A Mathematical Theory of Hints: An Approach to the Dempster-Shafer Theory of Evidence

430 Pages·1995·13.746 MB·English

by  Prof. Dr. Jürg Kohlas, Dr. Paul-André Monney (auth.)

Lecture Notes in Economics and Mathematical
Systems 425
Series Editors: Günter Fandel · Walter Trockel
Herbert Dawid · Dinko Dimitrov et. al.
Jürg Kohlas · Paul-Andre Monney
A Mathematical
Theory of Hints
An Approach to the Dempster-Shafer
Theory of Evidence
Lecture N otes in Economics
and Mathematical Systems 425
Founding Editors:
M. Beckmann
H. P. KUnzi
Editorial Board:
H. Albach, M. Beckmann, G. Feichtinger, W. Hildenbrand, W. Krelle
H. P. KUnzi, K. Ritter, U. Schittko, P. Schonfeld, R. Selten
Managing Editors:
Praf. Dr. G. Fandel
Fachbereich Wirtschaftswissenschaften
Fernuniversităt Hagen
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Prof. Dr. W. Trockel
Institut fUr Mathematische Wirtschaftsforschung (IMW)
Universităt Bielefeld
Universitătsstr. 25, D-33615 Bielefeld, Germany
Springer-Verlag
Berlin Heidelberg GmbH
Jiirg Kohlas Paul-Andre Monney
A Mathematical Theory
of Hints
An Approach to the
Dempster-Shafer Theory of Evidence
Springer
Authors
Prof. Dr. JUrg Kohlas
Institute of Informatics
University of Fribourg
CH-1700 Fribourg, Switzerland
e-mail [email protected]
Dr. Paul-Andre Monney
Insitute of Informatics
University of Fribourg
CH-1700 Fribourg, Switzerland
ISBN 978-3-540-59176-4
Ligrary of Congress Cataloging-in-Publication Data
Kohlas, JUrg, 1939-A mathematieal theory of hints: an approaeh to the Dempster
Shafer theory of evidenee / JUrg Kohlas, Paul-Andre Monney. p.em. - (Leeture notes
in eeonomies and mathematical systems; 425
ISBN 978-3-540-59176-4 ISBN 978-3-662-01674-9 (eBook)
DOI 10.1007/978-3-662-01674-9
Probabilities. 2. Mathematical statistics. 3. Evidence. 1. Monney, Paul-Andre.
II. Title. III. Series. QA273.K597 1995 519.5'4-dc20 95-12080
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© Springer-Verlag Berlin Heidelberg 1995
Originally published by Springer-Verlag Berlin Heidelberg New York in 1995
Typesetting: Camera ready by author
SPIN: 10486761 42/3142-543210 -Printed on acid-free paper
Preface
This book is about the theory of hints, a variant of the Dempster-Shafer
theory of evidence. Since the reader may already have heard of or even be
familiar with the the theory of evidence, let us begin by giving some historical
notes. Evidence theory started from a course on statistica! inference taught
by Arthur P. Dempster at Harvard University in the late sixties. Then Demp
ster's ideas were reinterpreted by Glenn Shafer in his book A Maihemaiical
TheoTY of Evidence published by Princeton University Press in 1976. Evidence
is an appealing term and it is no surprise that the theory of evidence found
much interest among knowledge engineers who have to model uncertain and
imprecise information. After a lot of ad hoc attempts to model uncertainty
in expert systems, a serious theory seemed finally available to treat this basic
problem of the field. However, as the theory is not really compatible with
the simple paradigms of early rule-based expert systems, misinterpretations
and oversimplistic misuses of the theory caused much deception. Only slowly
begins the true nature and the rea! meaning of evidence theory to emerge
and to be understood. And hopefully this book wiII contribute to its better
understanding as well demonstrate its usefulness and mathematica! elegance.
To put the subject of this book into perspective, it is a good idea to have a
closer look at Dempster's and Shafer's conceptions of the theory of evidence. It
will then be easier to explain what is the essence of the theory of hints, which is
our own view of the theory of evidence. Dempster originally developed a theory
of lower and upper probabilities in an attempt to reconcile Bayesian statistics
with Fisher's fiducial argument (Dempster, 1967, 1968). The following citation
(up to symbols) is taken from Dempster (1967): "Consider a pair of spaces fl
and e and a multivalued rnapping r which assigns a subset r( w) <;;; e to every
w E f2. Suppose that P is a probability measure which assigns probabilities to
the members of a class A of subsets of f2. If P is acceptable for the probability
judgements about an uncertain outcome w E f2, aud if this uncertain outcome
w is known to correspond to an uncertain outcome eE r(w), what probability
e
judgements may be made about the uncertain outcome Ee'?" As Dempster
argues, this leads to the concept of compatible probability measures and he
rcfers to the bounds as lower and upper probabilities.
On the other hand, Shafer states in the preface of his book: "It (his book)
offers a reinterpretation of Dempster's work, a reinterpretation that identifies
his lower probabilites as epistemic probabilities 01' degrees of belief, takes the
VI Preface
rule for combining such degrees of belief as fundamental, and abandons the
idea that they arise as lower bounds over classes of Bayesian probabilities."
Given a piece of evidence relative to a question whose possible answers form
the set e, Shafer postulates the existence of a function Bel on the power set
e
of satisfying some axiomatic conditions. He interprets the numerical value
Bel(H), a number between O and 1, as the degree of belief that H contains the
e
answer and calls the function Bel a belieffunction. When is finite he shows
that belief functions are generated by so-called basic probability assignments
which in turn are used to combine belief functions.
So where does the theory of hints stand with respect to these two points
of view ? The point of view of hints is closer to Dempster's approach than to
the one of Shafer. However it gives a new interpretation to the probabilistic
structure considered in Dempster. So we keep Dempster's original structure
( il, P, r, e) but we gi ve it the following meaning:
Suppose that a cert ain question, whose answer is unknown, has to be
e
studied. The elements of represent the possible answers to the question
and we assume that there is exactly one correct, but unknown answer in e.
The goal is to make assertions about the answer in the light of the available
information. We assume that this informat ion allows for several different inter
pretations, depending on some unknown circumstances. These interpretations
are regrouped into the set il and there is exactly one correct interpretation.
Not ali interpretations are equally likely and the known probability measure
P on il reflects our information in that respect. Furthermore, if the interpre
tation w E il is the correct one, then the answer is known to be in the subset
r(w) c;:;; e. Such a structure 11. = (il,p,r,e) is called a hint. The hint 11.
can then be used to judge the fact or the hypothesis, which can be true or
e
false, that the answer is in a subset H of by assinging to it a degree of
support and a degree of plausibility. An interpretation w E il supports the
hypothesis H if r( w) c;:;; H because in this case the answer is necessarily in
H. The degree of support of H, denoted by sp(H), is defined as the proba
bility of ali supporting interpretations of H. So sp(H) can be regarded as the
probability with which H can be infered ar proved from the available infor
mat ion encoded in the hint. In a similar way, if the interpretat ion w E il is
correct, the subset H possibly contains the answer if r(w) n H =f 0. Then the
degree of plausibility of H, denoted by pl(H), is defined as the probability of
all interpretations under which H is possible. Sa like in Shafer's theory it is
possible ta judge hypotheses about the answer, but in contrast to it this can
be dane only in a second stage, after the hint has been constructed. For us,
the degrees of support and plausibility are concepts which are derived from
the notion of a hint, and not primitive objects like in Shafer's approach. Also,
we speak of degrees of support and not degrees of belief because the word
belief suggests an idea of personal judgement which is absent as soon as the
e
hint is given. Considering ali possible subsets H c;:;; leads to support and
plausibility functions sp and pl.
Preface VII
Dempster mathematicaUy defines sp(H) and pl(H) exactly in the same
way as we clid above, but the interpretation he gives to these values is different
from ours. He points out that sp( H) can be regarded as the minimal amount
of probability which can be transfered from il to outcomes eE Hand pl(H) as
the largest possible amount of probability which can be transfered to outcomes
e
E H. Then he introduced the concept of compatible probability measures:
P is compatible if sp(H) ~ P(H) ~ p/(H) for aU H ~ 8. Ever since sp and
pl have oft en been looked at as lower and upper bounds for some unknown
probability distribution by many authors. This has had unfortunate effects
because it led to intriguing misinterpretations of the theory of evidence. In our
opinion, the question treated by the theory of evidence is whether a hypothesis
H is true or not, or more precisely, to what degree a piece of evidence supports
a hypothesis H. It is quite a different question to ask what could be the
probabiliiy of the hypothesis H. In the latter case the unknown is noi whether
H is true or not, but the value of its probability. So let us emphasize that
although approaches using partially specified probabilities may be very useful
in many circumstances, the theory of evidence as developed in this book as a
theory of hints does noi belong to this class of methods.
It is clear today that the theory of evidence can be given various dif
ferent forms. Some of them are based on probability theory and others are
axiomatic theories, a priori without reference to probability theory. The first
approach tries to integrate evidence theory in the framework of classical prob
ability theory, and the theory of hints belongs to this category, whereas the
second one deliberately goes beyond classical probability. Shafer's theory of
belieffunctions and Smets' transferable belief model (Smets, 1988) are the two
best known representatives of this non-standard probability approach. Despite
the differences in approach and interpretation, aU of them lead essentially to
mathematically equivalent theories in the finite case, which implies that they
share the same basic theorems and the same computational procedures apply.
However, substantial differences appear when the infinite case is considered.
This book is divided into three Parts: Part 1 (Chapters 1 to 8) presents
the elementary theory of hints, Part II (Chapters 9 to 12) discusses some
possible applications of this theory and Part III (Qhapters 13 to 16) develops
the rudiments of a general theory of hints.
By discussing some simple examples, Chapter 1 intuitively introduces the
idea of a hint and the related concepts of support and plausibility. These con
cepts are then defined and studied formally in Chapters 2 and 3. Chapter 4
explains how to combine hints, an operat ion that is fundamental to the whole
theory. Chapter 5 shows that the theory of hints provides for an adequate
framework for probabilistic assumption-based reasoning, thereby establishing
the link with propositional logic. The degree of support of a logical formula
h is then essentiaUy equal to the probability of the assumptions that per
mit to logicaUy infer h, given the knowledge base E. This idea goes back to
Pearl's notion of probability of provability (Pearl, 1988), an idea also present
in Laskey, Lehner (1989) and Provan (1990). The assumptions can be seen
Vlll Preface
as arguments in favour of hypotheses. These arguments are weighed by their
likelihoods and the degree of support can then be viewed as the reliability of
an associated proof system. This point of view is considered in Chapter 6. Let
us note that this approach can be developed beyond the material presented
in this book: The idea of arguments for hypotheses suggests that it might be
possible to develop a purely qualitative theory of evidence by dropping the
probabilities on the arguments. This has been do ne in Kohlas (1993 a) for
propositional logic and was generalized to Boolean algebras and lattices in
Kohlas (1994). It is even possible to go one step further by generalizing the
notion of provability to arbitrary consequence relations in the sense of Tarski
(Besnard, Kohlas, 1994). Of course, in a second step, it is always possible
to reintroduce probabilities into these qualitative or symbolic versions of evi
dence theory. Chapter 7 is concerned with the analysis of families of frames of
discernment. This done in the abstract setting of lattices and important no
tions like conditional independence of frames are discussed. Chapter 8 deals
with the problem of combining hints efficiently and local propagat ion schemes
are presented.
Chapter 9 shows how statistical inference can be developed from the point
of view of the theory of hints. This leads to interesting reinterpretations of clas
sical statistical results. Chapter 10 discusses uncertainty in dynamical systems
from the perspective of hints and Chapter 11 considers diagnostic problems
from an evidential point of view. The last chapter of Part II, Chapter 12,
explains how the theory of hints can be used to represent and support spatial
and temporal reasoning. A very general class of models based on groups is
analyzed and computational methods are presented.
Chapter 13 presents a general theory of hints, where the sets of interpreta
tions and the frames of discernment are not necessarily finite. The main result
in this chapter is that every support function is generated by a generalized
hint. Chapter 14 introduces the notion of inclusion between hints, as well as
the the notion of canonical hint of a support function. Chapter 15 shows how
to combine general hints by Dempster's rule and it is proved that inclusion
of hints is preserved by this operation. Finally, the last Chapter 16 studies
closed random intervals and random variables from the perspective of hints.
Part I should be read first, and then either Part II and then Part III,
or Part III and then Part II. \Vithin parts the chapters should be read in
sequence.
There is no doubt that theoretical foundations, practical use in modeling
and computational aspects represent some of the possible topics for future
research in evidence theory. On the theoreticallevel, evidence theory on topo
logical spaces is a large open field for investigation. AIso, a hint can be seen
as the generalization of a random variable and classical probability theory can
largely be considered as a theory of random variables. However, Dempster's
rule of combination of hints, which is a key concept in the whole theory of
evidence, does not appear as such in probability theory. What is the role of
this rule in probability theory ? Also, recently connections between evidence
Acknowledgments IX
theory and modallogic have been put forward, but there remains a lot of open
questions about the relationship between evidence theory and non~standard
logics. Furthermore, it seems that evidential information is essentially rela
tional in nature, but conditional information has been proved very important
in knowledge engineering (e.g. in Bayesian networks where the knowledge is
essentially encoded as condi tional probabili ty distributions). Does this aspect
really restrict the applicability of evidence theory ? Finally, from the point
of view of computations, there is clearly a need for efficient approximations
and heuristics for evidential calculations. In order to study a question, prob
ably not ali available evidence has to be considered because the inf1uence of
far-fetched information might not be substantial. So focussing strategies to
simplify the reasoning is another important topic for investigation.
Many of the developments in this book are tentative, especially in Parts II
and III. This is due to the fact that the theory is still new and many important
questions remain open. It is hoped that this text contributes to expand the
new ideas on evidence theory and its applications.
Fribourg, December 1994
J. Kohlas, P.A. Monney
Acknowledgments
We would like to thank a number of people who have helped to bring this book
into existence. We are deeply indebted to Miss Ruth Leu, who corrected a lot
of english mistakes and greatly improved the style of the text. We also wish
to express our appreciation to Prof. Prakash Shenoy for reading some chap
ters, and to Louis Cardona, Rolf Haenni, Urs Hănni and Norbert Lehmann
for implementing some of the ideas contained in the book. Also, we would
like to thank the Swiss National Foundation for Scientific Research and the
Swiss Federal Ministry of Science and Education for their financial support
(Grants Nr. 21-30186.90 and 21-32660.91). Finally, one of the authors (P.A.
Monney) expresses his gratitude to the Research Commission of the Univer
sity of Fribourg and the Swiss National Foundation for Scientific Research for
supporting him during his twelve months stay at the U niversity of Kansas. In
that regard, he wants to thank the entire Uni versi ty of Kansas communi ty for
its welcome, especially Prof. Prakash Shenoy for his invitation to be part of
this community.