Markov Bases in Algebraic Statistics
Algebraic statistics is a rapidly developing field, where ideas from statistics and algebra meet and stimulate new research directions. One of the origins of algebraic statistics is the work by Diaconis and Sturmfels in 1998 on the use of Gröbner bas
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Satoshi Aoki • Hisayuki Hara • Akimichi Takemura
Markov Bases in Algebraic Statistics
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Satoshi Aoki Department of Mathematics and Computer Science Graduate School of Science and Engineering Kagoshima University Kagoshima, Japan
Hisayuki Hara Faculty of Economics Niigata University Niigata, Japan
Akimichi Takemura Department of Mathematical Informatics Graduate School of Information Science and Technology University of Tokyo Tokyo, Japan
ISSN 0172-7397 ISBN 978-1-4614-3718-5 ISBN 978-1-4614-3719-2 (eBook) DOI 10.1007/978-1-4614-3719-2 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012938710 © Springer Science+Business Media New York 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Algebraic statistics is a rapidly developing field, where ideas from statistics and algebra meet and stimulate new research directions. Statistics has been relying on classical asymptotic theory as a basis for statistical inferences. This classical basis is still very useful. However, when the validity of asymptotic theory is in doubt, for example, when the sample size is small, statisticians rely more and more on various computational methods. Similarly, algebra has long b
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