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Reviews of Books
A Mathematical Theory of Evidence.
GLENN SHAFER.
Princeton University Press, Princeton, NJ, 1976.
The seminal work of Glenn Shafer-which is based on an
earlier work of Arthur Dempster-was published at a time
when the theory of expert systems was in its infancy and
there was little interest within the AI community in issues
relating to probabilistic or evidential reasoning.
Recognition of the relevance of the Dempster-Shafer
theory to the management of uncertainty in expert systems
was slow in coming. Today, it is the center of considerable
attention within AI due in large measure to (a) the emergence
of expert systems as one of the most significant areas of ac-
tivity in knowledge engineering, and (b) the important exten-
sions, applications and implementations of Shafer’s theory
made by John Lowrance at SRI International, Jeff Barnett
at USC/ISI, and Ted Shortliffe and Jean Gordon at Stanford
University.
What are the basic ideas behind the Dempster-Shafer
theory? In what ways is it relevant to expert systems? What
are its potentialities and limitations? My review of Shafer’s
book will be more of an attempt to provide some answers
to these and related questions than a chapter-by-chapter
analysis of its contents.
To understand Shafer’s theory, it is best to start with
a careful reading of Dempster’s original paper (Dempster,
1967) which provides the basis for it.
Stated in somewhat simplified terms, the central prob-
lem considered by Dempster is the following. If y is a func-
tion of 2, say y = f(s), and 5 has a specified probability
distribution, then an elementary result in probability theory
yields the probability distribution of y as a function off and
the probability distribution of x. But what if f is a relation
or, equivalently, a set-valued function, which implies that to
a given value of x corresponds a set of values of y? Suppose
that x and y take values in U and V, respectively, and A is
a specified subset of V-which is referred to as the frame of
discernment in Shafer’s book. Then the question is: What
is the probability that y is in A? If f is a point function,
the answer is a real-valued probability. But f is a set-valued
function, the answer is not unique and all that can be as-
serted is that the probability in question lies between two
bounds which are the lower and upper probabilities, P*(A)
and P*(A), respectively. The lower and upper probabilities
associated with the proposition “y is in A” correspond to
what in Shafer’s theory are called the degree of belief, Bel(A),
and the degree of plausibility, Pi(A). Do P*(A) and P*(A)
capture our intuitive perception of the meaning of belief and
plausibility? This is a semantic question that, objectively,
does not have an obvious answer. In my opinion, however,
the answer is in the negative.
What makes Shafer’s exposition much harder to un-
derstand than Dempster’s is that Dempster, starting with
the concept of a set-valued mapping, derives a number of
properties of P*(A)
and P*(A). Shafer, on the other hand,
uses the belief function, Bel(A), as the point of departure.
This is more elegant mathematically, but has the effect of
obscuring the motivation of the postulated properties of the
belief function, one of which is that of super-additivity, i.e.,
Bel(A
U
B) > Bel(A) + Bel(B) - Bel(A n B). Actually, the
Dempster-Shafer theory is a natural, important and useful
extension of classical probability theory in which the prob-
abilities may be assigned not just to points in the frame
of discernment-which is assumed to be a finite set-but,
more generally, to subsets of V. Viewed in this perspective,
the Dempster-Shafer theory is closely related to the theory
of random sets. Another related viewpoint which is devel-
oped in Zadeh (1979a) involves the notion of information
granularity-a notion which is partly probabilistic and partly
possibilistic in nature.
The basic ideas underlying the Dempster-Shafer theory
are actually quite simple and can readily be understood
through a concrete example. Specifically, assume that
Country X believes that a submarine, S, belonging to
Country Y is hiding in X’s territorial waters. The Minis-
ter of Defense of X summons a group of experts, Ei , , . . , E,,
and asks each one to indicate the possible locations of S.
Assume that the possible locations specified by the experts
El,. . ., E,,
m
4 n, are L1, . . ., L,, respectively, where Li,
i = 1,. . ., m, is a subset of the territorial waters; the remain-
ing experts, E,+I,. . ., E,, assert that there is no submarine
in the territorial waters, or, equivalently, that Lmfl = . . . =
L, = 0, where 0 is the empty set.
Now suppose that the question raised by the Minister of
Defense is: Is S in a specified subset, A, of the territorial
waters? In this regard, there are two cases to consider: (i)
EiCA, which implies that, in the view of E;, it is certain,
i.e., necessarzly true, that S is in A; and (ii) Ei f% A # 0,
i.e., Ei intersects A, in which case it is possible that S is in
A. Note that (i) implies (ii).
Furthermore, suppose that the Minister of Defense ag-
gregates his experts opinion by averaging. Thus, if k out of
n experts vote (i), the average certainty (or necessity) is /c/n;
and if 1 (1 > Ic) experts vote (ii), the average possibility is
l/n. Finally, if the opinion of those experts who believe that
there is no submarine in the territorial waters is disregarded,
the average certainty and possibility will be
k/m
and l/m,
respectively. The disregarding of those experts whose Li is
the empty set corresponds to what is called normalization in
the Dempster-Shafer theory. As we shall see at a later point,
normalization can lead to highly counterintuitive results, for
it suppresses an important aspect of the experts opinion.
THE AI MAGAZINE Fall 1984 81
AI Magazine Volume 5 Number 3 (1984) (© AAAI)
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