arXiv:1704.04000v1 [] 13 Apr 2017

Dempster-Shafer Belief
Function - A New
Interpretation
Mieczyslaw A. Klopotek
Institute of Computer Science, Polish Academy of Sciences
PL 01-237 Warsaw, 21 Ordona St., e-mail: klopotek@ipipan.waw.pl

1

Introduction

Dempster-Shafer theory of evidence has been found by many researchers
very attractive as a way of modeling reasoning behaviour under uncertainty
stemming from ignorance. It provides a framework for representation of
certainty of a logical formula without necessity of expressing commitment to
any of its consequences. E.g. we can express our 100 % belief in fact that
Tweedy’s wife is either Mary or Jane and at the same time express our total
ignorance to the fact who of them is actually his wife (zero belief attached to
the statement ”Mary is Tweedy’s wife” and zero belief in ”Jane is Tweedy’s

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MIECZYSLAW A. KLOPOTEK

wife”).
If a theory is to become of practical importance in expert systems application - as foundation for knowledge representation and reasoning, ar least
the following conditions must be fulﬁlled:
• there must exist an eﬃcient method for reasoning within this framework
• there must exist a clear correspondence between the contents of the
knowledge base and the real world
• there must be a clear correspondence between the reasoning method
and some real world process
• there must exist a clear correspondence between the results of the reasoning process and the results of the real world process corresponding
to the reasoning process.
Only under such circumstances we can say that the expert system is
helpful as it allows us either to predict or to follow retrospectively real world
processes.
Dempster initiated the theory of evidence in his paper [4] and other works,
and Shafer developed this theory in his book [21] and other publications.
Though it became obvious that the DST (Dempster-Shafer Theory) captures
many intuitions behind the human dealing with uncertainty (e.g. as mentioned above), it did not become a foundation for implementation of expert
systems with uncertainty due to claimed high computational complexity [9].

DS BELIEF FUNCTION - A NEW INTERPRETATION

3

In the recent years, however, a number of eﬃcient methods for dealing with
DS reasoning have been developed - see e.g. [23] and citations therein. So the
ﬁrst of the above mentioned conditions is met. Meeting of other conditions
proved to be more complicated.
Smets [26] and also initially Shafer [21] insisted on Bels (measures of
uncertainty in the DST) not being connected to any empirical measure (frequency, probability etc.) considering the domain of DST applications as the
one where ”we are ignorant of the existence of probabilities”, and warn that
the DST is ”not a model for poorly known probabilities” ([26], p.324). The
question may be raised, however, what practically useful can be obtained
from a computer reasoning on the basis of such a DST. It would have to be
demonstrated that humans indeed reason as DST. Then the computer, if fed
with our knowledge, would be capable to predict our conclusions on a given
subject. However, to my knowledge, no experiment conﬁrming that humans
actually use internally DST for reasoning under uncertainty has been carried
out. Under these circumstances the computer reasoning with DST would tell
us what we have to think and not what we think. Hence, from the point of
view of computer implementation the position of Smets and Shafer is not
acceptable.
The other category of DST interpretations, described by Smets as approaches assuming existence of an underlying probability distribution, which
is only approximated by the Bels (called by him PXMY models), is represented by early works of Dempster [4], papers of Kyburg [12], Fagin [7],

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MIECZYSLAW A. KLOPOTEK

[8],, Halpern [10], Skowron [24], Grzymala-Busse [9] and others. Both Smets
[26] and Shafer [22] consider such approaches as inadequate as most of them
give rise to contradictions and counter intuitiveness. As Smets states, ”Far
too often, authors concentrate on the static component (how beliefs are allocated?) and discover many relations between TBM (transferable belief
model of Smets) and ULP (upper lower probability) models, inner and outer
measures (Fagin and Halpern [6]), random sets (Nguyen [16]), probabilities
of provability (Pearl [17]), probabilities of necessity (Ruspini [19]) etc. But
these authors usually do not explain or justify the dynamic component (how
are beliefs updated?), that is, how updating (conditioning) is to be handled
(except in some cases by deﬁning conditioning as a special case of combination). So I (that is Smets) feel that these partial comparisons are incomplete,
especially as all these interpretations lead to diﬀerent updating rules.” ([26],
pp. 324-325). Smets attributes this failure to the very nature of attempts
of assigning a probabilistic interpretation. We disagree with Smets and will
show in this paper that creation of a probabilistic interpretation of the DST
incorporating the Dempster rule of combination is actually possible. However, this new interpretation indicates the need for a drastic change in viewing
the Dempster rule: it does not accommodate evidence, but prejudices. How
this statement is to be understood, will be visible later. Nonetheless our interpretation allows for assignment of an experimentally veriﬁable numerical
meaning to a DS knowledge base, assigns a numerical meaning to the reasoning process (the DS rule of combination) and yields agreement between

DS BELIEF FUNCTION - A NEW INTERPRETATION

5

numerical empirical interpretation of the results of DS reasoning and results
of a real world process. This means that we have an interpretation ﬁtting
formal interpretation of the DS theory to the largest extent ever achieved.

Smets ([26],p.327) subdivided the DST into two categories: a closed world
category (as if excluding the possibility of contradictions in the ”evidence”)
and an open world category of DST (as if allowing for this). Let us assume
that two independent experts elicited their beliefs concerning the event A:
both assigned beliefs of 0.7 to the event A, and beliefs of 0.3 to the event
¬A. The open world DST will lead us to a combined belief in A of 0.5 and
in ¬A of 0.1. The closed world assumption on the other hand will assign a
combined belief in A of 0.7 and in ¬A of 0.3. I ﬁnd it a dismaying property
of a theory if collecting agreeing information from independent expert shall
decline my belief in the opinions of both experts. Hence only closed world
theories are subject of this paper.

We ﬁrst recall the formal deﬁnition of the DS-Theory, then introduce some
notation used throughout the rest of the paper. Subsequently we develop our
interpretation of the joint belief distribution and of evidential updating. We
conclude with a brief comparison of our interpretation with other attempts.

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2

MIECZYSLAW A. KLOPOTEK

Formal Definition of the Dempster-Shafer
Theory of Evidence

Let us make the remark that if an object is described by a set of discrete attributes X1 , X2 , ..., Xn taking values from their respective domains
Ξ1 , Ξ2 , ..., Ξn then we can think of it as being described by a complex attribute X having vector values, that is the domain Ξ of X is equal:
Ξ = {(x1 , x2 , ..., xn )|xi ∈ Ξi , i = 1, ..., n}
.
So unless speciﬁed otherwise let us assume that we are talking of objects
described by a single attribute X taking its values from the domain Ξ. We
say that Ξ, the domain of X is our space of discourse spanned by the attribute
X. We shall also brieﬂy say that X is our space of discourse instead.
For the purpose of this paper we deﬁne the Bel-function as follows (compare also [10], [26], [22]):
Definition 1 The Belief Function in the sense of the DS-Theory is defined
as Bel:2Ξ → [0, 1] with Ξ = Ξ1 × Xi2 × ... × Ξn being the space spanned by
the attribute X = X1 × X2 × . . . × Xn with
∀A;A⊆Ξ

Bel(A) =

X

m(B)

B⊆A

where m(A) is a Mass Function in the sense of the DS-Theory (see Def.2
below).

DS BELIEF FUNCTION - A NEW INTERPRETATION

7

The function m is deﬁned as
Definition 2 The Mass Function in the sense of the DS-Theory is defined
as m:2Ξ → [0, 1] with
m(∅) = 0
X

m(A) = 1

A∈2Ξ

∀A∈2Ξ

m(A) ≥ 0

.
Definition 3 Whenever m(A) > 0, we say that A is the focal point of the
Bel-Function.
Let us also introduce the Pl-Function (Plausibility) as:
Definition 4 The Plausibility Function in the sense of the DS-Theory is
defined as
∀A;A⊆Ξ

P l(A) = 1 − Bel(Ξ − A)

Beside the above deﬁnition a characteristic feature of the DS-Theory is
the so-called DS-rule of combination of independent evidence:
Definition 5 Let BelE1 and BelE2 represent independent information over
the same space of discourse. Then:
BelE1 ,E2 = BelE1 ⊕ BelE2

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MIECZYSLAW A. KLOPOTEK

defined as:
mE1 ,E2 (A) = c ·

X

mE1 (B) · mE2 (C)

B,C;A=B∩C

(c - normalizing constant) represents the Combined Belief-Function of Two
Independent Beliefs

3

Denotation

F. Bacchus in his paper [2] on axiomatization of probability theory and ﬁrst
order logic shows that probability should be considered as a quantiﬁer binding
free variables in ﬁrst order logic expressions just like universal and existential
quantiﬁers do. So if e.g. α(x) is an open expression with a free variable x
then [α(x)]x means the probability of truth of the expression α(x). (The
quantiﬁer []x binds the free variable x and yields a numerical value ranging
from 0 to 1 and meeting all the Kolmogoroﬀ axioms). Within the expression
[α(x)]x the variable x is bound. See [2] on justiﬁcation why other types of
integration of probability theory and ﬁrst order logic or propositional logic
fail. Also for justiﬁcation of rejection of the traditional view of probability
as a function over sets. While sharing Bacchus’ view, we ﬁnd his notation a
bit cumbersome so we change it to be similar to the universal and existential
quantiﬁers throughout this paper. Furthermore, Morgan [14] insisted that
the probabilities be always considered in close connection with the population
they refer to. Bacchus’ expression [α(x)]x we rewrite as:
ProbP (x) α(x) - the probability of α(x)] being true within the population
x

DS BELIEF FUNCTION - A NEW INTERPRETATION

9

P. The P (population) is a unary predicate with P(x)=TRUE indicating
that the object x(∈ Ω, that is element of a universe of objects) belongs to
the population under considerations. If P and P’ are populations such that
∀x P ′(x) → P (x) (that is membership in P’ implies membership in P, or in
other words: P’ is a subpopulation of P), then we distinguish two cases:
P (x)
case 1: ( Prob
P ′ (x)) = 0 (that is probability of membership in P’ with
x
respect to P is equal 0) - then (according to [14] for any expression α(x) in
P ′ (x)
Prob
α(x)) = 1
free variable x the following holds for the population P’: (
x

P (x)
case 2: ( Prob
P ′ (x)) > 0then (according to [14] for any expression α(x)
x

in free variable x the following holds for the population P’:

P ′ (x)

(

Prob
x

α(x)) =

ProbP (x) (α(x) ∧ P ′ (x))
x

ProbP (x) P ′ (x)
x

We also use the following (now traditional) mathematical symbols:
∀x α(x) - always α(x) (universal quantiﬁer)
∃x α(x) - there exists an x such that α(x) (existential quantiﬁer)

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MIECZYSLAW A. KLOPOTEK

α∧β
V

B

- logical AND of expressions

α(B) - logical AND over all instantiations of
the expression α(B) in free variable B

α∨β
W

B

- logical OR of expressions

α(B) - logical OR over all instantiations of
the expression α(B) in free variable B

¬

- logical negation

P ∩Q

- intersection of two sets

P ∪Q

- union of two sets

4

A New Interpretation of Belief Functions

The empirical meaning of a new interpretation of the DS Belief function will
be explained by means of the following example:

Example 1 Let us consider a daily-life example. Buying a bottle of hair
shampoo is not a trivial task from both the side of the consumer and the
manufacturer. If the consumer arrives at the consciousness that the shampoos may fall into one of the four categories: high quality products (excellent
for maintaining cleanness and health of the consumer) (H), moderate quality
products (keeping just all Polish industry standards) (M), suspicious products (violating some industry standards) (S) and products dangerous for
health and life (containing bacteria or fungi or other microbes causing infectious or invasive diseases, containing cancerogenous or poisonous substances

DS BELIEF FUNCTION - A NEW INTERPRETATION

11

etc.) (D), he has a hard time upon leaving his house for shopping. Clearly,
precise chemical, biochemical and medical tests exist which may precisely
place the product into one of those obviously exclusive categories. But the
Citizen1 Coot2 usually neither has a private chemical laboratory nor enough
money to make use of required services. Hence Citizen Coot coins a personal
set of ”quality” tests M 1 mapping the pair (bottle of shampoo, quality) into
the set {TRUE, FALSE} (the letter O - object - stands for bottle of shampoo,
H, M, S, D indicate quality classes: high, moderate, suspicious, dangerous):

1. If the shampoo is heavily advertised on TV then it is of high quality
(M 1 (O, {H}) = T RUE) and otherwise not (M 1 (O, {H}) = F ALSE).
2. If the name of the shampoo was never heard on TV, but the bottle looks
fine (pretty colours, aesthetic shape of the bottle), then the shampoo
must be of moderate quality (M 1 (O, {M}) = T RUE) and otherwise
not (M 1 (O, {M}) = F ALSE).
3. If the packaging is not fine or the date of production is not readable
on the bottle or the product is out of date, but the shampoo smells
acceptably otherwise then it is suspicious (M 1 (O, {S}) = T RUE) and
otherwise not (M 1 (O, {S}) = F ALSE).
1

The term ”Citizen” was a fine socialist time descriptor allowing to avoid the cumber-

some usage of words like ”Mr.”, ”Mrs.” and ”Miss”
2

This family name was coined as abbreviation for ”Citizen Of Our Town”

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MIECZYSLAW A. KLOPOTEK
4. If either the packaging is not fine or the date of production is not
readable on the bottle or the product is out of date, and at the same
time the shampoo smells awfully, then it is dangerous (M 1 (O, {D}) =
T RUE and otherwise not (M 1 (O, {D}) = F ALSE).
Notice that the criteria are partially rational: a not fine looking bottle

may in fact indicate some decaying processing of the shampoo or at least
that the product remains for a longer time on the shelf already. Bad smell
is usually caused by development of some bacteria dangerous for human
health.Notice also that test for high and moderate quality are enthusiastic,
while the other two are more cautious.
Notice that the two latter tests are more difficult to carry out in a shop
than the leading two (the shop assistant would hardly allow to open a bottle
before buying). Also, there may be no time to check whether the shampoo
was actually advertised on TV or not (as the son who carefully watches all
the running advertisements stayed home and does his lessons). Hence some
simplified tests may be quite helpful:
• M 1 (O, {S, D}): If the packaging is not fine or the product is out of
date or the production date is not readable then the product is either
suspicious or dangerous (M 1 (O, {S, D}) = T RUE and otherwise not
(M 1 (O, {D, S}) = F ALSE). .
• M 1 (O, {H, M}): If the packaging looks fine, then the product is either
of high or moderate quality (M 1 (O, {M, H}) = T RUE and otherwise

DS BELIEF FUNCTION - A NEW INTERPRETATION

13

not (M 1 (O, {M, H}) = F ALSE)..
Clearly these tests are far from being precise ones, but for the Citizen Coot
no better tests will be ever available. What is more, they are not exclusive:
if one visits a dubious shop at a later hour, one may buy a product meeting
both M 1 (O, {H}) and M 1 (O, {D}) as defined above !
Let us assume we have two types of shops in our town: good ones
(G) and bad ones (B). (Let M 2 : Ω × 2{ G, B} → {T RUE, F ALSE} indicate for each shampoo in which shop type it was available. Further, let
M 3 : Ω × 2{H,M,S,D}×{G,B} → {T RUE, F ALSE} indicate for each shampoo
both its quality and the type of shop it was available from. Let clearly
M 1 (O, Quality) ∧ M 2 (O, Shop) = M 3 (O, Quality × Shop).
The good shops are those with new furniture, well-clothed shop assistants.
Bad ones are those with always dirty floor or old furniture, or badly clothed
shop assistants. Clearly, again, both shop categories may be considered
(nearly) exclusive as seldom well clothed shop assistants do not care of floors.
Let us assume we have obtained the statistics of shampoo sales in our town
presented in Table 1:

Rows and columns are marked with those singleton tests which were
passed (e.g. in the left upper corner there are 20 shampoo bottles sold in an
undoubtedly bad shop and having exclusively high quality, that is for all those
bottles (O) M 1 (O, {H}) = T RUE, M 1 (O, {M}) = F ALSE, M 1 (O, {S}) =
F ALSE, M 1 (O, {D}) = F ALSE, and M 2 (O, {B}) = T RUE, M 2 (O, {G}) =

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MIECZYSLAW A. KLOPOTEK

Table 1: Sold shampoos statistics
Quality true for Shop type B

G B,G

Total

H

20 100

70

190

M

80 100

110

290

S

50

5

15

70

D

10

1

3

14

H,S

15

10

14

39

M,S

30

20

25

75

H,D

8

2

3

13

M,D

15

7

10

32

228 245

250

723

total

DS BELIEF FUNCTION - A NEW INTERPRETATION

15

F ALSE.) The measurement of M 1 (O, {H}) would yield TRUE for 190+39+13
=242 bottles and FALSE for the remaining 581 bottles, the measurement of
M 1 (O, {D}) would yield TRUE for 14+13+32=59 bottles, and FALSE for
the remaining 664 bottles. The measurement M 1 (O, {S, D}) will turn true in
70+14+ 39+75+ 13+12 =343 cases and FALSE in the remaining 480 cases.✸
In general let us assume that we know that objects of a population can be
described by an intrinsic attribute X taking exclusively one of the n discrete
values from its domain Ξ = {v1 , v2 , ..., vn } . Let us assume furthermore that
to obtain knowledge of the actual value taken by an object we must apply a
measurement method (a system of tests) M
Definition 6 X be a set-valued attribute taking as its values non-empty subsets of a finite domain Ξ. By a measurement method of value of the attribute
X we understand a function:
M : Ω × 2Ξ → {T RUE, F ALSE}
. where Ω is the set of objects, (or population of objects) such that
• ∀ω;ω∈Ω

M(ω, Ξ) = T RUE (X takes at least one of values from Ξ)

• ∀ω;ω∈Ω

M(ω, ∅) = F ALSE

• whenever M(ω, A) = T RUE for ω ∈ Ω, A ⊆ Ξ then for any B such
that A ⊂ B M(ω, B) = T RUE holds,

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MIECZYSLAW A. KLOPOTEK
• whenever M(ω, A) = T RUE for ω ∈ Ω, A ⊆ Ξ and if card(A) > 1
then there exists B, B ⊂ A such that M(ω, B) = T RUE holds.
• for every ω and every A either M(ω, A) = T RUE or M(ω, A) =
F ALSE (but never both).

M(ω, A) tells us whether or not any of the elements of the set A belong to
the actual value of the attribute X for the object ω.
The measuring function M(O,A), if it takes the value TRUE, states for
an object O and a set A of values from the domain of X that the X takes for
this object (at least) one of the values in A.
It makes sense to talk of such measuring function assigning truth values
to sets of values of an attribute if it is possibly cheaper to measure M(O,A)
than to measure M(O,B) whenever B ⊂ A and we are interested in avoiding
measuring M(O,B) whenever possible, that is whenever measuring M(O,A)
suﬃces. For example, measuring pH-value with a pH-meter may turn out to
be more expensive than one with litmus paper, at the advantage of a higher
precision.
The above deﬁnition assumes that this measurement method is supersetand subset-consistent that is: Whenever M(object, A) = T RUE then
∀B;A⊂B

M(object, B) = T RUE

holds, and if card(A) > 1 then
∃B;B⊂A

M(object, B) = T RUE

DS BELIEF FUNCTION - A NEW INTERPRETATION

17

holds. The superset consistency means that if a test for larger set of values
indicates FALSE then it is not necessary to test its subsets as they will not
contribute to our knowledge of the value of X (cost savings). The subset
consistency means that if the M-test for a given value set gives true than in
end eﬀect at least of its singleton subsets would yield TRUE for the respective
M-test. It is clearly the matter of convention: we assume that we can always
provide the answer YES or NO, and whenever we are in doubt we still answer
YES.
Such a convention is not an unusual one: in various legal systems ”anything, that is not forbidden by law, is permitted”; in the default logics if a
default statement cannot be proven wrong, it is assumed correct.
In any case, this means that from the universe of all possible objects, a
concrete measurement method selects a population for which its assumptions
are satisﬁed. E.g. if we have a measurement method for measuring pH-values,
we surely consider an aqueous sodium solution as a member of our universal
population, but never a car as such (because then pH-value has no meaning
at all)..
Furthermore us consider this measurement method a stable one that is
whenever the same object is presented, the results are the same. However,
let us assume that the measurement method is not completely reliable: it
measures only quantities related to the quantity X and not X itself. So it
is conceivable that e.g. M(object, {v1 }) = T RUE and at the same time
M(object, {v2 }) = T RUE though both values of X are deemed to be exclu-

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MIECZYSLAW A. KLOPOTEK

sive. For practical reasons however it may not bother us at all as either the
true value of X may not be accessible at all (e.g. the true event of killing
or not killing a person by the suspect belongs to the past and can never be
recalled as such), may be too expensive to access (e.g. if the most reliable
method of checking whether a match can inﬂame or not it to inﬂame it, but
thereafter it would be useless, so we check only for its color, dryness etc.)
or it may be prohibitive to access it for other reasons, e.g. social (sex may
be treated with extremely high precision as an exclusive attribute taking
values male,female, but we would reluctantly check the primary features before deciding to call someone Mr, Miss or Mrs). Beside this it may prove
too expensive to check all the elementary hypotheses (e.g. in the medical
diagnosis) so that after stating M(object, {v1 }) = T RUE we do not bother
of other alternatives, that is of the degree of imprecision of the relationship
between the measured quantities and the real values of X. We assume that
the approximations of X achieved by the measurement method are in most
cases suﬃcient for our decision making (whatever its nature), so we do not
insist on closer knowledge of X itself.
So though we wish X to take singleton values only, we actually live with
the fact that for our practical purposes X is possibly set-valued.

Let us make at this point some remarks on practical relevance.
Example 2 If we are making statistical tests on equality or non-equality of
two quantities (means, variances, distributions), we can purely logically say

DS BELIEF FUNCTION - A NEW INTERPRETATION

19

that the quantities are either equal or not equal but never both. However,
the available indirect measurement method (by sampling) may lead to a
statement that there is neither evidence to reject equality nor to reject nonequality. So we say that in those cases both equity and inequity holds.
We still enjoy statistical inference because in sufficiently many other cases
statistics provides us with more precise results.✸
Example 3 Similarly if we consider components of a chemical substance,
the measurement methods for absence and presence of a component may
be different from one another depending whether or not we should be more
sensitive to its presence or absence and hence in some cases applying both
may lead to apparently contradicting results. ✸
Let us furthermore assume that with each application of the measurement
procedure some costs are connected, increasing roughly with the decreasing
size of the tested set A so that we are ready to accept results of previous
measurements in the form of pre-labeling of the population. So
Definition 7 A label L of an object ω ∈ Ω is a subset of the domain Ξ of
the attribute X.
A labeling under the measurement method M is a function l : Ω → 2Ξ such
that for any object ω ∈ Ω either l(ω) = ∅ or M(ω, l(ω)) = T RUE.
Each labelled object (under the labeling l) consists of a pair (Oj , Lj ), Oj the jth object, Lj = l(Oj ) - its label.
By a population under the labeling l we understand the predicate P : Ω →

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MIECZYSLAW A. KLOPOTEK

{T RUE, F ALSE} of the form P (ω) = T RUE if f l(ω) 6= ∅ (or alternatively, the set of objects for which this predicate is true)
If for every object of the population the label is equal to Ξ then we talk of an
unlabeled population (under the labeling l), otherwise of a pre-labelled one.
Let us assume that in practice we apply a modiﬁed measurement method
Ml being a function:
Definition 8 Let l be a labeling under the measurement method M. Let
us consider the population under this labeling. The modified measurement
method
Ml : Ω × 2Ξ → {T RUE, F ALSE}
where Ω is the set of objects, is is defined as
Ml (ω, A) = M(ω, A ∩ l(ω))
(Notice that Ml (ω, A) = F ALSE whenever A ∩ l(ω) = ∅.)
For a labeled object (Oj , Lj ) (Oj - proper object, Lj - its label) and a set
A of values from the domain of X, the modiﬁed measurement method tells
us that X takes one of the values in A if and only if it takes in fact a value
from intersection of A and Lj . Expressed diﬀerently, we discard a priori any
attribute not in the label.
Please pay attention also to the fact, that given a population P for which
the measurement method M is deﬁned, the labeling l (according to its deﬁnition) selects a subset of this population, possibly a proper subset, namely

DS BELIEF FUNCTION - A NEW INTERPRETATION

21

the population P’ under this labeling. P ′ (ω) = P (ω) ∧ M(ω, l(ω)). Hence
also Ml is deﬁned possibly for the ”smaller” population P’ than M is.

Example 4 In practice, we frequently have to do with pre-labelled population. The statistics of illnesses based on poly-clinical data are based on a
population pre-labelled by financial status (whether or not they are ready
to visit a physician with less serious disease due to economical background),
educational background (whether or not they estimate properly the seriousness of the disease, whether or not they care of symptoms) etc. Similarly in
chemical analysis knowledge of substrates pre-labels the tests on composition
of the product (not relevant measurements are a priori discarded) etc. ✸
Example 5 To continue Citizen Coot example, we may believe that in good
shops only moderate and high quality products are available, that is we assign to every shampoo ω the label l(ω) = ∅ (we discard it from our register)
if ω denies our belief that there are no suspicious nor dangerous products in
a good shop, and l(ω) = {H, M} if it is moderate or high quality product
in a good shop and l(ω) = Ξ to all the other products. After this rejection
of shampoos not fitting our beliefs we have to do with (a bit smaller) soldshampoos-population from Table 5 :

Please notice the following changes: Suspicious and dangerous products
encountered in good shops were totally dropped from the statistics (their

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MIECZYSLAW A. KLOPOTEK

Table 2: Modiﬁed sold shampoos statistics
Quality true for Shop type B

G B,G

Total

H

20 112

70

202

M

80 127

110

317

S

65

0

0

65

D

13

0

0

13

H,S

15

0

14

29

M,S

30

0

25

55

H,D

8

0

3

11

M,D

15

0

10

25

246 239

232

717

total

DS BELIEF FUNCTION - A NEW INTERPRETATION

23

existence was not revealed to the public). Suspicious and dangerous products
from shops with unclear classification (good/bad shops) were declared to
come from bad shops. Products from good shops which obtained both the
label high quality and dangerous were simply moved into the category high
quality products (the bad smelt was just concealed) etc. This is frequently
the sense in which our beliefs have impact on our attitude towards real facts
and we will see below that the Dempster-Shafer Theory reflects such a view
of beliefs. ✸
Let us now deﬁne the following function:
Definition 9
BelPM (A)

=

ProbP (O)

(¬M(O, Ξ − A))

O

which is the probability that the test M, while being true for A, rejects every
hypothesis of the form X=vi for every vi not in A for the population P. We
shall call this function ”the belief exactly in the the result of measurement”.
Let us deﬁne also the function:
Definition 10
P lPM (A)

=

ProbP (O)

(M(O, A))

O

which is the probability of the test M holding for A for the population P. Let
us refer to this function as the ”Plausibility of taking any value from the set
A”.

24

MIECZYSLAW A. KLOPOTEK
Last not least be deﬁned the function:

Definition 11
mM
P (A)

=

ProbP (O)
O

(

^

M(O, B) ∧

B;B={vi }⊆A

^

¬M(O, B))

B;B={vi }⊆Ξ−A

which is the probability that all the tests for the singleton subsets of A are
true and those outside of A are false for the population P.
Let us illustrate the above concepts with Citizen Coot example:
Example 6 For the belief function for sold-bottles-population and the measurement function M 3 , if we identify probability with relative frequency, we
have the focal points given in the Table 4:✸
It is easily seen that:
THEOREM 1 mM
P is the mass Function in the sense of DS-Theory.
PROOF: We shall recall the deﬁnition and construction of the DNF (Disjunctive Normal Form). If, given an object O of a population P under
the measurement method M, we look at the expression
expr(A) =

^

B;B={vi }⊆A

M(O, B) ∧

^

¬M(O, B)

B;B={vi }⊆Ξ−A

for two diﬀerent sets A1 , A2 ⊆ Ξ then clearly expr(A1 ) ∧ expr(A2 )
is never true - the truth of the one excludes the truth of the other.

DS BELIEF FUNCTION - A NEW INTERPRETATION

25

Table 3: Mass and Belief Function under Measurement Method M 3
Set
{(H,B) }
{(H,G) }

mM
P

3

20/723

BelPM

3

20/723

100/723 100/723

{(H,B),(H,G) }

70/723 190/723

{(M,B) }

80/723

80/723

{(M,G) }

100/723 100/723

{(M,B),(M,G) }

110/723 290/723

{(S,B) }

50/723

50/723

{(S,G) }

5/723

5/723

{(S,B),(S,G) }

15/723

70/723

{(D,B) }

10/723

10/723

{(D,G) }

1/723

1/723

{(D,B),(D,G) }

3/723

14/723

{(H,B),(S,B) }

15/723

85/723

{(H,G),(S,G) }

10/723 115/723

{(H,B),(S,B),(H,G),(S,G) }

14/723 299/723

{(M,B),(S,B) }

30/723 160/723

{(M,G),(S,G) }

20/723 125/723

{(M,B),(S,B),(M,G),(S,G) }

25/723 435/723

{(H,B),(D,B) }

8/723

{(H,G),(D,G) }

2/723 103/723

{(H,B),(D,B),(H,G),(D,G) }

3/723 217/723

38/723

{(M,B),(D,B) }

15/723 105/723

{(M,G),(D,G) }

7/723 108/723

{(M,B),(D,B),(M,G),(D,G) }

10/723 336/723

26

MIECZYSLAW A. KLOPOTEK
They represent mutually exclusive events in the sense of the probability
theory. On the other hand:
_

expr(A) = T RUE

A;A⊆Ξ

hence:
(

ProbP (O)
O

_

(

expr(A))) = (

ProbP (O)

T RUE) = 1

O

A;A⊆Ξ

and due to mutual exclusiveness:

X

(

ProbP (O)

expr(A)) = 1

O

A;A⊆Ξ

which means:

X

mM
P (A) = 1

A;A⊆Ξ

Hence the ﬁrst condition of Def.2 is satisﬁed.Due to the second condition of Def.6 we have

(

ProbP (O)

expr(∅)) = 1 − (

ProbP (O)

O

(M(O, Ξ))) =

O

= 1−(

ProbP (O)

T RUE) = 1 − 1 = 0

O

Hence
mM
P (∅) = 0
.The last condition is satisﬁed due to the very nature of probability:
Probability is never negative. So we can state that mM
P is really a Mass
Function in the sense of the DS-Theory.

Q.e.d.✷

DS BELIEF FUNCTION - A NEW INTERPRETATION

27

THEOREM 2 BelPM is a Belief Function in the sense of DS-Theory corresponding to the mM
P .
PROOF: Let A be a non-empty set. By deﬁnition
M(O, Ξ − A) =

_

M(O, C)

^

¬M(O, C)

C={vi }⊆Ξ−A

hence by de-Morgan-law:

¬M(O, Ξ − A) =

C={vi }⊆Ξ−A

On the other hand, ¬M(O, Ξ − A) implies M(O, A).
But :
M(O, A) =

_



B⊆A



^

M(O, C) ∧

C;C={vi }⊆B



^

C;C={vi }⊆A−B

¬M(O, C)

So .
¬M(O, Ξ − A) = ¬M(O, Ξ − A) ∧ M(O, A) =
^

=

¬M(O, C) ∧ M(O, A) =

C;C={vi }⊆Ξ−A



^

=

C;C={vi }⊆Ξ−A

¬M(O, C) ∧ 

∧

^

C;C={vi }⊆A−B

=

_

B⊆A




^

C;C={vi }⊆B

_

B⊆A




^

M(O, C)∧

C;C={vi }⊆B



¬M(O, C) =

M(O, C) ∧

^

C;C={vi }⊆Ξ−A

¬M(O, C)∧

28

MIECZYSLAW A. KLOPOTEK

∧

^

C;C={vi }⊆A−B



_

=

B⊆A



^



¬M(O, C) =

M(O, C) ∧

C;C={vi }⊆B



^

C;C={vi }⊆Ξ−B

¬M(O, C)

Hence

_

¬M(O, Ξ − A) =

expr(B)

B⊆A

and therefore:
(

ProbP (O)

¬M(O, Ξ − A)) = (

O

ProbP (O) _
O

expr(B))

B⊆A

expr(A) being deﬁned as in the previous proof. As we have shown in
the proof of the previous theorem, expressions under the probabilities
of the right hand side are exclusive events, and therefore:
(

ProbP (O)

¬M(O, Ξ − A)) =

O

ProbP (O)

X

(

X

mM
P (B)

B⊆A

expr(B))

O

that is:
BelPM (A ∈ 2Ξ ) =

B⊆A

As the previous theorem shows that mM
P is a DS Theory Mass Function,
it suﬃces to show the above.

Q.e.d.✷

THEOREM 3 P lPM is a Plausibility Function in the sense of DS-Theory
and it is the Plausibility Function corresponding to the BelPM .

DS BELIEF FUNCTION - A NEW INTERPRETATION

29

PROOF: By deﬁnition:
P lPM (A)

=

ProbM (O)

(O, A)

O

hence
P lPM (A)

= 1−(

ProbP (O)

¬M(O, A))

O

But by deﬁnition:
(

ProbP (O)

¬M(O, A)) = (

O

ProbP (O)
O

¬M(O, Ξ−(Ξ−A))) = BelPM (Ξ−A)

hence
P lPM (A) = 1 − BelPM (Ξ − A)
Q.e.d.✷
Two important remarks must be made concerning this particular interpretation:
• Bel and Pl are both deﬁned, contrary to many traditional approaches,
as THE probabilities and NOT as lower or upper bounds to any probability.
• It is Pl(A) (and not Bel(A) as assumed traditionally) that expresses
the probability of A, and Bel(A) refers to the probability of the complementary set AC .

30

MIECZYSLAW A. KLOPOTEK
Of course, a complementary measurement function is conceivable to re-

vert the latter eﬀect, but the intuition behind such a measurement needs
some elaboration. We shall not discuss this issue in this paper.

Let us also deﬁne the following functions referred to as labelled Belief,
labelled Plausibility and labelled Mass Functions respectively for the labeled
population P:
Definition 12 Let P be a population and l its labeling. Then

BelPMl (A) =

P lPMl (A)

l
mM
P (A)

=

ProbP (ω)
ω

(

^

ProbP (ω)

¬Ml (ω, Ξ − A)

ω

=

ProbP (ω)

Ml (ω, A)

ω

Ml (ω, B) ∧

B;B={vi }⊆A

^

¬Ml (ω, B))

B;B={vi }⊆Ξ−A

Let us illustrate the above concepts with Citizen Coot example:
Example 7 For the belief function for sold-bottles-population P and the
measurement function M 3 , let us assume the following labeling:
l(ω) ={(H,G),(H,B),(M,G),(M,B),(S,B),(D,B)}
for every ω ∈ Ω, which means that we are convinced that only high and
moderate quality products are sold in good shops.For the population P’ under

DS BELIEF FUNCTION - A NEW INTERPRETATION

31

this labeling, if we identify probability with relative frequency, we have the
focal points given in the Table 4:✸

It is easily seen that:
l
THEOREM 4 mM
P is the mass Function in the sense of DS-Theory.

PROOF: To show this is suﬃces to show that the modiﬁed measurement
method Ml possesses the same properties as the measurement method
M.
Let us consider a labeling l and a population P under this labeling.
Let O be an object and L its label under labeling l (L = l(O)). Always
Ml (O, Ξ) = T RUE because by deﬁnition Ml (O, Ξ) = M(O, Ξ ∩ L) =
M(O, L) and by deﬁnition of a labeled population for the object’s O
label L M(O, L) = T RUE.
Second, the superset consistency is satisﬁed, because if A ⊂ B then
if Ml (O, A) = T RUE then also Ml (O, A) = M(O, A ∩ L) = T RUE,
but because A ∩ L ⊆ B ∩ L then also M(O, B ∩ L) = T RUE, but by
deﬁnition M(O, B ∩ L) = Ml (O, B) = T RUE and thus it was shown
that Ml (O, A) = T RUE implies Ml (O, B) = T RUE for any superset
B of the set A.
Finally, also the subset consistency holds, because if M(O, L ∩ A) =
T RUE then there exists a proper subset B of L∩A such that M(O, B) =
T RUE. But in this case B = L∩B so we can formally write: M(O, L∩

32

MIECZYSLAW A. KLOPOTEK

Table 4: Mass and Belief Function under Modiﬁed Measurement Method Ml3
Set
{(H,B) }
{(H,G) }

M3

M3

mP ′l

BelP ′l

20/717

20/717

112/717 112/717

{(H,B),(H,G) }

70/717 202/717

{(M,B) }

80/717

80/717

{(M,G) }

127/717 127/717

{(M,B),(M,G) }

110/717 317/717

{(S,B) }

65/717

65/717

{(D,B) }

13/717

13/717

{(H,B),(S,B) }

15/717 100/717

{(H,B),(S,B),(H,G) }

14/717 184/717

{(M,B),(S,B) }

30/717 175/717

{(M,B),(S,B),(M,G) }

25/717 387/717

{(H,B),(D,B) }

8/717

{(H,B),(D,B),(H,G) }

3/717 114/717

41/717

{(M,B),(D,B) }

15/717 108/717

{(M,B),(D,B),(M,G) }

10/717 228/717

DS BELIEF FUNCTION - A NEW INTERPRETATION

33

B) = T RUE. Hence we see that Ml (O, A) = T RUE implies the existence of a proper subset B of the set A such that Ml (O, B) = T RUE.
M
Hence considering analogies between deﬁnitions of mM
as
P and mP l

well as between the respective Theorems we see immediately that this
Theorem is valid.
Q.e.d.✷
THEOREM 5 BelPMl is a Belief Function in the sense of DS-Theory corl
responding to the mM
P .

PROOF: As Ml is shown to be a DS Theory Mass Function and considering
analogies between deﬁnitions of BelPM and BelP M
l as well as between
the respective Theorems we see immediately that this Theorem is valid.
Q.e.d.✷
THEOREM 6 P lPMl is a Plausibility Function in the sense of DS-Theory
and it is the Plausibility Function corresponding to the BelPMl .
PROOF: As Ml is shown to be a DS Theory Mass Function and considering
analogies between deﬁnitions of P lPM and P lP M
l as well as between the
respective Theorems we see immediately that this Theorem is valid.
Q.e.d.✷
This does not complete the interpretation.
Let us now assume we run a ”(re-)labelling process” on the (pre-labelled
or unlabeled) population P.

34

MIECZYSLAW A. KLOPOTEK

Definition 13 Let M be a measurement method, l be a labeling under this
measurement method, and P be a population under this labeling (Note that
the population may also be unlabeled). The (simple) labelling process on the
population P is defined as a functional LP : 2Ξ × Γ → Γ, where Γ is the
set of all possible labelings under M, such that for the given labeling l and a
given nonempty set of attribute values L (L ⊆ Ξ), it delivers a new labeling
l′ (l′ = LP (L, l)) such that for every object ω ∈ Ω:
1. if Ml (ω, L) = F ALSE then l′ (ω) = ∅
(that is l’ discards a labeled object (ω, l(ω)) if Ml (ω, L) = F ALSE
2. otherwise l′ (ω) = l(ω) ∩ L (that is l’ labels the object with l(ω) ∩ L
otherwise.
Remark: It is immediately obvious, that the population obtained as the
sample fulﬁlls the requirements of the deﬁnition of a labeled population.

The labeling process clearly induces from P another population P’ (a population under the labeling l′ ) being a subset of P (hence perhaps ”smaller”
than P) labelled a bit diﬀerently. Clearly if we retain the primary measurement method M then a new modiﬁed measurement method Ml′ is induced
by the new labeling. The (re-)labelling process may be imagined as the diagnosis process made by a physician. A patient ”labelled” with symptoms
observed by himself (many symptoms remain hidden for the physician, like
the body temperature curve over last few days) is relabeled by the physician
when being ill (labelled with the diseases suspected) or rejected (declared

DS BELIEF FUNCTION - A NEW INTERPRETATION

35

healthy due to symptoms not matching physician’s diagnostic procedure).
Let us deﬁne the following
Definition 14 ”labelling process function” mLP ;L : 2Ξ → [0, 1]: is defined
as:
mLP ;L (L) = 1
∀B;B∈2Ξ ,B6=L mLP ;L (B) = 0
It is immediately obvious that:
THEOREM 7 mLP ;L is a Mass Function in sense of DS-Theory.
Let BelLP,L be the belief and P lLP,L be the Plausibility corresponding
to mLP,L . Now let us pose the question: what is the relationship between
M

BelP ′l′ , BelPMl , and BelLP,L . It is easy to show that
THEOREM 8 Let M be a measurement function, l a labeling, P a population under this labeling. Let L be a subset of Ξ. Let LP be a labeling process
and let l′ = LP (L, l). Let P’ be a population under the labeling l′ . Then
M

BelP ′l′ is a combination via DS Combination rule of BelMl , and BelLP ;L .,
that is:
M

BelP ′l′ = BelPMl ⊕ BelLP ;L
.
PROOF: Let us consider a labeled object (Oj , Lj ) from the population P
(before re-labeling, that is Lj = l(Oj )) which passed the relabeling and

36

MIECZYSLAW A. KLOPOTEK
became (Oj , Lj ∩ L), that is Lj ∩ L = l′ (Oj ).. Let us deﬁne exprB
(before relabeling) and exprA (after labeling) as:
^

exprB ((Oj , Lj ), A) =

Ml (O, B)∧

B;B={vi }⊆A

^

∧

¬Ml (O, B)

B;B={vi }⊆Ξ−A

and
^

exprA ((Oj , Lj ), A) =

Ml′ (O, B)∧

B;B={vi }⊆A

^

∧

¬Ml′ (O, B)

B;B={vi }⊆Ξ−A

Let exprB ((Oj , Lj ), C) = T RUE and exprA ((Oj , Lj ), D) = T RUE for
some C and some D. Obviously then for no other C and no other D the
respective expressions are valid. It holds also that:
^

exprB ((Oj , Lj ), C) =

M(Oj , Lj ∩ B)∧

B;B={vi }⊆C

∧

^

¬M((Oj , Lj ∩ B)

B;B={vi }⊆Ξ−D

and
exprA ((Oj , Lj ), D) =

^

M(Oj , Lj ∩ L ∩ B)∧

B;B={vi }⊆D

∧

^

¬M(Oj , Lj ∩ L ∩ B)

B;B={vi }⊆Ξ−D

In order to get truth on the ﬁrst expression, C must be a subset of Lj ,
and for the second we need D to be a subset of Lj ∩ L. Furthermore,
for a singleton F ⊆ Ξ either M(Oj , Lj ∩ F ) = T RUE, M(Oj , Lj ∩ L ∩
F ) = T RUE, and then it belongs to C, L and D, orM(Oj , Lj ∩ F ) =

DS BELIEF FUNCTION - A NEW INTERPRETATION

37

T RUE, M(Oj , Lj ∩ L ∩ F ) = F ALSE, and then it belongs to C, but
not to L and hence not to D, orM(Oj , Lj ∩ F ) = F ALSE, so due to
superset consistency also M(Oj , Lj ∩ L ∩ F ) = F ALSE, and then it
belongs neither to C nor to D (though membership in L does not need
to be excluded). So we can state that D = C ∩ L,
So the absolute expected frequency of objects for which exprA (D)
holds, is given by:
X

l
samplecardinality · mM
P (C)

C;D=C∩L

that is:

X

LP ;L
l
(L)
samplecardinality · mM
P (C) · m

C;D=C∩L

which can be easily re-expressed as:

X

LP ;L
l
samplecardinality · mM
(G)
P (C) · m

C,G;D=C∩G

So generally:

M

mP ′l′ (D) = c ·

X

LP ;L
l
(G)
mM
P (C) · m

C,G;D=C∩G

with c - normalizing constant.

Q.e.d.✷

Example 8 To continue Citizen Coot example let us recall the function
BelPM from Example 6 which is one of an unlabeled population. Let us

38

MIECZYSLAW A. KLOPOTEK

define the label
L = {(H, G), (H, B), (M, G), (M, B), (S, B), (D, B)}
as in Example 7. Let us define the labeling process function as
mLP ;L (L) = 1
∀B;B∈2Ξ ,B6=L mLP ;L (B) = 0
. Let us consider the function BelPM′l from Example 7. It is easily seen that:
BelPM′l = BelPM ⊕ BelLP ;L
✸
Let us try another experiment, with a more general (re-)labeling process.
Instead of a single set of attribute values let us take a set of sets of attribute values L1 , L2 , ..., Lk (not necessarily disjoint) and assign to each one
a probability mLP,L

1 ,L2 ,...,Lk

(Ai ) of selection.

Definition 15 Let M be a measurement method, l be a labeling under this
measurement method, and P be a population under this labeling (Note that
the population may also be unlabeled). Let us take a set of (not necessarily
disjoint) nonempty sets of attribute values {L1 , L2 , ..., Lk } and let us define
the probability of selection as a function mLP,L
X

A;A⊆Ξ

mLP,L

1 ,L2 ,...,Lk

1 ,L2 ,...,Lk

(A) = 1

: 2Ξ → [0, 1] such that

DS BELIEF FUNCTION - A NEW INTERPRETATION
∀A;A∈{L1 ,L2 ,...,Lk } mLP,L

1 ,L2 ,...,Lk

∀A;A6∈{L1 ,L2 ,...,Lk } mLP,L

1 ,L2 ,...,Lk

39

(A) > 0
(A) = 0

The (general) labelling process on the population P is defined as a (randomΞ

ized) functional LP : 22 × ∆ × Γ → Γ, where Γ is the set of all possible
labelings under M, and ∆ is a set of all possible probability of selection functions, such that for the given labeling l and a given set of (not necessarily
disjoint) nonempty sets of attribute values {L1 , L2 , ..., Lk } and a given probability of selection mLP,L

1 ,L2 ,...,Lk

it delivers a new labeling l” such that for

every object ω ∈ Ω:
1. a label L, element of the set {L1 , L2 , ..., Lk } is sampled randomly according to the probability distribution mLP,L

1 ,L2 ,...,Lk

; This sampling is done

independently for each individual object,
2. if Ml (ω, L) = F ALSE then l”(ω) = ∅
(that is l” discards an object (ω, l(ω)) if Ml (ω, L) = F ALSE
3. otherwise l”(ω) = l(ω) ∩ L (that is l” labels the object with l(ω) ∩ L
otherwise.)
Again we obtain another (”smaller”) population P” under the labeling l”
labelled a bit diﬀerently. Also a new modiﬁed measurement method Ml” is
induced by the ”re-labelled” population. Please notice, that l” is not derived
deterministicly. Another run of the general (re-)labeling process LP may
result in a diﬀerent ﬁnal labeling of the population and hence a diﬀerent
subpopulation under this new labeling.

40

MIECZYSLAW A. KLOPOTEK

Example 9 The (re-)labelling process may be imagined as the diagnosis
process made by a team of physicians in a poly-clinic. A patient ”labelled”
with symptoms observed by himself is (a bit randomly) directed by the ward
administration to one of the available internists each of them having a bit
different educational background and/or a different experience in his profession, hence taking into consideration a bit different set of working hypotheses.
The patient is relabeled by the given physician being ill (labelled with the
diseases suspected) or rejected (declared healthy) according to the knowledge of this particular physician. The final ward statistics of illnesses does
not take into account the fact that a physician may have had no knowledge
of a particular disease unit and hence qualified the patient either healthy or
ill of another, related disease unit. And it reflects the combined processes:
of random allocations of patients to physicians and of belief worlds of the
physicians rather then what the patients were actually suffering from. (We
are actually satisfied with the fact that both views of ward statistics usually
converge).✸

Clearly:

THEOREM 9 mLP,L
Let BelLP ;L

1 ,...,Lk

sponding to mLP,L

1 ,...,Lk

is a Mass Function in sense of DS-Theory.

be the belief and P lLP,L

1 ,...,Lk

1 ,...,Lk

be the Plausibility corre-

. Now let us pose the question: what is the relation-

ship between BelPM”l” , BelPMl , and BelLP,L

1 ,...,Lk

. It is easy to show that

DS BELIEF FUNCTION - A NEW INTERPRETATION

41

THEOREM 10 Let M be a measurement function, l a labeling, P a population under this labeling. Let LP be a generalized labeling process and
let l” be the result of application of the LP for the set of labels from the
set {L1 , L2 , ..., Lk } sampled randomly according to the probability distribution mLP,L

1 ,L2 ,...,Lk

;. Let P” be a population under the labeling l”. Then The

expected value over the set of all possible resultant labelings l” (and hence
populations P”) (or, more precisely, value vector) of BelPM”l” is a combination
via DS Combination rule of BelPMl , and BelLP,L
M′

1 ,...,Lk

E(BelP ”l ) = BelPMl ⊕ BelLP,L

., that is:

1 ,...,Lk

.
PROOF: By the same reasoning as in the proof of Theorem 8 we come to
the conclusion that for the given label Li and the labeling l” (instead of
l′ the absolute expected frequency of objects for which exprA (D) holds,
is given by:
X

LP ;L
l
samplecardinality · mM
P (C) · m

1 ,...,Lk

(Li )

C;D=C∩Li

as the process of sampling the population runs independently of the
sampling the set of labels of the labeling process.
But exprA (D) may hold for any Li such that C ⊆ Li , hence in all the
exprA (D) holds for as many objects as:

X

X

i;i=1,...,k C;D=C∩Li

LP ;L
l
samplecardinality · mM
P (C) · m

1 ,...,Lk

(Li )

42

MIECZYSLAW A. KLOPOTEK
which can be easily re-expressed as:

X

LP ;L
l
samplecardinality · mM
P (C) · m

1 ,...,Lk

(G)

C,G;D=C∩G

So generally:

E(mPM”l” (D)) = c ·

X

LP ;L
l
mM
P (C) · m

1 ,...,Lk

(G)

C;D=C∩G

with c - normalizing constant.Hence the claimed relationship really
holds.

Q.e.d.✷

Example 10 The generalized labeling process and its consequences may be
realized in our Citizen Coot example by randomly assigning the sold bottles
for evaluation to two ”experts”, one of them - considering about 30 % of
the bottles - is running the full M test procedure, and the other - having to
consider the remaining 70 % of checked bottles - makes it easier for himself
by making use of his belief in the labeling l of Example 7. ✸

4.1

Summary of the New Interpretation

The following results have been established in this Section:
• concepts of measurement and modiﬁed measurement methods have
been introduced
• a concept of labelled population has been developed

DS BELIEF FUNCTION - A NEW INTERPRETATION

43

• it has been shown that a labelled population with the modiﬁed measurement method can be considered in terms of a Joint Belief Distribution
in the sense of DS-Theory,
• the process of ”relabeling” of a labelled population has been deﬁned
and shown to be describable as a Belief Distribution.
• it has been shown that the relationship between the Belief Distributions of the resulting relabeled population, the basic population and
the relabeling process can be expressed in terms of the Dempster-Ruleof-Independent-Evidence-Combination.
This last result can be considered as of particular practical importance.
The interpretation schemata of DS Theory made by other authors suﬀered
from one basic shortcoming: if we interpreted population data as well as
evidence in terms of their DS schemes, and then combine the evidence with
population data (understood as a Dempster type of conditioning) then the
resulting belief function cannot be interpreted in terms of the population
data scheme, with subsequent updating of evidence making thinks worse
till even the weakest relation between the belief function and the (selected
sub)population is lost.
In this paper we achieve a break-through: data have the same interpretation scheme after any number of evidential updating and hence the belief
function can be veriﬁed against the data at any moment of DS evidential
reasoning.

44

MIECZYSLAW A. KLOPOTEK
The above deﬁnition and properties of the generalized labeling process

should be considered from a philosophical point of view. If we take one by
one the objects of our domain, possibly labelled previously by an expert in
the past, and assign a label independently of the actual value of the attribute
of the object, then we cannot claim in any way that such a process may be
attributed to the opinion of the expert. Opinions of two experts may be
independent of one another, but they cannot be independent of the subject
under consideration. This is the point of view with which most people would
agree, and should the opinions of the experts not depend on the subject, then
at least one of them may be considered as not expert.
This is exactly what we want to point at with our interpretation: the
precise pinpointing at what kind of independence is assumed within the
Dempster-Shafer theory is essential for its usability. Under our interpretation, the independence relies in trying to select a label for ﬁtting to an
object independently of whatever properties this object has (including its
previous labeling). The distribution of labels for ﬁtting is exactly identical
from object to object. The point, where the dependence of object’s labeling
on its properties comes to appearance, is when the measurement method
states that the label does not ﬁt. Then the object is discarded. From philosophical point of view it means exactly that we try to impose our philosophy
of life onto the facts: cumbersome facts are neglected and ignored. We suspect that this is exactly the justiﬁcation of the name ”belief function”. It
expresses not what we see but what we would like to see.

DS BELIEF FUNCTION - A NEW INTERPRETATION

45

Our suspicion is strongly supported by the quite recent statement of Smets
that ”authors (of multiple interpretations in terms of upper lower probability
models, inner and outer measures, random sets, probabilities of provability,
probabilities of necessity etc.) usually do not explain or justify the dynamic
component, that is, how updating (conditioning) is to be handled (except in
some cases by deﬁning conditioning as a special case of combination. So I
(that is Smets) feel that these partial comparisons are incomplete, especially
as all these interpretations lead to diﬀerent updating rules. ” Our interpretation explains both the static and dynamic component of the DST, and does
not lead to any other but to the Dempster Rule of Combination, hence may
be acceptable from the rigorous point of view of Smets. As in the light of
Smets’ paper [26] we have presented the only correct probabilistic interpretation of the DS theory so far, we feel to be authorized to claim that our
philosophical assessment of the DST is the correct one.

We have seen from the proofs of the theorems of this paper, that our
interpretation may be called a true one. The paper of Smets [26] permits us
to claim that we have found the true interpretation.

5

Belief from Data

As the DS-belief function introduced in this paper is deﬁned in terms of frequentist measures, there exists a direct possibility of calculating the belief

46

MIECZYSLAW A. KLOPOTEK

function from data.

It has to be assumed that we have a data set for which the measurements
of type Ml have been carried out for each singleton subset of the space of
discourse Ξ. The results of these measurements may be available for example
as a set-valued attribute associated with each object in such a way that the
values actually appearing are those for which the singleton set tests were positive (i.e. TRUE). In this case if for an object the attribute X has the value
X = A with A ⊆ Ξ then this object increases the count for the DS-Mass
Function m(A) (and for no other m).

Whenever any statistical quantity is estimated from data, there exists
some risk (uncertainty) about unseen examples. If we assume some signiﬁcance levels, we can complete the estimation by taking the lower bounds as
actual estimates of m’s and shifting the remaining burden (summing up to 1)
onto the m(Ξ) just taking for granted that doubtful cases may be considered
as matching all the measurements.

6

Discussion

In the past, various interpretations have been sought for the Dempster-Shafer
Bel-Functions. Two main steams of research were distinguished by Smets
[26]: probability related approaches and probability discarding approaches

DS BELIEF FUNCTION - A NEW INTERPRETATION

47

(the former disguised, the latter welcome by Smets). Let us make some
comparisons with our interpretation and its underlying philosophy.

6.1

Shafer and Smets

Shafer [22] and Smets [26] have made some strong statements in defense
of the Dempster-Shafer theory against sharp criticism of this theory by its
opponents as well as unfortunate users of the DST who wanted to attach
it to the dirty reality (that is objectively given databases). Smets [26] and
also initially Shafer [21] insisted on Bels not being connected to any empirical measure (frequency, probability etc.) considering the domain of DST
applications as the one where ”we are ignorant of the existence of probabilities”, and not one with ”poorly known probabilities” ([26], p.324). The basic
property of probability, which should be dropped in the DST axiomatization,
should be the additivity of belief measures. Surely, it is easily possible to
imagine situations where in the real life additivity is not granted: imagine
we have had a cage with 3 pigs, we put into it 3 hungry lions two hours ago,
how many animals are there now ? (3 + 3 < 6). Or ten years ago we left one
young man and one young woman on an island in the middle of the atlantic
ocean with food and weapons suﬃcing for 20 years. How many human beings
are there now ? (1 + 1 > 2).
The trouble is, however, that the objects stored in databases of a computer
behave usually (under normal operation) in an additive manner. Hence the
DST is simply disqualiﬁed for any reasoning within human collected data on

48

MIECZYSLAW A. KLOPOTEK

real world, if we accept the philosophy of Smets and Shafer.
The question may be raised at this point, what else practically useful can
be obtained from a computer reasoning on the basis of such a DST. If the
DST models, as Smets and Shafer claim, human behaviour during evidential
reasoning, then it would have to be demonstrated that humans indeed reason
as DST. We take e.g. 1000 people who never heard of Dempster-Shafer
theory, brieﬂy explain the static component, provide them with two opinions
of independent experts and expect of them to answers what are their ﬁnal
beliefs. Should their answers correspond to results of the DST (at least
converge toward them), then the computer, if fed with our knowledge, would
be capable to predict our conclusions on a given subject. However, to my
knowledge, no experiment like this has ever been carried out. Under these
circumstances the computer reasoning with DST would tell us what we have
to think and not what we think. But I don’t suspect that anybody would be
happy about a computer like this.
Hence, from the point of view of computer implementation the philosophy
of Smets and Shafer is not acceptable.Compare also Discussion in [10] on the
subject.
Both of them felt a bit uneasy about a total loss of reference to any scientiﬁc experiment checking practical applicability of the DST and suggested
some probabilistic background for decision making (e.g. the pigeonistic probabilities of Smets), but I am afraid that by these interpretations they fall
precisely into the same pitfalls they claimed to avoid by their highly abstract

DS BELIEF FUNCTION - A NEW INTERPRETATION

49

philosophy.
As statistical properties of Shafer’s [21] notion of evidence are concerned,
suﬃcient criticism has been expressed by Halpern and Fagin ([10] in sections
4-5). Essentially it is pointed there at the fact that ”the belief that represents
the joint observation is equal to the combination is in general not equal to the
combination of the belief functions representing the individual (independent)
observations” (p.297). The other point raised there that though it is possible
to capture properly in belief functions evidence in terms of probability of
observations update functions (section 4 of [10]), it is not possible to do the
same if we would like to capture evidence in terms of beliefs of observations
update functions (section 5 of [10]).
As Smets probabilistic interpretations are concerned, let us ”continue” the
killer example of [26] on pages 330-331. ”There are three potential killers, A,
B, C. Each can use a gun or a knife. I shall select one of them, but you will
not know how I select the killer. The killer selects his weapon by a random
process with p(gun)=0.2 and p(knife)=0.8. Each of A, B, C has his own personal random device, the random devices are unrelated. ...... Suppose you are
a Bayesian and you must express your ”belief” that the killer will use a gun.
The BF (belief function) solution gives Bel(gun) = 0.2 × 0.2 × 0.2 = 0.008.
..... Would you defend 0.2 ? But this applies only if I select a killer with a
random device ...... But I never said I would use a random device; I might be
a very hostile player and cheat whenever I can. ... . So you could interpret
bel(x) as the probability that you are sure to win whatever Mother Nature

50

MIECZYSLAW A. KLOPOTEK

(however hostile) will do.”
Yes, I will try to continue the hostile Mother Nature game here. For completeness I understand that Bel(knif e) = 0.83 = 0.512 and Bel({gun, knif e}) =
1. But suppose there is another I, the chief of gangster science ﬁction physicians, making decisions independly of the chief I of the killers. The chief I
of physicians knows of the planned murder and has three physicians X,Y,Z.
Each can either rescue a killed man or let him die. I shall select one of
them, but you will not know how I select the physician. The physician, in
case of killing with a gun, selects his attritude by a random process with
p(rescue|gun) = 0.2 and p(let die|gun) = 0.8 and he lets the person die
otherwise. Each of X, Y, Z has his own personal random device, the random
devices are unrelated. ...... Suppose you are a Bayesian and you must express your ”belief” that the physician will rescue if the killer will use a gun.
The BF (belief function) solution gives Bel1 (rescue|gun) = 0.23 = 0.008.
Bel1 (let die|gun) = 0.83 = 0.512, Bel1 ({recue, let die}|gun) = 1. Also
Bel2 (let die|knif e) = 1. As the scenarios for Bel1 and Bel2 are independent, let us combine them by the Dempster rule: Bel12 = Bel1 ⊕ Bel2 . We
make use of the Smets’ claim that ”the de re and de dicto interpretations
lead to the same results” ([26], p. 333), that is Bel(A|B) = Bel(¬B ∨ A).
Hence
m12 ({(gun, let die), (knif e, let die), (knif e, rescue)}) = 0.480
m12 ({(gun, rescue), (knif e, rescue)}) = 0.008

DS BELIEF FUNCTION - A NEW INTERPRETATION

51

m12 ({(knif e, rescue), (gun, let die)}) = 0.512
Now let us combine Bel12 with the original Bel. We obtain:

m ⊕ m12 ((gun, let die) = 0.008 · 0.480 + 0.008 · 0.512 = 0.008 · 0.992
But these two unfriendly chiefs of gangster organizations can be extremely
unfriendly and in fact your chance of winning a bet may be as bad as
0.008 · 0.512 for the event (gun, let die). Hence the ”model” proposed by
Smets for understanding beliefs functions as ”unfriendly Mother Nature” is
simply wrong. If the Reader ﬁnds the combination of Bel2 with the other
Bels a little tricky, then for justiﬁcation He should refer to the paper of Smets
and have a closer look at all the other examples.

Now returning to the philosophy of ”subjectivity” of Bel measures: Even
if a human being may possess his private view on a subject, it is only after
we formalize the feeling of subjectiveness and hence ground it in the data
that we can rely on any computer’s ”opinion”. We hope we have found
one such formalization in this paper. The notion of labeling developed here
substitutes one aspect of subjective human behaviour - if one has found one
plausible explanation, one is too lazy to look for another one. So the process
of labeling may express our personal attitudes, prejudices, sympathies etc.
The interpretation drops deliberately the strive for maximal objectiveness
aimed at by traditional statistical analysis. Hence we think this may be a
promising path for further research going beyond the DS-Theory formalism.

52

MIECZYSLAW A. KLOPOTEK
Smets [26] views the probability theory as a formal mathematical appa-

ratus and hence puts it on the same footing as his view of the DST. However,
in our opinion, he ignores totally one important thing: The abstract concept
of probability has its real world counterpart of relative frequency which tends
to behave approximately like the theoretical probability in suﬃciently many
experimental settings as to make the abstract concept of probability useful
for practical life. And a man-in-the-street will expect of the DST to possess
also such a counterpart or otherwise the DST will be considered as another
version of the theory of counting devils on a pin-head.

Let us also have a look at interpretations disguised by Shafer and Smets
(i.e. all the mentioned below):

6.2

DST and Random Sets

The canonic random set interpretation [16] is one with a statistical process
over set instantiations. The rule of combination assumes then that two such
statistically independent processes are run and we are interested in their
intersections. This approach is not sound as empty intersection is excluded
and this will render any two processes statistically dependent. We overcome
this diﬃculty assuming in a straight forward manner that we are ”walking”
from population to population applying the Rule of Combination. Classical
DS theory in fact assumes such a walk implicitly or it drops in fact the

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53

assumption that Bel() of the empty set is equal 0. In this sense the random
set approaches may be considered as sound as ours.
However, in many cases the applications of the model are insane. For
example, to imitate the logical inference it is frequently assumed that we
have a Bel-function describing the actual observed value of a predicate P(x),
and a Bel-Function describing the implication ”If P(x) then Q(x)” [13]. It is
assumed further that the evidence on the validity of both Bel’s has been collected independently and one applies the DS-rule of combination to calculate
the Bel of the predicate Q(x). One has then to assume that there is a focal m
of the following expression: m({(P (x), Q(x)), (¬P (x), Q(x)), (¬P (x), ¬Q(x))})
which actually means that with non-zero probability at the same time P (x)
and ¬P (x) hold for the same object as we will see in the following example:
Let Bel1 represent our belief in the implication, with focal points:
m1 (P (x) → Q(x)) = 0.5, m1 (¬(P (x) → Q(x))) = 0.5,
Let further the independent opinion Bel2 on P(x) be available in the form of
focal points:
m2 (P (x)) = 0.5, m2 (¬P (x)) = 0.5
Let Bel12 = Bel1 ⊕ Bel2 represent the combined opinions of both experts.
The focal points of Bel12 are:
m12 ({(P (x), Q(x))}) = 0.33, m12 ({(P (x), ¬Q(x))}) = 0.33,
m12 ({(¬P (x), Q(x)), (¬P (x), ¬Q(x))}) = 0.33

54

MIECZYSLAW A. KLOPOTEK
m12 ({(P (x), Q(x))}) = 0.33 makes us believe that there exist objects for

which both P(x) and Q(x) holds. However, a sober (statistical) look at expert
opinions suggests that all situations for which the implication P (x) → Q(x)
holds, must result from falsity of P (x), hence whenever Q(x) holds then
¬P (x) holds. These two facts combined mean that P(x) and its negation
have to hold simultaneously. This is actually absurdity overseen deliberately. The source of this misunderstanding is obvious: the lack of proper
deﬁnition of what is and what is not independent. Our interpretation allows
for sanitation of this situation. We are not telling that the predicate and its
negation hold simultaneously. Instead we say that for one object we modify
the measurement procedure (set a label) in such a way that it, applied for
calculation of P (x), yields true and at the same time for another object, with
the same original properties we make another modiﬁcation of measurement
procedure (attach a label to it) so that measurement of ¬P (x) yields also
true, because possibly two diﬀerent persons were enforcing their diﬀerent beliefs onto diﬀerent subsets of data.

Our approach is also superior to canonical random set approach in the
following sense: The canonical approach requires knowledge of the complete
random set realizations of two processes on an object to determine the combination of both processes. We, however, postpone the acquisition of knowledge of the precise instantiation of properties of the object by interleaving
the concept of measurement and the concept of labeling process. This has a

DS BELIEF FUNCTION - A NEW INTERPRETATION

55

close resemblance to practical processing whenever diagnosis for a patient is
made. If a physician ﬁnds a set of hypotheses explaining the symptoms of
a patient, he will usually not try to carry out other testing procedures than
those related to the plausible hypotheses. He runs clearly at risk that there
exists a diﬀerent set of hypotheses also explaining the patients’s symptoms,
and so a disease unit possibly present may not be detected on time, but usually the risk is suﬃciently low to proceed in this way, and the cost savings
may prove enormous.

6.3

Upper and Lower Probabilities

Still another approach was to handle Bel and Pl as lower and upper probabilities [4]. This approach is of limited use as not every set of lower and
upper probabilities leads to Bel/Pl functions [12], hence establishing a unidirectional relationship between probability theory and the DS-theory. Under
our interpretation, the Bel/Pl function pair may be considered as a kind of
interval approximations to some ”intrinsic” probability distributions which,
however, cannot be accessed by feasible measurements and are only of interest as a kind of qualitative explanation to the physical quantities really
measured.

Therefore another approach was to handle them as lower/upper envelops
to some probability density function realization [12], [8]. However, the DS

56

MIECZYSLAW A. KLOPOTEK

rule of combination of independent evidence failed.

6.4

Inner and Outer Measures

Still another approach was to handle Bels/Pl in probabilistic structures
rather than in probabilistic spaces [7]. Here, DS-rule could be justiﬁed as one
of the possible outcomes of independent combinations, but no stronger properties were available. This is due to the previously mentioned fact that exclusion of empty intersections renders actually most of conceivable processes
dependent. Please notice that under our interpretation no such ambiguity
occurs. This is because we not only drop empty intersecting objects but also
relabel the remaining ones so that any probability calculated afterwards does
not refer to the original population.
So it was tried to drop the DS-rule altogether in the probabilistic structures, but then it was not possible to ﬁnd a meaningful rule for multistage
reasoning [10]. This is a very important negative outcome. As the DempsterShafer-Theory is sound in this respect and possesses many useful properties
(as mentioned in the Introduction), it should be sought for an interpretation
meeting the axiomatic system of DS Theory rather then tried to violate its
fundamentals. Hence we consider our interpretation as a promising one for
which decomposition of the joint distribution paralleling the results for probability distributions may be found based on the data.

DS BELIEF FUNCTION - A NEW INTERPRETATION

6.5

57

Rough Set Approach

An interesting alternative interpretation of the Dempster-Shafer Theory was
found within the framework of the rough set theory [24], [9]. Essentially the
rough set theory searches for approximation of the value of a decision attribute by some other (explaining) attributes. It usually happens that those
attributes are capable only of providing a lower and upper approximation to
the value of the decision attribute (that is the set of vectors of explaining
attributes supporting only this value of the decision variable, and the set
of vectors of explaining attributes supporting also this value of the decision
variable resp.- for details see texts of Skowron [24] and Grzymala-Busse [9]).
The Dempster Rule of combination is interpreted by Skowron [25] as combination of opinions of independent experts, who possibly look at diﬀerent sets
of explanation attributes and hence may propose diﬀerent explanations.
The diﬀerence between our approach and the one based on rough sets
lies ﬁrst of all in the ideological background: We assume that the ”decision attribute” is set-valued whereas the rough-set approach assumes it to
be single-valued. This could have been overcome by some tricks which will
not be explained in detail here.But the combination step is here essential:
If we assume that the data sets for forming knowledge of these two experts
are exhaustive, then it can never occur that these opinions are contradictory.
But the DST rule of combination uses the normalization factor for dealing
with cases like this. Also the opinions of experts may have only the form
of a simple (that is deterministic) support function. Hence, rough-set in-

58

MIECZYSLAW A. KLOPOTEK

terpretation implies axioms not actually present in the DST. Hence rough
set interpretation is on the one hand restrictive, and on the other hand not
fully conforming to the general DST. From our point of view the DST would
change the values of decision variables rather then recover them from expert
opinions.
Here, we come again at the problem of viewing the independence of experts. The DST assumes some strange kind of independence within the data:
the proportionality of the distribution of masses of sets of values among intersecting subsets weight by their masses in the other expert opinion. Particularly unhappy is the fact for the rough set theory, that given a value of
the decision variable, the respective indicating vectors of explaining variables
values must be proportionally distributed among the experts not only for this
decision attribute value, but also for all the other decision attribute values
that ever belong to the same focal point. Hence applicability of the rough set
approach is hard to justify by a simple(, ”usual” as Shafer wants) statistical
test. On the other hand, statistical independence required for Dempster rule
application within our approach is easily checked.
To demonstrate the problem of rough set theory with re combination of
opinions of independent experts let us consider an examle of two experts
having the combined explanatory attributes E1 (for expert 1) and E2 (for
expert 2) both trying to guess the decision attribute D. Let us assume that
D takes one of two values: d1 , d2, E1 takes one of three values e11 , e12 , e13 ,
E2 takes one of three values e21 , e22 , e23 . Furthermore let us assume that the

DS BELIEF FUNCTION - A NEW INTERPRETATION

59

rough set analysis of an exhaustive set of possible cases shows that the value
e11 of the attribute E1 indicates the value d1 of the decision attribute D,
e12 indicates d2 , e13 indicates the set {d1 , d2}, Also let us assume that the
rough set analysis of an exhaustive set of possible cases shows that the value
e21 of the attribute E2 indicates the value d1 of the decision attribute D, e22
indicates d2 , e32 indicates the set {d1 , d2 }, From the point of view of bayesian
analysis four cases of causal inﬂuence may be distinguished (arrows indicate
the direction of dependence).
E1 → D → E2
E1 ← D ← E2
E1 ← D → E2
E1 → D ← E2
From the point of view of bayesian analysis, in the last case attributes
E1 and E2 have to be unconditionally independent, in the remaining cases:
E1 and E2 have to be independent conditioned on D. Let us consider ﬁrst
unconditional independence of E1 and E2 . Then we have tthat:
(

ProbP (ω)

E1 (ω) = e11 ∧ E2 (ω) = e22 ) =

ω

=(

ProbP (ω)
ω

E1 (ω) = e11 ) · (

ProbP (ω)

E2 (ω) = e22 ) > 0

ω

P (ω)
E1 (ω) = e11 ∧ E2 (ω) = e22 ) > 0
However, it is impossible that ( Prob
ω

because we have to do with experts who may provide us possibly with information not speciﬁc enough, but will never provide us with contradictory

60

MIECZYSLAW A. KLOPOTEK

information. We conclude that unconditional independence of experts is impossible.
Let us turn to independence of E1 and E2 if conditioned on D. We introduce
the following denotation:

p1 =

ProbP (ω)

D(ω) = d1

ω

p2 =

ProbP (ω)

D(ω) = d2

ω

e′1
e′3

=

Prob(D(ω)=d1 )∧P (ω)

E1 (ω) = e11

ω

=

f1′ =
f3′ =
e2 ” =

Prob(D(ω)=d1 )∧P (ω)

E1 (ω) = e13

ω

Prob(D(ω)=d1 )∧P (ω)

E2 (ω) = e21

ω

Prob(D(ω)=d1 )∧P (ω)

E2 (ω) = e23

ω

Prob(D(ω)=d2 )∧P (ω)

E1 (ω) = e12

ω

e3 ” =

Prob(D(ω)=d2 )∧P (ω)

E1 (ω) = e13

ω

f2 ” =

Prob(D(ω)=d2 )∧P (ω)

E2 (ω) = e22

ω

f3 ” =

Prob(D(ω)=d2 )∧P (ω)

E2 (ω) = e23

ω

Let Bel1 and m1 be the belief function and the mass function representing
the knowledge of the ﬁrst expert, let Bel2 and m2 be the belief function
and the mass function representing the knowledge of the second expert. Let

DS BELIEF FUNCTION - A NEW INTERPRETATION

61

Bel12 and m12 be the belief function and the mass function representing the
knowledge contained in the combined usage of attributes E1 , E2 if used for
prediction of D - on the grounds of the rough set theory. It can be easily
checked that:

m1 ({d1 }) = e′1 · p1 , m1 ({d2 }) = e2 ” · p2 , m1 ({d1 , d2}) = e′3 · p1 , +e3 ”′ · p2
m2 ({d1}) = f1′ · p1 , m2 ({d2 }) = f2 ” · p2 , m2 ({d1 , d2}) = f3′ · p1 , +f3 ”′ · p2
and if we assume the conditional independence of E1 and E2 conditioned on
D, then we obtain:
m12 ({d1 }) = e′1 · f1′ · p1 + e′1 · f3′ · p1 + e′3 · f1′ · p1
m12 ({d2 }) = e2 ” · f2 ” · p2 + e2 ” · f3 ” · p2 + e3 ” · f2 ” · p2
m12 ({d1 , d2}) = e′3 · f3′ · p1 + e3 ” · f3 ” · p2
However, the Dempster rule of combination would result in (c - normalization constant):

m1 ⊕m2 ({d1 }) = c·(e′1 ·f1′ ·p21 +e′1 ·f3′ ·p21 +e′1 ·f3 ”·p1 ·p2 +e′3 ·f1′ ·p21 +e3 ”·f1′ ·p1 ·p2 )
m1 ⊕m2 ({d2 }) = c·(e2 ”·f2 ”·p22 +e2 ”·f3′ ·p1 ·p2 +e2 ”·f3 ”·p22 +e′3 ·f2 ”·p1 ·p2 +e3 ”·f2 ”·p22 )
m1 ⊕ m2 ({d1 , d2 }) = c · e′3 · f3′ · p21 + e3 ” · f3 ” · p22 + e′3 · f3 ” · p1 · p2 + e3 ” · f3′ · p1 · p2 )
Obviously, Bel12 and Bel1 ⊕ Bel2 are not identical in general. We conclude
that conditional independence of experts is also impossible. Hence no usual

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staatistical indeperndence assumption is valid for the rough set interpretation of the DST. This fact points at where the diﬀerence between rough
set interpretation and our interpretation lies in: in our interpretation, traditional statistical independence is incorporated into the Dempster’s scheme
of combination (labelling process).
By the way, lack of correspondence between statistical independence and
Dempster rule of combination is characteristic not only of the rough set interpretation, but also of most of the other ones. The Reader should read
carefully clumsy statements of Shafer about DST and statistical independence in [22].

6.6

General Remarks

The Dempster-Shafer Theory exists already over two decades. Though it was
claimed to reﬂect various aspects of human reasoning, it has not been widely
used in expert systems until recently due to the high computational complexity. Three years ago, however, an important paper of Shenoy and Shafer
[23] has been published, along papers of other authors similar in spirit, which
meant a break-through for application of both bayesian and Dempster-Shafer
theories in reasoning systems, because it demonstrated that if joint (bayesian
or DS) belief distribution can be decomposed in form of a belief network than
it can be both represented in a compact manner and marginalized eﬃciently
by local computations.
This fact makes them suitable as alternative fundamentals for represen-

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63

tation of (uncertain) knowledge in expert system knowledge bases [11].
Reasoning in bayesian belief networks has been subject of intense research
work also earlier [20], [23], [15], [17]. There exist methods of imposing various logical constraints on the probability density function and of calculating
marginals not only of single variables but of complicated logical expressions
over elementary statements of the type X =x (x belonging to the domain of
the variable X ) [17]. There exist also methods determining the decomposition of a joint probability distribution given by a sample into a bayesian
belief network [3], [18], [1], [27].
It is also known that formally probability distributions can be treated as
special cases of Dempster-Shafer belief distributions (with sinngleton focal
points) [10].
However, for application of DS Belief-Functions for representation of uncertainty in expert system knowledge bases there exist several severe obstacles. The main one is the missing frequentist interpretation of the DS-Belief
function and hence neither a comparison of the deduction results with experimental data nor any quantitative nor even qualitative conclusions can
be drawn from results of deduction in Dempster-Shafer-theory based expert
systems [13].
Numerous attempts to ﬁnd a frequentist interpretation have been reported (e.g. [7], [8], [9], [10], [12], [22], [24]). But, as Smets [26] states,
they failed either trying to incorporate Dempster rule or when explaining
the nature of probability interval approximation. The Dempster-Shafer The-

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MIECZYSLAW A. KLOPOTEK

ory experienced therefore sharp criticism from several authors in the past
[17], [10]. It is suggested in those critical papers that the claim of DST to
represent uncertainty stemming from ignorance is not valid. Hence alternative rules of combination of evidence have been proposed. However, these
rules fail to fulﬁll Shenoy/Shafer axioms of local computation [23] and hence
are not tractable in practice. These failures of those authors meant to us
that one shall nonetheless try to ﬁnd a meaningful frequentist interpretation
of DST compatible with Dempster rule of combination.
We have carefully studied several of these approaches and are convinced
that the key for many of those failures (beside those mentioned by Halpern
in [10]) was: (1) treating the Bel-Pl pair as an interval approximation and (2)
viewing combination of evidence as a process of approaching a point estimation. In this paper we claim that the most reasonable treatment of Bel’s Pl’s
is to consider them to be POINT ESTIMATES of probability distribution
over set-valued attributes (rather then Interval estimates of probability distribution over single valued attributes). Of course, we claim also that Bel-Pl
estimates by an interval some probability density function but in our interpretation that ”intrinsic” probability density function is of little interest for
the user. The combination of evidence represents in our interpretation manipulation of data by imposing on them our prejudices (rather then striving
for extraction of true values).
Under these assumptions a frequentionistically meaningful interpretation
to the Bel’s can be constructed, which remains consistent under combination

DS BELIEF FUNCTION - A NEW INTERPRETATION

65

of joint distribution with ”evidence”, giving concrete quantitative meaning
to results of expert system reasoning. Within this interpretation we were
able to prove the correctness of Dempster-Shafer rule. This means that this
frequentist interpretation is consistent with the DS-Theory to the largest
extent ever achieved.

7

Conclusions
• According to Smets [26] there has existed no proper frequentist interpretation of the Dempster-Shafer theory of evidence so far.
• In this paper a novel frequentist interpretation of the Dempster-ShaferTheory has been found allowing for close correspondence between Belief
and Plausibility functions and the real data.
• This interpretation ﬁts completely into the framework of Bel/Pl deﬁnitions and into the Dempster rule of combination of independent evidence relating for the ﬁrst time in DST history this rule to plain statistical independence just overcoming diﬃculties of many alternative
interpretations of the Dempster-Shafer-Theory. Hence this interpretation dismisses the claim of Smets [26] that such an interpretation
cannot exist.
• It is distinguished by the fact of postponing the moment of measuring object properties behind combination of evidence leading even to

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MIECZYSLAW A. KLOPOTEK
dropping some costly measurements altogether.
• The interpretation allows for subjective treatment of Bel’s and Pl’s as
some approximations to unknown probability distribution of an intrinsic, but not accessible, attribute.
• The introduced concept of labeled population may to some extent represent subjectivity in viewing probabilities.
• This interpretation questions the common usage of the DST as a mean
to represent and to reason with uncertainty stemming from ignorance.
This view has been already shaken by works of Pearl [17] and Halpern
and Fagin [10]. What our interpretation states clearly is that the DST
should be viewed as a way to express unwillingness to accept objective
facts rather than as a mean to express ignorance about them. Hence it
should be called a theory of prejudices rather than a theory of evidence.
Finally, I feel obliged to apologize and to say that all critical remarks

towards interpretations of DST elaborated by other authors result from deviations of those interpretations from the formalism of the DST. I do not
consider, however, a deviation from DST as a crime, because modiﬁcations
of DST may and possibly have a greater practical importance than the original theory. The purpose of this paper was to shed a bit more light onto the
intrinsic nature of pure DST and not to call for orthodox attitudes towards
DST.

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67

Acknowledgements
I am indebted to anonymous referees who greatly contributed to enhancement
of the quality of presentation of this paper.

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