Information, Weight of Evidence, the Singularity between Probability Measures and Signal Detection
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376 David Bridston Osteyee Office of Naval Research, Pasadena, CNUSA
Irving John Good Virginia Polytechnic Institute and State University, Blacksburg, VA/USA
Information, Weight of Evidence, the Singularity between Probability Measures and Signal Detection
Springer-Verlag Berlin. Heidelberg • New York 1 974
A M S Subject Classifications (1970):
60-G-15, 60-G-30, 62-F-15, 62-H-15, 94-02, 94-A-05
60-G-35, 62-M-10,
ISBN 3-540-06?26-4 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06726-4 Springer-Verlag New Y o r k " Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1974. Library of Congress Catalog Card Number "74-393. Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
The technologies and sciences of information processing and communication have made great advances during the last thirty years and have also stimulated much first-class theory.
This monograph contains a survey of the part of this
theory dealing with measures of information and evidence and especially with the detection of signals in noise.
Although the theory arises directly from
practical problems and realistic models, some of it turns out to be surprisingly advanced mathematically. If all the proofs were included the book would be very long, and we decided that it would be better to omit most of them. Acknowledsements.
The work was supported in part by the Department of
Health, Education and Welfare, Grants #ES 00033 and #I ROI GM18770. We wish to thank Becky Clevenger for her expert typing.
December 1973
D.B.Osteyee, l.J.Good,
Pasadena,
Blacksburg,
California Virginia
TABLE OF CONTENTS CHAPTER
I.
II.
III.
INTRODUCTION
1
LIST OF sYMBOLS
3
INFORMATION IN EVENTS AND WEI(~T OF EVIDENCE
8
1.1
Information in Events l.l.1 Properties of Information
8 9
1.2
Conditional Information in Events 1.2.1 Properties of Conditional Information
9 9
1.3
Mutual Information between Events 1.3.1 Properties of Mutual Information
i0 i0
1.4
Weight 1.4.1 1.4.2 1.4.3
ii Ii II 12
of Evidence Definition and Expressions Another Expression of the Weight of Evidence A Special Case
ENTROPY
13
2.1
Entropy of Discrete Random Variables 2.1.1 Properties of Entropy 2.1.2 Quantum Mechanics Model
13 13 14
2.2
Conditional Entropy for Discrete Random Vax.iables 2.2.1 Some Properties of Conditional Entropy
15 15
2.3
Entropy of Random Variables with Continuous Probability Density Functions
15
2.4
Conditional Entropy of Random Variables with Continuous Density Functions
17
SINGULARITY BETWEEN TWO PROBABILITY MEASURES
18
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