BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 83, Number 4, July 1977
BOOK REVIEWS
A mathematical theory of
evidence,
by Glenn Shafer, Princeton Univ. Press,
Princeton, New Jersey, 1976, xiii + 297 pp., $17.50 (cloth) and $8.95
(paper).
This is an aptly titled effort to supplement probability theory as developed
for chance/aleatory devices by a parallel, but distinct, epistemically oriented
quantitative theory of evidence for, and evidential support of, our opinions,
judgements of facts, and beliefs. That probability takes its meaning from and
is used to describe such diverse phenomena as propensities for physical
behavior, propositional attitudes of
belief,
logical relations of inductive
support, and experimental outcomes under prescribed conditions of unlinked
repetitions, has long been the source of much of the controversy and vitality
in the development and application of probability theory and its associated
concepts. Ian Hacking in his recent book The
emergence
of
probability
[1]
attempted to trace and explain this intertwining of belief/knowledge and
physical (objective) behavior in terms of a conceptual transformation of the
categories of knowledge and opinion that was mainly completed by the early
18th century. Hacking's historical/philosophical analysis aims to explain what
he holds to be our present dualistic conception of probability as being jointly
epistemic (oriented towards assessment of knowledge/belief) and aleatory
(oriented towards the objective description of the outcomes of 'random'
experiments) with most of the present-day emphasis on the latter. Historically,
however, the epistemic component was initially dominant in conceptions of
probability.
Probability through the Renaissance applied only to opinions/beliefs and
was based upon authoritative testimony in support of these opinions/beliefs.
The 19 year-old Leibniz writing in 1665 wished to formalize the evidential
support for beliefs by a numerical assignment on a scale of [0, 1] of what he
referred to as 'degrees of
proof'.
The object of this exercise was to be a
rationalized jurisprudence. Key to such assignments was an analysis into
equally possible (likely) cases.
The growth of an aleatory notion of probability concerning inductive
relations between physical signs and physical phenomena starts in the
Renaissance. The extent to which the aleatory notion was dependent upon the
epistemic notion (there was also a strong converse dependence) is apparent in
the posthumously published (1713) A rs
conjectandi
of J. Bernoulli. In Part IV
of the Ars [2] we find the first statement and proof of a law of large numbers,
the first firm step on the road to the frequentist/aleatory concepts dominant
today. Significantly though, J. Bernoulli was not a frequentist. For Bernoulli,
frequency of occurrence was only a clue to the enumeration of the equally
possible cases that was the basis of quantitative epistemic probability. Much
667
668
BOOK REVIEWS
of Part IV is given over to discussions of evidence and evidential support and
how to deal with pure and with mixed evidence; pure evidence either
confirmed or discontinued the hypothesis. Bernoulli's analysis of this situation
led him to be willing to assign probabilities P(A) to hypothesis A and P(A
C
)
to false A that violated the usual assumption of P(A) + P(A
C
) = 1, although
Bernoulli, as Leibniz, accepted 0 < P(A) < 1.
While there have been sporadic efforts to deal with the problems of evidence
and evidential support since J. Bernoulli, these efforts have generated little
momentum. The explanation for the slight impact of these attempts to
advance probabilistic reasoning seems to lie in the sociology/psychology of
mathematics and philosophy and lacks any substantial intellectual basis.
Shafer's book is a welcome contribution to the effort to continue the
Leibnizian-Bernoullian line, and redress the intellectual imbalance that has
developed over the last 200 years by reintroducing issues of practical and
intellectual importance for inductive inference. Shafer's approach to the
characterization and quantification of evidential reasoning follows suggestions
advanced by Dempster [3] and should appeal to the mathematical community
as it is a self-contained mathematical theory of evidence, related to Choquet's
study of alternating and monotone capacities, that can be viewed as a
generalization of probability theory. A relation of this theory to parameter
estimation is sketched in Chapter II.
Several of the terms basic to Shafer's discussion of evidence are the
following.
(a) Frame of discernment 0-counterpart to a sample space. List of
possibilities relative to our knowledge with distinctions based on our interests.
Not a logically exhaustive list descriptive of our best resolving power
concerning possibilities. In regard to expanding 0, Shafer (p. 276) notes "it is
always possible to enlarge a frame so as to reduce one's evidence to a
collection of nullities". © is taken to be finite throughout the discussions.
(b) Basic probability function m: 2
e
-»
[0,1],
subject to
m(<t>)
= 0,
2,4 ce
m
(A) ^ *•
m
C^) ^fleets the degree of belief exactly committed to A.
(c) The degree of belief or support function Bel: 2
e
-» [0,1] and satisfies:
Bel(0) - 1; Bel(<|>) = 0; (Vn)(\/A
v
...
%
A
n
C 0) Bel( Û A\
Bel is a set function that is monotone of order infinity. Bel relatés to m through
BelOi) =
ZBCA
«(*)•
(d) Bayesian belief function is one for which m is positive only on singleton
sets and thus probability measure.
(e) Degree of plausibility or upper probability P*(A) = 1 - Be\(A
c
).
In the case of infinite 0, discussed in Shafer [4], the function m is of less
importance and the argument is based on a representation theorem for
BOOK REVIEWS
669
monotone set functions showing that they are a composition of a probability
measure and an intersection homomorphism.
This framework enables Shafer to reasonably formalize a total absence of
relevant evidence bearing on a frame of discernment © through the belief
function
-
UA
. CO iîA^e,
Beiw =
(i iM-e.
This characterization of ignorance is preferable to any that has been attempt-
ed in the usual setup of probability theory. In a probability setup the only
alternatives seem to be to either take no position (e.g., invoke an unknown, as
distinct from random, parameter), or to assign a uniform distribution to the
elements of ©, a device with well-known problems.
Central to the theory Shafer develops is a rule of combination of belief
functions that appeared in Dempster and a special case of which is credited to
J. H. Lambert (1764). From two belief functions
Belj,
Bel
2
on a frame 0, with
associated basic probability functions m
v
m
2
we can form the combined belief
function Bel
12
on 0 with basic probability function m
12
through
2
wM^iB.)
2 ^i(Ai)m
2
(B:)
Bd
l2
(A) = 2 m
n
(B).
BaA
This rule of combination is applicable when the component belief functions
are (p. 57) "based on entirely distinct bodies of evidence" and "the frame of
discernment discerns the relevant interaction of the bodies of evidence". Much
of the text concerns the mathematical implications of this definition of
combination. In terms of it Shafer defines conditional belief functions
Bel(^4|jB) and assessments of evidence w(A).
A conditional belief function Bel(v4|i?) is viewed as the combination of a
belief function Bel(^4) and the degenerate belief function
if
,4
D B,
0 if other.
Equivalent^, if P*(A\B) = 1 - Bel(^l
c
|J5) then
p+(A\n\-
p
*(
A
nB)
P (A\B) -
p
*
(5)
.
Dempster in [3] presents several different definitions of what amounts to
Bel(^4|5), his preferred one being the one Shafer adopts.
The assessment of evidence function w: 2
0
-»
[0,
oo]
is meant to measure
the weight of evidence pointing to any subset of the fraane ©. An elementary
belief function S
B
, called a simple support function, is defined by
670
BOOK REVIEWS
1 if ^ = O,
S
B
(
A
) = ! s if A D B,A # 0,
0 if
^4
other,
where set B is called its focus and 0 < s < 1. The corresponding weight of
evidence function w
s
is then argued to be given by:
I
oo if A = 0,
-log(l - s) if A D B, A ^ 0,
0 if A other.
Curiously, the Bayesian belief functions then turn out to be pathological in
that they can be viewed as arising in the limiting case of infinite contradictory
weights of evidence pointing to the atoms of 0.
While Shafer provides a number of homely examples illustrative of the
definitions and their consequences, and some philosophical/interpretive dis-
cussion concerning the nature and typology of evidence collections, none of it
elaborates how we are to transfer from an evidence collection to the numerical
assessments of support, weight of evidence, or degree of
belief.
Perhaps the
absence of elaboration is purposeful, for as Shafer remarks at the close (p.
285),
"The construction of a frame of discernment is a creative act... The
translation of our vague and amorphous knowledge and experience into
degrees of support within our frame of discernment can be challenge to the
reason and judgement of our astutest minds." The title of this work accurately
reflects its mathematical emphasis and its concern with explicating the formal
structure of numerical measures of evidential support, albeit Shafer also
believes, and I agree, that numerical measures are an idealization. However, a
purely mathematical treatment of this subject may be premature if it precedes
a sound intuitive grasp of this complex and significant problem. I do not hold
with confirmed personalists who might maintain that the relation between a
quantitative measure of belief and the basis for this belief is intuitive,
primitive, and a priori.
It is particularly important that the notion of distinct or separate bodies of
evidence be clarified as it is the basis for the essential operation of combining
belief functions. The situation here is analogous to that of stochastic inde-
pendence in probability theory. Stochastic independence is an essential notion
of unlinkedness or the uninformativeness of one outcome about another.
While it has been explicated mathematically, via the probability of a joint
event formed from independent events being equal to the product of their
individual probabilities, the adequacy of this explication of our intuitive
concept has been questioned [5], and the importance of this issue has been
noted by Kolmogorov [6] when he said "... one of the most important
problems in the philosophy of the natural sciences is ... to make precise the
premises which would make it possible to regard any given real events as
independent." Dempster's approach to the combination of bodies of evidence
better illustrates this parallel between stochastic independence and distinct
bodies of evidence.
BOOK REVIEWS
671
This issue of the nature of separate bodies of evidence and the Dempster
combination rule also impacts on Shafer's selection of a definition of
conditional degree of
belief.
There is evidently a philosophical issue here as to
whether in conditioning on a proposition B we need to think of the knowledge
(possibly hypothetical or even counterfactual) that B is true as being based on
a separate body of evidence from that which went into the belief function we
are conditioning. At any rate this issue is glossed over in the usual probabilis-
tic approach to conditional probability.
The significance and obscurity of the notion of distinct bodies of evidence
is also brought out by the possibility of having two distinct bodies of evidence
which individually give rise to the same belief function. Yet when we combine
the bodies of evidence, they give rise to a different belief function; e.g. if the
original basic probability function m was such that
m(A)
> 0, and m{B) = 0
if B C A, then the new function
rri
may now be positive on subsets of A. The
remarks in §8.2 bear on this issue.
Furthermore the issue skirted by renormalizing the joint basic probability
function m
l2
to account for the seeming assignment of support to the
impossible proposition
(<f>)
suggests a defect in the rule of combination. At first
reflection an ideal combination rule would not attempt to provide support for
<t>
and then have to be adjusted to eliminate this possibility. Admittedly, from
the perspective of P*(A\B) this problem seems less important.
The matter of a decision-making role for belief functions is not addressed.
Some discussion of inference, wherein likelihoods are converted to belief
functions, is provided in Chapter 11. However, this discussion is flawed (e.g.
11.3),
suggesting that the author has not pursued the issue of the utilization of
belief functions as closely as he has that of the mathematical characterization
of belief functions.
Nonetheless, Shafer's A
mathematical theory
of
evidence
is a lucid introduc-
tion to the unfortunately neglected study of epistemic probability and
evidential reasoning. While it is clear that the relations between evidence and
beliefs and the classification of types of evidence are more complex than yet
accounted for by any formal theory, he at least treats these issues more
carefully than is done in the standard probabilistic treatment of inference.
Other recent attempts to deal with evidence and epistemic probability would
include those centered around Carnap's logical probability [7], I. J. Good's
many attempts to mathematicize reasoning
[8],
[9], and Kyburg's epistemolog*
ical probability [10]. Hopefully, Shafer's worthy effort will stimulate mathe-
maticians and philosophers to expand their efforts until this subject is at least
worthy of the attention of lawyers, as Leibniz hoped it would be 300 years
ago!
REFERENCES
1. I. Hacking, The
emergence
of
probability,
Cambridge Univ, Press, London and New York,
1975.
2.
J. Bernoulli, Ars
conjectandi,
Pars quarta
(1713); English transi, by Bing Sung, Tech. Report
12,
February 1966, Dept. of Statistics, Harvard Univ., Cambridge, Mass.
672
BOOK REVIEWS
3.
A. Dempster,
Upper
and
lower probabilities induced
by a
multivalued
mapping,
Ann. Math.
Statist. 38 (1967), 325-339. MR 34 #6817.
4.
G. Shafer, A
theory
of
statistical
evidence,
Foundations of Probability Theory, Statistical
Inference, and Statistical Theories of Science (W. Harper and C. Hooker, Editors), Reidel,
Dordrecht, Holland (to appear).
5. T. Fine,
Theories
of
probability,
Academic Press, New York, 1973, pp. 90-94.
6. A. Kolmogorov,
Foundations
of
the theory
of
probability,
2nd éd., English transi, Chelsea,
New York, 1956, p. 9. MR 18, 155.
7. R. Carnap,
Logical foundations
of
probability,
2nd éd., Univ. of Chicago Press, Chicago,
1962.
MR 32 #2310.
8. I. J. Good,
Probability
and the
weighing
of
evidence,
Griffin, London, Hafner, New York,
1950.
MR
12,
837.
9.
f
Th
e
probabilistic explication
of
information,
evidence,
surprise,
causality,
explanation,
and utility, Foundations of Statistical Inference (V. Godambe and D. Sprott, Editors), Holt,
Rinehart, and Winston, Toronto, Canada,
1971,
pp. 108-141. MR 51 #9280.
10.
H. Kyburg, Jr.,
Logical foundations
of
statistical
inference,
Reidel, Dordrecht, Holland,
1974.
TERRENCE
L. FINE
BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 83, Number 4, July 1977
Order and potential resolvent families of kernels, by Aurel Cornea and
Gabriela Licea, Lecture Notes in Mathematics, no. 494, Springer-Verlag,
Berlin, Heidelberg, New York, 1975, 154 pp., $7.40,
The first title of this book is
Order
and potential If the nonspecialist reader
opens it at any page, just looking for familiar words, he can be sure to see
some mention of order, and has reasonable chances to find
potentials,
but
may wonder whether the use of the latter word has anything to do with
newtonian potential, harmonic functions and similar things. After all, the
word potential has different connotations in different contexts (the military
potential of the United States, the industrial potential of Europe) and the
recurrent mention of a mysterious "domination principle" might lead to
further political misinterpretations. So let me tell first what the subject of the
book really is.
We must come back to the early history of the subject. Between 1945 and
1950,
H. Cartan proved some fundamental results in classical potential
theory, which were rapidly digested, generalized and improved by the French
school of potential theory around M. Brelot, G. Choquet and J. Deny. The
axiomatic trend had always been felt in potential theory (the use of the old
word "principle" to mean "axiom" may be good evidence for it), and anyhow
the years 1950 were those of the big axiomatic boom in mathematics. Hence
it is entirely natural that the interest shifted from potential
theory
to potential
theories
defined by suitable axioms. Among the interesting features of classi-
cal potential theory, the so called
complete maximum principle
came to play a
leading role. It can be easily stated and understood, as follows. Let u and v be
two newtonian potentials of positive measures
A
and
/x,
and let a be a positive
constant. Assume that
(1) a + u > v on the closed support F on the measure fi corresponding to
v.
Then the same inequality takes place everywhere. This is almost obvious.
In the open set F
c
complement of F, the function a + u
—
v is super-
