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.
On the justification of
Dempster's rule of combination
Frans Voorbraak
Department of Philosophy, University of Utrecht
Heidelberglaan 2, 3584 CS Utrecht, The Netherlands
Abstract
In Dempster-Shafer theory it is claimed that the pooling of evidence is
reflected by Dempster's rule of combination, provided certain requirements
are met. The justification of this claim is problematic, since the existing
formulations of the requirements for the use of Dempster's rule are not
completely clear. In this paper, randomly coded messages, Shafer's
canonical examples for Dempster-Shafer theory, are employed to clarify
these requirements and to evaluate Dempster's rule. The range of
applicability of Dempster-Shafer theory will turn out to be rather limited.
Further, it will be argued that the mentioned requirements do not guarantee
the validity of the rule and some possible additional conditions will be
described.
Acknowledgment: I would like to thank Gerard Renardel de Lavalette for drawing my
attention to the problem of justifying Dernpster's rule of combination.
December 1988
Contents
1 Introduction
3
2 Dempster-Shafer theory
4
2.1 Belief functions
2.2 Dempster's rule of combination
4
5
3 Arguments against Dempster's rule
3.1 Lemmer's counterexample
3.2 An amended example
9
9
11
4 Discerning the relevant interaction
13
5 Independent bodies of evidence
16
5.1 Shafer's constructive probability
5.2 DS-independence
5.3 The interaction of evidence
16
17
19
6 Assumptions underlying Dempster's rule
6.1 Conditions for the validity of Dempster's rule
6.2 Bayesian approximations
21
21
23
7 Conclusion
25
Appendix: propositions and proofs
26
References
30
2
1 Introduction
Dempster.'s rule of combination is the most important tool of Dempster-Shafer theory (also
known as;.eyidence. theory or--,.- theory, of belief functions). ;This theory; which originates
from Arthur Dempster [2] and has been developed to its present form, by Glenn Shafer[ 10,11,12,13], is meant to be a theory of evidence and probable reasoning and has recently
received much attention as a promising theory for the handling of uncertain information in
expert systems. Mathematically, Dempster's rule of combination_is=simply a =rule for
computing a belief function from two other belief functions. However, it is claimed that the
rule reflects, the pooling of, evidence. within De mpster-Shaferittheory, provided certain
requirements are met: The justification of this claim presents a problem which is the subject
of this paper.
Unfortunately,. Shafer's; first; formulation-. oft the requirements for the use of
Dempster's, rule was rather .vague-and- despite, several- attempts to clarify the
of,
the requirements, the existing formulations are still not completely clear. This obscurity`
may have contributed to the fact that in the literature these requirements are often more or
less ignored: frequently, the only proviso mentioned is that the bodies of evidence to be
combined have to be independent, which may at best be considered to be a very superficial
account of the requirements. In fact, the useof the term "independent" may .be misleading,
since in the context of Dempster-Shafer theory independence cannot be equated with
stochastical independence.
One of the main objectives of this paperis
to the clarification of the
requirements for the use of Demp,stef s-rule and,irf particular to the clarification of the
Dempster-Shafer theory notion of independence. -Further, it will be argued that Shafer's
requirements are not sufficient to guarantee;-,the validity of the rule and some possible
additional conditions for the -use of the rule will be. described.
Section 2 consists of a brief .introductionto.:Dempster-Shafer-theoryIn section 3 we
the applicability of Dempster's
describe and amend John F . Lemmer'ss argument
rule under a sample space interpretation of ,Dempster-Shafer theory: he gives an example in
which the application of Dempster's rule yields an undesirable result. This (amended)
example is used to give concrete form to the considerations of sections 4 and 5 where the
requirements for the use of Dempster's rule are studied in detail. In section 6 the additional
assumptions underlying the rule are
3
2 Dempster-Shafer theory
In this section we "briefly explain. some notions and terminology °of De-tnpstef-S hafer
theory. For,amore detailed exposition and some background information-see Shafer [10]
or Gordon and S-hortliffe-[4]: w
21 Belief, functions
.
Let, O be a set of rnutually.exclusive.. and ex haust ve,h-ypbtheses-about-some problem
domain. (For simplicity, O will always be assumed to. be finite.). Relevant propositions are
represented as subsets of this set e which is called the frame of discernment, or
simply frame. Suppose: for example :0= {:a;b,c;d}, then. the proposition- `(ave)r-bn-id
is, represented by the set {a;c}.- In Fig:-,.1 the hierarchy of propositions is depicted. Notice
that set theoretical inclusion between sets corresponds -with logical implication between the
represented propositions
-e
(a,b,d)
(a,b,c)
(a, c, d)
E {a,dj
{a}
(b)
(b,c,d)
-
{b,d} -
(c)
{c,d}
(d)
Fig. 1. Hierarchy of propositions for the frame {a;b,"c,d}
Definition 2.1 Let e be a frame. A basic probability "Assi-gnrnent (bpa) on O is a
function m from 2e, the powerset of U, to [0,1] such that
m(o) = 0 and I m(A) = 1
Ac O
4
The .quantity m(A), called A 's basic probability number, corresponds to the measure
of belief that is committed exactly to the proposition (represented" by the set) A° and in
general not to the total belief committed to A, since the total belief includes the measures of
belief committed to subsets of A. This explains the following:
Definition 2.2 The belief function Bel induced by the bpa m'6n-19 is defined by
E
Bel(A) =
m(B)
(AB c e)
BcA
Bel(A) measures the total belief committed to A. Each belief function Bel is induced by a
unique bpa m which can be recovered from Bel as follows:
m(A) _
I
(-1) IA-BIBe1(B)
(A CJA-t, e)
BcA
A proper evaluation of the degree of belief in a proposition A takes -into consideration not
only the measure of belief committed to A, but also that committed to the negation of A.
Information concerning this latter measure is, indirectly coded in the following. function:
,
Definition 2.3 The plausibility function PI induced by the bpa m is defined by
Pl(A) = I m(B) w
Br-A#0
It is easy to see that Bel(A) < Pl(A) and thatPl(A)-=->l Bel(AC), where Ac denotes the set-
theoretical complement of A [Bel(A}.P1(A)} is
belief interval ofA.,
Definition 2.4 Let m be the bpa of Bel..
(1) If m(A) > 0, then A is called a focal element -of Bel.
(2) The union of all focal elements of gel is called the core of Bel.
(3) A belief function is called vacuous if i9 is its only focal element.
r
(4) If all focal elements of Bel are singletons, then Bel is called Bayesian:
2.2 Dempster's rule of combination
The following explanation of Dempster's rule of combination is essentially the one given in
Glenn Shafer's A Mathematical Theory of Evidence.
5
Let m and tn' be the bpa's of the belief functions Bel and Bel' with cores [A,...,Ap} and
(B1,,:.-.,Bq) respectively. The probability masses assigned by the bpa's can, be
as segments of the unit interval, as :shown in Fig. =2 below,,,-
1) .......
m(
I
I
m( A i
......
I
1
m( Ap)
1
1,
1
0
1
m"(B1)
......:.
I
m'(Bj )..... m'(Bq)
I
I
m
I
I
Q
m'
1
Fig. 2. Probability masses assigned by, m and m:
Fig. 3 shown how the 'two intervals can be ortHogona11y' combined
a square
representing` the total probability mass assignedby'' BelDB'el , -the cornbinati`on of Bel and
s.,
Bel', where Bel commits vertical` strips to its focal elements and Bbl' horizontal ones.
M'
1
m' ( B17 )
probability mass
m( Ai )tn' ( B j
committed to
in' ( B j
Ai n Bj
)
m' ( B 1 )
0
m( A 1
)
.......
m(Ai )
......
Fig. 3: Probability masses assigned by the
6
m(AP)
1
m
of Bel and Bel'.
The exact commitment of ,the intersection of the vertical strip of measure m(Ai) and the
horizontal. strip of measure m.'(B); .has-measure m(Ai)=m'(By) and, since it is committed
both to Ai, and BP we may say ,that, it i5 committed AinB :v A given :subset A of O ,may, of
course have more. than one of these rectangles exactly committed to it. Hence to obtain the
total probability ,mass exactly committed to A: by the combination m- and m' (notation
m(@m') .we have to take, the sum; of all m(A.)-m'(Bp.) such that A = AZcn&J.
However, in this way some probability mass may be committed to 0, since there
_ o .But
may be a focal element Ai of m,and ;a focal, element B of m' such that A r-B
then m m' would fail to. be a bpa. Therefore, all rectangles- committed to-the empty- set are
discarded and .the measures of the remaining rectangles:are rescaled by dividing through
the sum of all m
such that A1nB 0, .provided this sum,.does..not.equal 0;
otherwise we say that BeIO+Bel' does not exists _or that Bel and Bel' are not combinable..
Hence we arrive at the following definition:
Definition 2.5, (Dempster's. rule of, combination) =Let,B,e-l and Bel' be belief function
induced by the bpa's m. and, m' respectively. If 1: m (Ai} m:'(B) 1 AinB1 Q }: = 0; , then
Bel and Bel' are called not combinable. Otherwise, BeIO+Bel', the combination of Bel,
and Bel' by Demps-ter's rule, is the belief function; induced by mO+ m',. where
Z m(Ad -m'(BJ -)
AinB -A
J
mO+ m'(A) _
(if A#o;m(m'(o) .= 0.)
Z
Air, s
The factor [Y-{m(A) m'(B) I
and Bel'.
VV)
QS}]
is called renormalizm
constant of B
Shafer exhibits some "sensible and intuitive results of Dempster's rule in the simplest
cases, but he' does -not give an a priori justification of the rule:
Given several belief functions over the same frame of discernement but based on distinct bodies
of evidence, Dempster's rule of combination enables us to compute their orthogonal sum, a
new belief function based on the combined evidence. Though this essay provides no conclusive
a priori argument for Dempster's rule, we will see in the following chapters that the rule does
seem to reflect the pooling of evidence, provided only that the belief functions to be combined
are actually based on entirely distinct bodies of evidence and that the frame of discernment
discerns the relevant interaction of these bodies of evidence. (Shafer [10], p. 57)
What it means for a frame of discernment to discern the relevant interaction was adequately
explained in chapter 8 of Shafer [10]. (See also section 4 of this paper.) But the meaning
of "entirely distinct bodies of evidence" remained obscure.
7
.
It seems obvious that "one should hot"ttse Dempster's rule to combine a body of
evidence X with` itself; or more generally, `with some-body of evidence Y which is implied
by X (i.e. with some body of evidence Y for =which=x-4y holds, where x (Y)denotes the
proposition which which, expresses the -evidence X (-Y)) the reason for this is that if X implies
Y, then the evidence: Y is already taken. info account in` the basic probability assignment
mX. Hence one should have
whereas in general this does not hold:
Example 2.6 Let e be' the frame {A HA } 'where A denotes the proposition "patient P
has the flue". Suppose that X representsthe `observati'on that P has a fever > 39°C, that Y
represents the observation that P has 'a fever > 38.5° C and''that tle`basic probability`
assignments -of,X andY are: =mx(A) `-0:6; mx(e) = 0.4, 'my(A) = 0.4and my((9) =-0.6''
Then>BeIX+OBely(A) =`0.76-,°which does not equal Belx(A),= 0.6.
A natural following case to be considered is that where, although X does not (logically)
imply. Y, X considerably increases the `probability of Y, i.e. P(y [x) » P(y) It' might
seem plausible that also in this case the effect of the body' of evidence Y is (for a large part,
at_least) already taken into account in the°basic`proabability assignment mX. In that case my
should not be allowed to function as an' equal` partner of `inx in- Denlpster'"s rule of
combination. But this would suggest that the rule is only justified if the bodies of evidence
to be combined are independent in the sense that P(xAy) =
Indeed, the requirement that the belief functions to be combined have to be based on
independent evidence is often mentioned in the literature as a proviso on the application of
Dempster's rule. However, authors largely ignore Shafer's opinion that the belief function,
concept of independence, which will be referred to as DS-independence, differs from
the usual probability theory concept. (Cf. Shafer [11].) The inadequacy of the usual
interpretation of independence is illustrated in the following sections by a description of a
proposed "counterexample" to Dempster's rule followed by a detailed study of
;r's
requirements for the use of the rule.
8
3 Arguments against Demp-sterhss -r
Dempster-Shafer theory has been criticized by means of arguments purporting to show that`
probability theory is the unique correct= description of`unceriainty-(see e g. Lindley' [9]).
These arguments which are based on the work of Savage and De Finetti; undoubtedly
have some appeal, but they are not entirely convincing. In spite of them, several authors
have argued for a relaxation of the probability axioms. In particular, the additivity axiom is
felt to be too restrictive in case one has to deal with uncertainty deriving from (partial)
ignorance. (See Kyburg [7] orFishburn [3].)
In this. paper, we will focus on more concrete arguments against Dempster-Shafer
theory, which do notfind fault with the failure of additivity in'the theory, but question-the
validity of Dempster's rule by claiming that in some situations its application yields
undesirable results. (see Kyburg [7], Lemmer-[8] or Zadeh -[ 16].) These arguments are at
least incomplete, since it is not shown that the requirements for the use of Dempster's rule
are satisfied in the particular situations under consideration. Below we 'desribe' (and amend)
one such argument which is relatively explicit about the kind of independence that is
assumed.
3.1 Lemmer's counterexample
In Lemmer [8] John Lemmer proposes the following`- sample space interpretation of
Dempster-Shafer theory: Imagine balls in an urn which have a single "true" label. The set
of these labels (or rather, the set{ of. propositions expressing that a ball has a particular label
from the set of these labels) functions as the frame of discernment e. Henceforth, we will
call these "true" labels frame labels. Belief functions are formed empirically on the basis
of evidence acquired from observation processes which are called sensors. These sensors
attribute to the balls labels which are subsets of D.
The labelling of each sensor is assumed to be accurate in the sense that the frame
label of a particular ball is consistent with the attributed label. (In other words, a sensor s is
accurate if s attributes to each ball with frame label t a label U such that { t}C U. Each
sensor s gives rise to a bpa m by defining for all, UC e m(U) to be the fraction of balls
labelled U by sensors. Then, due to the assumed accurateness of the sensors, Bel(U)
corresponds to the minimum fraction of balls' for with a frame label tE U and Pl(U)
corresponds-to the maximum fraction of balls which 'could have a frame label which is an
element of U.
Lemmer proceeds by giving the following example which shows that the
9
combination by Dempster's rule-,of belief functions which ;derive from accurate labelling
processes does not necessarily yield a belief function which assigns accurate probability
ranges. to gash proposition, even if, the labelling processes arer-independent when
conditioned by the value of the frame label. It is not clear why he chooses for conditional
independence, but he claims that, unconditional :independence, of labelling processes is
undesirable in real applications since it would imply anon-zero probability of inconsistent
labelling by the two sensors,
Example 3.1 Let e = { a,b,c } and assume than balls labelled a are light in weight while
balls labelled either b or c areheavy and that balls labelled either a or b are;red,while balls
labelled c are blue Let swbe a sensor which can classify weight and labels balls having
frame label a with f a } 4,and, balls having either b or c as frame label with. {b,c.}. -and`s' a
sensor which can classify color, and attributes the label {a,b) to balls with either a or b as
frame label. and the label, {c} to balls with frame label :c.=LetP.(t),.represenx the .actual
fraction of balls with frame label,
bpa's m and m' resulting from the, sensors s- and -s-'
respectively are given by.
m({a}) = P(a), m({b,c}) = 1 - P(a), m'({ c}) = P(c) and m'({a,b}) = 1 - P(c).
It is easy to see that although the labelling processes are unconditionally independent
only if P(a) = 0 or P(c) = 0, they are independent when conditioned by the value of the
frame label. Dempster's rule yields
mO+m'(U)
(a) nU#O
1 - rn({a}) m'({c},)
P(a)-P(c)
For M GM' to be-accurate,'P(a)=<_ Pl PI'({a})must be `valid. Hence the following
inequality' must hold:
<
P(a) = Q v P(c) ==G.
Hence if neither P(c) nor P(a) equals zero, then the belief function BelO+Bel' does not
correctly bound the value of P(a).
Lemmer concludes that Dempster 's rule is not applicable in situations which can be
modelled by sample spaces in which sample points have true labels independent of the,
labelling process. However, we do not believe that his example provides strong evidence
for this conclusion, since we doubt the adequacy of the described sample space
interpretation of Dempster-Shafer theory. Consider Lemmer's.own words on the sample
10
space. interpretation of classical -and Bayesian probability theory:The problem in classical probability theory into estimate, before drawing the ball, the
probability of drawing, a_ ball labeled "a,". This probability, under appropriate assumptions
about how the ball is drawn, .is usually taken to .be the fraction, of balls in. the urn which are
The. problem in Bayesian probability theory is to estimate, having drawn the
labeled "a
ball and made an observation about label "a", the probability of the ball also being marked
with label "b". Under Bayes' rule this probability is, taken to be. the fraction of balls :having,
label "b" in that portion of the sample defined by balls having label "a". (Lemmer [8], p.1,19)
Likewise, the problem in Dempster-Shafer theory is to attribute, having drawn the ball and
obtained some information about .it,, pro ability -masse-s -,to! the .different possible
propositions concerning the frame label of°the.ball. In this way'Dempster-Shafer theory
can be seen as a generalization of Bayesian probability 'theory whereas Lemmer's sample
space interpretation is a generalizaton of the interpretation of classical probability theory:
although he describes an experiment in his, model to be the "drawing of a ball and
assigning a probability range to each possible proposition about the true label of the ball",
the result of the experiment is in fact independent of the chosen-ba.11.
Below we adjust Lemmer's sample.dspace..-interpretation in a more or less obvious
manner so that the evidence that a particular ball is drawn is taken into account and we
reformulate Lemmer's argumentin`terms of this adjusted interpretation
3.2 An amended example'
Consider an urn containing: balls.w=ith,multiple labels: Each ball has among its multiple
labels exactly one label,- from some set.'of frame labels, say ;19 =J a,b}.-Sup'pos`e X denotes
the evidence that =a particular`ball-drawnrfrom the urn has label x.
the probability=
that the ball has label a (b) equals P(a I x) (P(b I x)), i.e. the fraction of the x-labelled balls
which are labelled a (b). Let BeIX be a belief function on the frame of discernment { a,b }
has
based on the evidence that the drawn
and- possibly on some evidence
concerning the (fraction of the) frame labels of the x-labelled balls
Definition 3.2 Let X be the evidence that x is the case, BeIX a belief fuction based on X
on a frame e and P a probability measure on { ut.x I WE O} u { un-,x I uE (9). BeIX is called
conditionally accurate w.r.t. P if for all Uc1:
BeIX(U) S P(U I x) S PIX(U).
The notion of conditional accurateness is of course closely related to Lemmers notion of
accurateness: one can think of a BeIX which is conditionally accurate w.r.t. a P measuring
1L 11
actual fractions of balls as deriving 'from an accurate =sensor whose working is restricted to
the x-labelled balls. The following example shows that the conditional accurateness of two
belief functions BeIX and Belt' does not imply-the conditional accurateness=of'Bei. Bely,
even in case X and Y are independent in the sense that P(xny) = P(x);P(y)
Example 33 Consider an urn containing 100 balls which are labelled according to
Table 1 below.
Table 1. Contents of urn of example 3.3. Each string of labels is
followed by the number of ballsthat-liavexexactly `those labels.'
labels:. number of balls
k4
axy
ax
ay
4g
16.
-a
b xy
1610
by..
-20
b
20
bx
10-
Let the bpa's mx and my,be given by: mX({.a)) = 2/7, mX({.b )) = 5/7, my({ a}) =r2/5 and
any({ b)) = 3/5 and let P measure the actual fractions of balls in the urn. Then BeIX and
Belt' are conditionally accurate w.r.t to P since for all Ue { a,b 1:
Belx(U) <_ P(U I x) <_ PIX(U) and Bely(U) S P(UI y) <Ply(U).
Notice that in this case BeIX and Bely are even Bayesian belief functions and that P(xAy) _
0.14;=
We also: have conditional independence: P(x I
a)
I a) and P(x.wl
l ,la) ,=f (xny ), b): Demp;ster's rule yields Belt' Belt'({b }) _
15/19 = 0.79, while P(b I xAy) =. 5/7,F ;0:;71 Hence BeIXO$Bely is not conditionally
accurate, since BelxQ±Bely({ b }) > P(b ,xny).
I
Before drawing any conclusion from this example, we will have a closer look at the
requirements for the use of Dempster's rule.
11,
4 Discerning the -relevant ,interactionDempster's rule should only beused -if the frame of=discernment- Oris fine -enough to
discern all relevant interaction of the evidence. to be combined. The following example
shows what can go wrong if a frame does not satisfy this requirement:
Example 4.1 During his investigation of a burglary Sherlock Holmes has come up with
the following -two pieces of evidence:
El:
from °his inspection of the safe Holmes was 'able to conclude, with a high
degree of certainty, say- 0:7 that the thief was? left-handed,
from the state of the door which gives access -to the room with the safe he
E2
was able to_ conclude, again with a high degree of certainty,
that the
theft was an "inside; job"
Let O _ {LI,LOjRI-,RO },where LI. stands for the proposition -'that: the thief was a lefthanded insider, etc., Then the, bpa'&-. ml and m2, ,based on El and E2 respectively, are given
m1({LI,LO)) = 0.7
m2({LI,RI}) = 0.8
ml((9) = 0.3
M209)"= 0.2
Dempster's rule yields: m10+m2({LI}) = 0.56 m1(@m2({LI,LO)}) = 0.14
= 0.24m1(Dm2(e) = 0 06
However, if one takes as frame of discernment S2 = {LI,L I }, then both El and E2 give
rise to a vacuous belief function and thus Bel1$$Be12 is also vacuous on12, although the
combination of E1 and E2 does support r{-L1}. S2 fails to discern this support because it can
discern El's support for {LI,LO } and E2's support, for {LI,RI } only as supportfor S2 and
S2nQ= D # {LI} = {LI,LO}n{LI,RI}.
by:
Before we can give. an exact description of what it means for a frame to discern all relevant
interaction of two bodies of evidence, welneed-some technical definitions ,from chapters 6
and 8 of Shafer [10]:.
"°
Definition 4.2 Let O and S2 be frames of discernment.
(1) o):219--* 2
(2) w : 219
2
is a-refining if for all AC-& oXA) denotes the same -propositio`as A.
ultimate
if for ,each refining 0: 219 - 2-there exists a
-
refining yf : 2 `y 2 such :that w = -VfO.
If there exists =a refining a 2
244, then S2 is called a refinement of O and- O is called
a coarsening of S2. If there exists an ultimate refining -ct :,,20 --* 2Q; then S2 is called -an
ultimate
of O:
,
-
13
wx
In fact, Shafer defines a refining to be a function ao-: 2 -4 2P such t
(i)
w(A) = U6e A
w( {e})
(ii)'- the sets Of 6)) with BE O constitute: a disjoint partition of S2, ire:
(,1) co({ 8 })
# ,QS for all all, Oe (9
(2) co({ 8})nco({ 8'}) = 0 -if 6 #8'
(3) UOC ©w({ 6))
Uee©O({9})=S2
Although it is clear that if co is a refining according to-definition 4.2, then . to' satisfies
Shafer's requirements for a refining, the two definitions are not equivalent, since. the fact
that w is a refining according- to:Shafer2s definition does not imply that-w A=) expresses the
same proposition as A (For -example,.:.if c9 = >{ 0-',-1'0 ) and° w'is defined` by co(Q) = 0,
co({--9}) = {@}and w(O)=.-(J then according to-Shafer's -definition, 0) is
a refining 219 -+ 219.) We believe that definition 4.2 is to be preferred, since Shafer
implicitly ;assumes: that, if co is, a refining, then w(A) and A denote' the same proposition:
The following useful. properties of a refining co,: 2 - 2V follow trivially from definition
4.2: (i)
co(A) = 0 iff A = QS
(ii)
for all A,BC8 w(AnB) = w(A)nw(B)
Definition 4.3 Let w : 29 - 20 be a refining..,
The inner reduction of wis'_the mapping coti 2Q-+ 2 ;given by
:
ooi(A )a = (6E O1 w(:{ 9.} )9-A }
The outer reduction; of co is the mapping .ciP `20-z 219 given by
O(A) =.{ 9E e 1,°w({ 9})ciA:
} :.
Definition 4.4 A frame O is said to discern the relevant information of the
bodies of -evidence E1 and E2 if cifl(AriB=):=
whenever
0 is the outer
reduction of a:refining w: 2- 20 and A and _B are focal elements -of Bell and Belt
respectively, where Beli is the belief function on S2 which arises from evidence Ei.
In general there exists no method to decide conclusively whether a- particular framediscerns all relevant interaction of the evidence to be combined, butt according to, Shafer
one can often.acquire. confidence, that the frame is fine enough from a geiieral`appraisal of
the evidence. E.g. he claims that one does not need -a detailed analysis to-become= confident
that the details which will:deceive, support-from. Ej after refining the .frame O in example
4.1-are: independent from those which will receive supporr from E2.
In case of the (highly) idealized example 3.3 we are able- to =beimore definite, sincehere the frame (9 = {a,b) has an ultimate refinement. The following proposition shows that
example.
14:'
the existence of an ultimate refinement considerably simplifies the matter:
Proposition 4.5 If co.: 2Q_ z22is an ulit
and ctP(AnB)= (A)naP(B)
based- on evidence E 1 and Bel2'based on
whenever A :and B are, focal elements of
evidence E2 respectively; then O discerns the re1evantinteraction of El and E2.
(Proofs of the propositions can be found in the: appendix:)'
Consider example 3.3.
Write 0 for { axy,ax,ay,a,bxy,bx,by,b } and let the refining co : 219 _ 20 be given by:
oX { a )) _ {axy,ax ay,a}
oX t b)) _ {bxy,bx,by,b}.
a inatterof routine to check that
It is easy to see.-that c)'is
a (Ar B} = ,c -(A)nc '(B)°, for all A and Be2S2 which are focal elements of Beix and
= {a} ={ a}n{a}
Bely:respectively For example,
= cP({ axy,ax } )noP ({-axy,ay } ). We, may conclude that in example 3.31 the requirement that
the frame has to discern, all relevant interaction of the evidence to be combined is met. In
the. following section ,we =investigate the other- (less clear) requirement'-for the 'use of
Dempster-s rule.
1 5
5 Independent, ho.: Lies.., o °s evidence
The, first serious attempt toexplain-what.itrnepans for two-bodies=of evidence, tobe "entirely
distinct" :or independent appeared in Shafer [111 This. explanation, which was further
developed in subsequent papers (Shafer [12, 13]), was placed in the context of what Shafer
called the "constructive interpretation of probability". Hence before describing the
Dempster-Shafer theory notion; of, independence., we, first give an. outline of this
constructive interpretation.
5.1 Shafer's constructive probability
According to the constructive interpretation of probability, probability judgments are made
by comparing the particular situation at hand to abstract canonical examples- in which the
uncertainties= are governed by known chances. These canonical examples matching the
particular, situation may e.g., be games which can be played repeatedly and for which the
long.-tun frequencies of the possible outcomes are known. In this .case one arrives -at
"classical" probability judgments. Shafer claims that -sometimes-, in particular in those
situations where one does not have sufficient information to make a comparison to
examples like these games, other kinds of examples may be judged to be appropriate and
that the use of these other canonical examples may give rise to judgments formulated in
terms of Dempster-Shafer theory. We will describe the most general kind of these
examples: randomly coded messages.
Let O be a frame of discernment. Suppose someone sends us an infallible encoded
message X, where the code is randomly taken from the list c1,c2,...,cn and the chance that
code ci is used is pi. (Both the list and the associated chances are supposed to be known by
us.) Further suppose that, for all i, decoding the encoded message using ci yields the truth
is in Ai where Aic O. (Notation: ci(X) = Ai.) Since in addition to the message both the
list and the chances of the codes are supposed to be known by us, evidence which is
judged to be like the receipt of such a message may be represented by a pair (X,c), where
X denotes an encoded message and c is a probability space ({cl,..:,cn},Pc). The evidence
(represented by) (X,c) may then be expressed in terms of Dempster-Shafer theory by
defining the basic probability assignment m(X c) as follows:
m(X c)(A) = f {Pc(ci) f ci(X) = A}.
Indeed, m(X,c)(A) is, in a certain sense, the chance the message was the truth is in
16
A" and Bel(X,c)(A), with Bel(X c) the belief function induced by m(X c), is the chance that
the truth is in A" is implied by the message. Hence-if the =coded-message is, our -only
evidence, then we will want to call Bel(X c)(A) our degree of -belief that the truth is i'A By
choosing the right probability space c, any belief function can be ,obtained by the above
described method. E.g. the evidence X of example 3.3 ma' y be represented by, BelMc),
where X denotes the message that the ball has .label xx,, c is the probability space
({ci,c2},Pc), Pc(cl) = 2/7, Pc(c2) = 5/7, c1(X) = {a} and c2(X) = {b}.
({dl,-... d;n}, Pd); represent
If (X,c) and (Y;d); with, c = ({cl;:
bodies of evidence, then the. combination: of those .bodies .of-evidence may be represented
by ((X,Y),(c;d)), where (X,Y). denotes the conjunction of, the encoded messages X and Y
and (c,d) is a, probability space.:({(cj,dj)-1:1<i<n, l<Jcy :};;P((c,dj) I (X,Y))) :(Notice that the,
probability measure o -(c,d) depends on X and Y. The reason for- choosing the notation`
P((c,d)) 1-(X J)) will be given below.)
2.,
,
DSndependeance
Although still not completely -clear,; the following-reformulation of the requirement that the
bodies of evidence to,-be, combined with >Deinpster's rue must, be-° entirely distinct is
somewhat more informative than: the original one:' "The uncertainties in` the arguments
being combined,..:, must be independent when the =arguments are viewed abstractly= i.e ,
before the interaction.s.oftheir conclusions are taken..>into°account . ([Shafer [11]; p-. 49.)
The kind of independence informally characterized by the above formulation will be called
D$-independe ce:a,
In Shafer [13] (X,c) and (Y,d) are.trea--ted as-DS-.-independent=if their combination
would be represented by ((X,Y),(c,d)) with (c,d) the product probability space of c and d,
in case X and Y were considered to be messages concerning totally unrelated questions.
(Let C =aUc1,C) a d d>= ({dl,::.;d, };:Pd):be probability spaces, then
1 <i<n, 1><j<_m }.,--. P) is, the product; probability space of c and d if P (cL;dj) _
P:c(cj) Pd(dj).)
Under this interpretation X and Y of example 3.3 would clearly be DS-independent
since the probability that an.. -labelled,ba1JB.1 has label a (b) does not depend on whether
an .y-labelled ball B2. (# B ),has -.,a or b. B:ut. also X and Y of example 2.6 tivould=be°
DS-independent, since the probability of Pi's fever > 39° C being caused by the flue does
not depend on whether the fever, > .3&::5°;C of P2 is being caused by the flue or not.
Therefore; we prefer the following related, but= more :cautious;. interpretation,- of
DS-independence.:
17
Definition $.i , (X,c) and (Y,d) with c: = ({_v"i;. ,cn},Pc) and d = ({'di :..,dm) Pd) are
called DS-independent if for al codes cj-and:dj 1(C,d)(cj I dj) = Pc(cj) and P d)(dj l c) =
(a priori) probability measure on (C;D) = -{(cj,`dj) 1<i<_n,
Pd(dj), where
1Sj<m } which does not take (the interaction: of) X and. Y into account and P -, (c) is an
abbreviation for P(Cd)({(cj,dj)-l1Sj5m}). '
I
Example 5.2 Suppose that two (rather untrustworthy)-: persons X and Y both answer
"yes" tothe question "is. it slippery, outside?" and
truthfully 80% of the
time, but that -the other 20% of the time he: is careless and answers `yes" or="no" without
taking into account, what is actually the case while Y answers truthfully 70% of the time
{slippery, not slippery}. Since a-carelessly given
and carelessly
the time. Let
yes" does not provide support for either "slippery" or not slippery the answers give
rise to the following bpa's: mX({slippery }) = 0.8, mx(e) = 0.2, m y({slippery }) = 0.7
and my((9) = 0.3.
According to def. 5.1 above, the answers of X and Y constitute DS-independent
bodies of evidence if, viewed abstractly, i.e. before considering the (interaction of the)
given answers; the, probability of Y being, careless is (probabilistically) independent of X
either being careless -orstru-thfj1°;(i e. P(Y careless) = P(Y careless I X truthful) =-P(Y
careless I X careless));
[12], p.132.) Notice that this does not imply thatthe'
answers are independent in the ;sense that P(X answers '=yes and Y answers "yes") = P(X
answers ."yes")T(Y answers "Yes,") Further notice that if viewed abstractly P(Y careless, I
X truthful) = P-(,Y,
then this -equality:does:anot necessarily remain true after
considering the given answers. E.g. if X and Y contradict each other, then they cannot
both be truthful
P(Y careless I X truthful) = 1 > 0.2 ='P(Y careless).In definition 5,1 it is presupposed thatone scan speak of tthe'probability P(C d)((cj ,dj)) that -a
is being, used without taking the: received messages X and Y into
pair of codes (cj
account. In fact, we.will assume; that the, probability measurePC of cand Pd of d do not
depend on the messages X and Y. (Notice that this does not necessarily mean that the
probability spaces,c and d are independent of X and Y,, since= the. answers might affect the
it does .meanthat if
list of possible
X and X' give rise to the same
set_gf possible codes, then: the probability of a
being used does- not depend on
whether X or X' is the message.)
Under this assumption,: which is also A(irnplicitly) made in Shafer [13], it makes
sense to speak of the a priori probability of `a pair of 'codes.
section
for some
remarks on the relation between the a priori probability measure P(c d) and P &,d))
18
I (X,Y)).)
However, this assumption puts rather strong constraints on the kind of evidence that can
be compared with a randomly coded message: For, instance; it-becomes far froth obvious
whether the coded messages are adequate canonical examples for the bodies of evidence X
and Y of example 3'.3: let ca (cb). be the':code translating message X into "the drawn ball
has= Label a-(b)
(db) the code translating message. Y into he drawn ball has label a
(b)";, then the probability of the codes do snot seem:to be independent bf X and`Y I 'any
case, X and Y could not be considered to be DS-independent bodies -of evidence, since
P(c, (ca,db) =.P(c,d) (cb,dQ). = 0. (Similar remarks hold for example 2.6.)=''
Hence:. we arrive. at the same conclusion as -Lerrimer: Dempster's rule is- not
applicable in situations. like example 3.3. However, =we do not base this conclusion on the
intuition that Bel-(A) (Pl(A)) should be interpreted, as something like a lower (upper)
probability, but on the fact that X and Y canni ot-be considered to be DS-independent, in
spite of all kinds of (irrelevant) indep_endence._properties.
5.3 The interaction of evidence:.
In order to clarify the relation between P((c,d)) I (X,Y)) and the a priori probability measure
P(c d) we first digress on the meaning of conditional probabilities and on the notion of a
partially specified probability measure.
In general, evidence can be evaluated on different levels of abstraction. Therefore, a
probability measure P on a frame O also has the meaning of a collection of constraints on
probability measures on refinements of O: If co : 219 --* 212 is a refining and P and Pare
probability measures induced by the evidence E on (9 and S2 respectively, then Pis called
a refinement of P and for all AC e P'(cu(A)) = P(A). If S2 is a proper refinement of O,
i.e. S2:;& O, then P only partially specifes a probability measure on S2. E.g. P(c d) partially
specifies a probability measure on any coded messages frame, where f2 is a coded
messages frame if it discerns codes and messages, i.e. if there exists a refining co :
-
A (X,Y) E co({
2(C,D)
2f2 such that
1).
More generally, any consistent set of probabilities may be interpreted as a partially
specified probability measure. (A set S of probabilities is called consistent if there exists
(i) a frame O which is a superset of every set for which a probability is given by S and (ii)
a probability measure P on O such that P agrees with S. In Van der Gaag [14] these
notions of consistency and of a partially specified probability measure are introduced and
employed in the context of Boolean algebras.) In particular, this set of probabilities may
19
contain a conditional probability without:containing the relevant absolute probabilities:
Traditionally, P(A I B) is defined o lyif (i) A and B are (elements of the a -field of)
subsets of the sample :space on which P is defined and (ii) P(B) > 0. (P(A 1 B) may then- be
calculated by means of the well,-known formula P(A I.B) = P(Ar B)/P(B).) However, one
probabilities
can argue that conditional probabilities, are not always obtained'
and that often absolute probabilities are in fact probabilities conditional upon some data for
which the probabilities are unknown.
For instance, the Probability measure. P((c,d)) (x>y)) measures the probabilities of
conditi-pnaL up on, ::the =messages X and Y. Intuitively, =these
pairs of codes (e
probabilities: are, the same the, conditional -probabilities.of (ci d.) given (X,Y) measured
by a ,probability measure which does not take (X,Y) into account, Hence P((c,d)) I (x,Y)j
puts some additional constraints';o ,the partially specified probability measure.P(c d on any
coded messages frame: P((c,d)) I (x,Y))((c,r5dj)) =P c,d)((ci,d) I (X;Y)). In the following it
will be shown that these are indeed additional constraints, although in Dempster's rule it is
assumed that P((c d)) I (x Y)) is uniquely determined by P(c,d). In other words, the particular
way (the interaction of) the evidence is taken into account in Dempster's rule corresponds
,r
to an additional assumption next to DS-independence.
`
P((c,d)).I
20
6 Assumptions underlying Ue
.steer s rule
If one applies Dempster's rulexof combination,, then one (implicitly) assumes more than
that, the bodies of evidence to be combined are DS-independent and that the frame- discerns
their interaction. Ire ,6..1 this will be shown in, the context of the ,coded messages, while in
6.2 it ,is argued more informally that the rule is somewhat biassed: probability masses
assigned to sets of a relatively large cardinality are in some sense given more weight, than
probability masses assigned to sets of a lower cardinality.
6.1 .Conditions -:for. the, validity ,of Dempst.er's :ruleIn this section we give conditions for the validity of Dempster's rule in the context of the
randomly coded messages as canonical examples-for.-Dempster-,p
theory.
Definition 6.1 Bel(X C)O+Bel(y d) is called valid. in the, context of ,.the coded
messages if for all Ac e
m(X,C)$+m(y,d)(A) = m((X,f,(c,d))(A)
;
(=
l
-{.P((c,d))
I
(Xfl)((crdj)) l ci(X)ndj(Y) =
=A))_
Definition 6.2 ((X,Y),(c,d)) is called simple if for all Ai,Aj in the core of Bel(x,c) and
all Bk,Bl in the core of Bel(y d): AinBk =AjnBt.-+: AinBk = 0 v (Ai =°Aj n Bk =B1).
Proposition 6.3 Let ((X,Y),((c,d)) be simple. Bel(X C)O+Bel(y d) is valid in the context
of the coded messages iff
(1) P((c,d))
Q whenever ci(X) c dj(Y) =,Q
i
(2) for all ij,k,l such that ci(X) n dk(Y) 0# cj(X) n d1(Y)
P((c,d)) I
(X,y))((ci,dk))
: P((c,d))
(X,y))((cj,dl)) = Pc(ci).Pd(dk) : Pc(cj).Pd(dl)
I
The proof is= straightforward (see .appendix) ° iid, shows that even if ((X,Y),(c,d)) is not
(1) is necessary for the
simple, then conditions (1) and (2) are
validity of Dempster's rule. And although, in general; .(2) "is not strictly- necessary for, the
validity of Bel(X C)$Bel(Yj) -(see example: 6.4), conditions {1') and (2) do seem to represent
the assumptions underlying Dempster's rule in the context of the coded messages, viz.
giving probability; Q to ''impossible!' pairs
codes and uniformly resealing the a priori,;
probability of the possible pairs.
21
Example 6.4 Let ,,&# {a.,j3,y};-`and. let (X,c,} be given by c-=`({c°1,c2},PC) PC(ci) _
Pc(c2) = 0.5, cl(X) _ {a}, c2(X) = O and (Y,d) by d = ({dl,d2,d3},Pd), Pd(dl) - Pd(d2)
= ,.Pd(d3)
3-, dtil(Y). = { c ,J3}.,. d2(Y) -- { a,y}° and
F1na11y al tP((C d))
= 1/3,
(X;Y))(_(
l
= P((c,d)) "l
Then for all ACE)
(X;fl)((C2,d1-))
= P((c,d)) I (X,Y))((c2,d2))='- P((c,d))
m- (X
P((c,d)) i (X,y)7((cld3)) =
d3(Y) = O.
2
C)e$m(y,d)(A)'`
:
1#1
:
0' P((c,d)) I (X,y))((c1,d3))
d3)) = 1/6.
I
(X,Y))((cl,dl))
m((X,Y),(c,d))(A),
m((X,y,(C,d))(A),-'but P((C,d))
I
1 = PC(c)).Pd(di)- :PC(cjl)`Pd(d3).
The conditions of proposition 6.3 imply the necessary and sufficient conditions for the
validity of Dempster's rule in a concrete example mentioned by Henry E. Kyburg Jr. [6,
p. 254]. However, theconditions are note implied .by DS-independence "Below we
characterize the assumption which is needed in addition to DS-independence.
.
Definition 6.5 Let P be a.(p-artially,,specified) probability measure and let P(E) > 0.
A 1,...,An are equally confirmed by E if
P(AI)P(A I.E) = P(A1
,(>1 <.i, < n)
Proposition 6.6 Let P be a (partially specified) probability measure and let P(E) > 0,
and P(Aj) > 0 (1 <- i <- n). The following are equivalent:
(i)
A1,,..,A arreequ-ally confirmed
(ii)
For some ,- P(Ar I E) _ A,P(Aj) (1 < i <. n).
(iii)
P(E I Ai) = P(E I A1) (1 5 ij <- n).
Proposition 6.7 Let (X,c) and (Y,d) with c = ({ci,...,cn},Pc) and d = ({di,...,drn},
Pd) be DS-independent. Condition (2) of prop. 6..3 holds iff all (ci,dj) with ci(X) n dj(Y)
# 0 are equally confirmed by -(X,Y).
Example 5.2 shows that DS-independence does not imply equal confirmation, since one
can imagine the answers of X and Y to be DS-independent without P(X answers "yes" aridY answers "yes" I X truthful and Y truthful) being equal to P(X answers "yes" and`Y
answers "yes" I X careless
Y truthful) and to P(X answers "yes" and Y answers- "yes"
I.X careless and Y careless). Hence Dempster's rule implicitly employs the often unrealistic"
assumption assumption. that : the., given answers equally,
pairs of codes 'which are-possible
given the answers.:(The observation that in Dempster's rude the given answersare taken
into account in a rather questionable way is also made in [1]:
22,
6.2 Bayesian{ approximation
for the
In Voorbraak [15] Bayesian approximations of belief functions
pupose of achieving possible computational savings in some applications of Dempster
Shafer theory. In this paper they are used to show that Dempster's rule is biassed in some
sense.
Definition 68 Let Bel be', a belief function °-induced by the bpa M.- The
.,
approximation [Bel] of Bel is induced by the bpa [m] defined- by
I m(B)
ACB
if A is a singleton; otherwise, [m](A) = 0.
[m](A) =
Cc e
The factor [I
I Cc
1 will be called the Bayesian constant ofBel.
Notice that in general the Bayesian approximation of Bel differs from the Bayesian belief
function obtained from Bel by distributing uniformly all probability mass assigned by m to
a subset of e over its elements. Probability mass assigned to a subset is not divided among
its elements, but in some sense assigned completely to all its elements.
Example 6.9 Let e = {a,b,c}, m({a}) = 0.4 and m({b,c}) = 0.6. Then [m]({a}) _
0.4/(0.4.1 + 0.6.2) = 0.25 and [m]({b}) = [m]({c}) = 0.6/(0.4.1 + 0.6.2) = 0.375.
Proposition 6.10
(i)
(ii)
Bel = [Bel] iff Bel is Bayesian iff the Bayesian constant is I.
If Bel and Bel' are combinable, then [Bel]O+ [Bel'] = [BelO+Bel']
Corollary 6.11
If BeIO+Bel' is Bayesian, then BelO+Bel' = BelO+ [Bell = [Bel]O+ [Bel'].
Hence Dempster's rule agrees with the assignment of the total probability mass committed
to a subset of the frame to all the elements of this subset. Example 6.12 shows that this
might give counterintuitive results in case some elements profit more from this assignment
than other elements.
Example 6.12 Let e= {a,b,c} and m({a}) = m({b,c}) = m'({a,b}) = m'({ c}) = 0.5.
23
Then we have [m]({a}) = [m]({b}) = [m]({c}) _ [m]-({a}_)_={m']({b}-) =
m +Om'({ a)) = m 0+ m'({ b }) = m +$m'({ c)) = 1/3. This result is counterintuitive, since
intuitively b seems to "share" twice a probability mass of 0.5, while both a=and c = only
v
have to -share once-0:5 with b and are once rassigned, 0.5 individually.
This counterintuitive result is of course related to the fact that Dempster's rule implicitly
assumes that all possible pair of focal elements are equallly confirmed by the combined
evidence, while intuitively in example. 6.10 ° ({ b,c } ; { a;,b }) -is less= confirmed thane
Qa},{Q,b}) and ({b,c},{c}):
24-
.,
7 Conclusion
We may conclude that the usual independence notions are inadequate to express the
requirements,, for the use'.of Dempster's rule.of combination A detailed- study of these
requirements, ;and the notion of DS-independence in particular, has shown that the range of
applicabiliy of. the Dempster-Shafer theory is rather limited, since particular bodies of
evidence have to satisfy very strong constraints in order to be called DS-independent.
Further it has been shown that applying Dempster's rule involves some additional and
often unrealistic assumptions.
Most of these conclusions: rely heavily on the assumption that a`judgments in terms
of Dempster-S:hafer theory arises outof`a'judginent that in a particular situation the
evidence may-be compared to- an abstract canonical example like the receipt of a randomly
coded message. This still leaves room for 'a justification of Dempster-Shafer theory
independent of the coded messages. However, we do not believe in the possibility of such
u justification, since; as is argued in the, -last. section, Dempster's rule seems to be
inherently biassed towards a counterintuitive distribution of probability mass m(A) among
the elements of A.
4
25,'-
Appendix: propositions and proofs
Proposition 4.5 If co : 20 -*,2-9- is an ultimate refining and cc': AnB)
El and Belt based-on'
whenever A and B are-focal=elements -ofuBel1 based
evidence E2xespect vely, then O discerns the relevant interaction of El and E2.
Proof
Suppose 0: 2(9-4 2`0 is a refining, then there is a refining it: 20- 2 such that co =;iiO.
# Z A 0(101)r)B # 0
eE 0°(a) A 6E (B) 0([0-j)(7-)A
0})r # 0. n 0({ 0})r # 0
#O
'W( ({ 0}))nyr(A)) # :Q
i8,E 0(V A))
A BEEaPt(tlr(B))
8E of (Vr(A)r)Vr(B))
V O({ 6) )) n VA) n tV(B) # 0
=> Vr(O({9))nAnB) # 0
O({ e } )nAnB # 0
BE OP (A r-B)
Hence c °(A)nc °(B) C O" (AnB). The inclusion O°(AnB) C O°(A)nO (B) is trivial.
Proposition 6.3 Let ((X,Y),((c,d)) be simple. Bel(X c)O+Bel(y d) is valid in the context
of the coded messages iff
(1) P((c,d)) I (x,Y))((ci,dj)) = 0 whenever ci(X) n dj(Y) = 0
(2) for all ij,k,l such that ci(X) n dk(Y) # Q # cj(X) n dl(Y)
P((c,d)) I
(X,Y))((ci,dk))
: P((c,d)) I (X,Y))((cj,dl)) = Pc(ci)-Pd(dk) : Pc(cj)-Pd(dl)
Proof
LetI(A) _ {(ixj) I ci(X)ndj(Y) =A}.
only if.- Assume Bel(X C)+$Bel(Y is valid in the context of the coded messages.
m(X,c)O+ m(Y d)(o) = 0 implies
(ij) E 1(0) P((c,d)) I (X,Y))(ci,dj) = 0, which implies (1).
To prove condition (2), suppose ci(X) n dk(Y) # 0 # cj(X) n dl(Y).
On the one hand,we have m(X C)O+m(Y d)(ci(X)ndk(Y)) =
K is the renormalizing constant. (Here we need the assumption that
((X,Y),((c,d)) is simple.)
On the other hand, m(X c)$m(Y d)(ci(X)ndk(Y)) = P((c,d)) I (X,Y))(ci,dk)26
Hence P((X 1) (c d))(cl,dk) =
Similarly one obtains P((c,d)) I (X,Y))(cj,dl) _
and condition (2) follows.
if. Asumme conditions (1) and (2).
(1) implies
(i,1)EI(o)P((c',dj)
I
Let A # 0. Then m(X c)O+m(Y,d)(A)
Since-
A#Q
=
(X,Y))(cl,d`)
I
,
which--implies
(X,c)+Om(Yd)(0)
= 0.
=K
(1,1)EI(A) Pc(cj) Pd(dj)
m (X,c) $rn (Y,d)(A) =_ '1;'we-have- K
j( ) PC(cl) Pd(dj) = L
U
We also have 2: (I&I(0) P((c,d)) I (X,Y))(ci,dj) = 1.
By condition (2), we may conclude -P((c,dj I (x,Y))(cl,d) = "--K
Hence m(X c)O+m(Yd(A)
= 2:.,(hi) E I(A) P((c,d)) I
=
Proposition 6.6 Let P be a (partially specified) probability measure and let P(E) > 0
and P(A) > 0 (1 S i <_ n). The following are equivalent:
(i)
A 1,...,An are equally confirmed by E.
(ii)
For some 2 P(Al I E) = ).P(Ai) (1 <_ i <_ n).
P(E I Al) = P(E I AJ-) (11,,5'ij <_ n),-
(iii)
Proof
(ii)
----->
(i): Let for all i P(Ai I E) = IIP(Al). Then for all i and j P(Ai)P(A1 II
P(Al)2,P(Aj) =
=:P(Aj I
(i) =* (iii): Suppose (`) for all i and j
°P(Ai)P(Aj I E) = P(A I E)P(A
P(Ai)P(E-I Al) =P(Al I E)P(E) (Bayer)
=>
P(Aj)P(Al)P(E I Al) = P(Aj)P(AZ I E)P(E)
P(Aj)P(A1)P(E I Aj), = ?(Ai)P(Aj I E)P(E)
P(Aj)P(E.I Ai),P(Aj;I E)P(E)
P(Aj)P(E I Ad = P(AI)P(E I Aj) (Bayes)
P(E I Al) = P(E I Aj).
(iii) =* (ii): Define ):=:P(A I E)/P(Ai) for some `i. `.
P(E I Al) = P(E I Aj),= ,P(E) = AP(E)-.(P(Aj LE)P(E))/P(Aj)
Proposition ,6:7 Let (X,c) and (Y,d) with c = (:
=*
P(Aj I E) =_;,P(Aj).
and d'`= ({-dl,..:,dm},
Pd) be- DS,-independent. Condition (2) of prop. 6-3 holds iff all (ci,dj) with ci(X) r-) dj(Y)
# 0 are equally confirmed by (X Y).
27-
Proof
i f. P((c,d))
(X,Y))((ci,dk))
I
= P(C d)((ci,dk)
: P((c,d)) I (X,Y))((cj,dl))
MY)) : P(c,d)((cj,dl) I (X,Y))
= kP(c,d)((ci,dk)).. 2' P(c,d)((cj,dl)), (by:assumption, of equal; confirmation)
I
=
Pc(cJ)3-
(d) (by assumption elf DS-independence)
d1
(2 # 0, since otherwise for all i j P((X,Y),(c,d))((ci,dj)) = O)
=
0 and define
only if.- Choose i and j such that
;.:= P((c,d))
P(X,Y))((ci,c).)/,(Pc(ci) Pd(dj))...'
I
(2) implies that for all ij with ci(X) n dj(Y) # 0 P((c d)) I (X Y))((ci,dj)) _
Hence, by assumption of DS-independence, for all i,j with ci(X) n dj(Y) # 0:
P(c,d)((ci,dj)
Pd(dj).
'-P(c,d)((ci,dj)) ).
1
11
Proposition 6.10
(i)
(ii)
Bel = [Bel] iff Bel is Bayesian iff the Bayesian constant is: [:
If Bel and Bel' are combinable, then [Bel]$[Bel'] = [Bel$$Belj
Proof
(IAI > 0 implies m(A) = [m](A) =)r= Bel is Bayesian.
Bel is Bayesian =* (ICI > 1 -> m(C) =:O) =* IC#om(C)I.CI =-IC#Om(C) = l:
zC Om(C)ICI =1 =:* (ICI > 1 -* m(C) = 0) and [m].(A) _ Y-AcBr(B) =* Be[jBel].
(i) Bel is [Bel]
=*
(ii) It is clear that if A is not a singleton, then [ln] O+ [m'] (A) = 0 = [m..©m ] (A). Let c
denote the Bayesian constant of Bel (Bel') and let k be the
constant of Bel
and Bel'. Then we have:
{a}=BnC
[rn]O+ [mI({a}) =
Ybe e
IBnCIO
aeC m(C))'(
m'(D,))
Ebe © (c-(EbeE
mV)))
bE © ((Y-b(=-E
28,
a,D')n!(D))
bEF m'(F)))
EC(D m(C)*,m'(D)
Cc©
lE(F
l
c m(E) ni (F) .IE( FI)
aEBmO+ m'(B)
YCc O m0+ m'(C)
= [m( mj(tta)).
Omm'(C).
I CI,
(D))
nF'-C
E,,B (kEB=CnD m(C). '(D))
k'l aEB ,(EB=Cr'D
k'l cce.
rrt:(C)
aEB
I Cce
(ICV kEr,F=C m(E)`m'(F))
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Heidelberglaan 2
3584 CS Utrecht
The Netherlands
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