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Mathias Drton Bernd Sturmfels Seth Sullivant

Lectures on Algebraic Statistics

Birkhäuser Basel · Boston · Berlin

Mathias Drton University of Chicago Department of Statistics 5734 S. University Ave Chicago, IL 60637 USA e-mail: [email protected]

Bernd Sturmfels Department of Mathematics University of California 925 Evans Hall Berkeley, CA 94720 USA e-mail: [email protected]

Seth Sullivant Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 USA e-mail: [email protected]

2000 Mathematics Subject Classification: 62, 14, 13, 90, 68

Library of Congress Control Number: 2008939526

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 978-3-7643-8904-8 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8904-8

e-ISBN 978-3-7643-8905-5

987654321

www.birkhauser.ch

Contents Preface

vii

1 Markov Bases 1.1 Hypothesis Tests for Contingency Tables . . . . . . . . . . . . . . . 1.2 Markov Bases of Hierarchical Models . . . . . . . . . . . . . . . . . 1.3 The Many Bases of an Integer Lattice . . . . . . . . . . . . . . . .

1 1 11 19

2 Likelihood Inference 2.1 Discrete and Gaussian Models . . . . . . . . . . . . . . . . . . . . . 2.2 Likelihood Equations for Implicit Models . . . . . . . . . . . . . . 2.3 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 40 48

3 Conditional Independence 3.1 Conditional Independence Models . . . . . . . . . . . . . . . . . . . 3.2 Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Parametrizations of Graphical Models . . . . . . . . . . . . . . . .

61 61 69 79

4 Hidden Variables 4.1 Secant Varieties in Statistics . . . . . . . . . . . . . . . . . . . . . . 4.2 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 99

5 Bayesian Integrals 105 5.1 Information Criteria and Asymptotics . . . . . . . . . . . . . . . . 105 5.2 Exact Integration for Discrete Models . . . . . . . . . . . . . . . . 114 6 Exercises 6.1 Markov Bases Fixing Subtable Sums . . . . 6.2 Quasi-symmetry and Cycles . . . . . . . . . 6.3 A Colored Gaussian Graphical Model . . . 6.4 Instrumental Variables and Tangent Cones . 6.5 Fisher Information for Multivariate Normals 6.6 The Intersection Axiom and Its Failure . . . 6.7 Primary Decomposition for CI