bounds on Cell Entries

Abstract

10.1. Motivating Applications -- 10.2. Integer Programming and Gröbner Bases -- 10.3. Quotient Rings and Gröbner Bases -- 10.4. Linear Programming Relaxations -- 10.5. Formulas for Bounds on Cell Entries -- 10.6. Exercises -- Chapter 11. Exponential Random Graph Models -- 11.1. Basic Setup -- 11.2. The Beta Model and Variants -- 11.3. Models from Subgraphs Statistics -- 11.4. Exercises -- Chapter 12. Design of Experiments -- 12.1. Designs -- 12.2. Computations with the Ideal of Points -- 12.3. The Gröbner Fan and Applications -- 12.4. Two-level Designs and System Reliability -- 12.5. Exercises -- Chapter 13. Graphical Models -- 13.1. Conditional Independence Description of Graphical Models -- 13.2. Parametrizations of Graphical Models -- 13.3. Failure of the Hammersley-Clifford Theorem -- 13.4. Examples of Graphical Models from Applications -- 13.5. Exercises -- Chapter 14. Hidden Variables -- 14.1. Mixture Models -- 14.2. Hidden Variable Graphical Models -- 14.3. The EM Algorithm -- 14.4. Exercises -- Chapter 15. Phylogenetic Models -- 15.1. Trees and Splits -- 15.2. Types of Phylogenetic Models -- 15.3. Group-based Phylogenetic Models -- 15.4. The General Markov Model -- 15.5. The Allman-Rhodes-Draisma-Kuttler Theorem -- 15.6. Exercises -- Chapter 16. Identifiability -- 16.1. Tools for Testing Identifiability -- 16.2. Linear Structural Equation Models -- 16.3. Tensor Methods -- 16.4. State Space Models -- 16.5. Exercises -- Chapter 17. Model Selection and Bayesian Integrals -- 17.1. Information Criteria -- 17.2. Bayesian Integrals and Singularities -- 17.3. The Real Log-Canonical Threshold -- 17.4. Information Criteria for Singular Models -- 17.5. Exercises -- Chapter 18. MAP Estimation and Parametric Inference -- 18.1. MAP Estimation General Framework -- 18.2. Hidden Markov Models and the Viterbi Algorithm

Abstract

18.3. Parametric Inference and Normal Fans -- 18.4. Polytope Algebra and Polytope Propogation -- 18.5. Exercises -- Chapter 19. Finite Metric Spaces -- 19.1. Metric Spaces and the Cut Polytope -- 19.2. Tree Metrics -- 19.3. Finding an Optimal Tree Metric -- 19.4. Toric Varieties Associated to Finite Metric Spaces -- 19.5. Exercises -- Bibliography -- Index -- Back Cover