Comparisons of Stochastic Matrices with Applications in Information Theory, Statistics, Economics and Population / Edition 1

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Overview

The focus of this work is on generalizing the notion of variation in a set of numbers to variation in a set of probability distributions. The authors collect some known ways of comparing stochastic matrices in the context of information theory, statistics, economics, and population sciences. They then generalize these comparisons, introduce new comparisons, and establish the relations of implication or equivalence among sixteen of these comparisons. Some of the possible implications among these comparisons remain open questions. The results in this book establish a new field of investigation for both mathematicians and scientific users interested in the variations among multiple probability distributions. A great strength of this text is the resulting connections among ideas from diverse fields - mathematics, statistics, economics, and population biology. In providing this array of new tools and concepts, the work will appeal to the practitioner. At the same time, it will serve as an excellent resource for self-study or for a graduate seminar course, as well as a stimulus to further research.
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Editorial Reviews

From the Publisher
"This book gives a mathematical treatment of a variety of methods for quantifying divergence or similarity between sets of proability distributions on a common space. Classical metrics, such as total variation and Kullback—Liebler divergence, are generalized. Such problems arise in statistics, economics and information theory. The book gives Brief but useful treatments of these and other applications with abundant references... well written with pointers to history and other methods of proof."

—Journal of the American Statistical Assoc.

"The book is compact and is dense with information. There is considerable motivation in the introduction... Contain[s] much material for the specialist."

—Zentralblatt Math

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Product Details

  • ISBN-13: 9780817640828
  • Publisher: Birkhauser Verlag
  • Publication date: 9/29/1998
  • Edition description: 1998
  • Edition number: 1
  • Pages: 158
  • Product dimensions: 0.44 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface
Pt. I Comparing Partial Orderings Among Stochastic Matrices 1
1 Introduction 3
2 Notation and definitions 19
3 Generalizations of classical channel comparisons 42
4 Degradation is the same as increasing density 50
5 Shannon's inclusion implies smaller capacity 54
6 A simple case: matrices A and B have only two columns 71
7 Open problems 79
Pt. II Divergence and Contraction Coefficients 83
1 Introduction, definitions, and notation 85
2 A generalization of an inequality of Dobrushin 88
3 The divergence 94
4 Divergence between images of measures via Markov kernels. Contraction coefficients 106
5 A particular case: At most countable spaces 119
6 Behavior of [actual symbol not reproducible] for a fixed Markov kernel T 125
7 Applications of global divergence to comparison of experiments 136
8 History of the problem 147
References 149
Index of principal notations 155
Index 157
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