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On the Combinability of Evidence
in the Dempster-Shafer Theory
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Lotfi A. Zadeh t
and
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Anca Ralescut
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Abstract
In the current versions of the Dempster-Shafer theory, the only essential restriction
on the validity of the rule of combination is that the sources of evidence must be statistically independent.
Under this assumption, it is permissible to apply the Dempster-Shafer
.
rule to two or more distinct probability distributions.
An essential step in the Dempster-Shafer rule of combination of evidence is that of
normalization.
The validity of normalization is open to question, particularly in application
to probability distributions (Zadeh, 1976).
At this juncture, the validity of normalization is
a controversial issue.
In this paper, we construct a relational model for the Dempster-Shafer theory which
greatly simplifies the derivation of its main results and cast considerable light on the valldity of the rule of combination.
The relational model is augmented with what is called the
ball-box analogy, yielding an intuitively simple way of visualizing the concepts of belief
and plausibility.
t Computer Science Division, University of California, Berkeley, CA 94720.
* Department of Computer Science, University of Cincinnati, Cincinnati. OH 45221.
Research supported in part by NSF Grant DCR-8513139, NASA Grant NCC-2-275 and NESC Contract
N00039·84-C-0243.
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In the relational model, a multivalued mapping is represented at' a second-order relation in which the attributes are granular, that is, set-valued.
In this model, a Dempster-
Shafer distribution is a granular distribution which may be interpreted as a summary of
the parent relation, that is, the relation which represents the multivalued mapping from
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individuals to their attributes.
Given a set-valued query, Q, the number of individuals
.
'
:.(_
'
.. . �
whose attribute is certain to satisfy Q is the necessity of Q, while the number of individuals
�
.
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whose attribute may possibly satisfy Q is the possibility of Q. Necessity and possibility are
"\
:
the counterparts of belief and plausibility in the Dempster-Shafer theory (Zadeh, 19i9a,
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1986).
In the relational model, two distinct sources of evidence are rep resen ted as two distinct columns for a single attribute in the parent relation.
A parent relation is said to be
conflict-free if the intersection of set-valued entries in the two columns is non-empty for
each individual.
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From this model, it follows that, in order to be combinable, the sources of
=
{(A1,
Pl),
granular distributions in which Pi• i
=
. ,<Am,
PmH
..
and H
=
{(B1, ql), ... , <Bn, q11)}
be two
1, . , m is the relative count of individuals in the
..
parent relation whose set-valued attribute is A, in one column, and
relative count of individuals whose set-valued attribute is
Bi
qi, j
=
1, . .. , n, is the
in another column, with the
understanding that the columns in question represent two distinct sources of evidence.
Then, a sufficient condition for noncombinality is that there is a granule A, in G which is
disjoint from all granules in H, or vice-versa.
is demonstrated by the following example: G
2/3)}, in which A 1 and A2 are disjoint.
This condition, however, is not necessary, as
=
{(Alt 2/3), <A2, 1/3)}.
H
=
{(Ab 1/3), (A2,
In this case, it is evident that there does not exist a
parent relation which is conflict-free.
A necessary and sufficient condition for noncombinability which is computationally
much more efficient than a direct test based on the definition of noncombinability, is formulated for the case where the granules in each distribution are disjoint.
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evidence must have a common parent relation which is conflict-free.
More generally, let G
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In addition, the
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ball-box analogy is employ�, to derive some of the basic results of the Dempster-Shafer
theory, and to extend it to cases in which the sources have unequal credibilities.
References and Related Publications
1
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,·tr•
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2
.
;'-"•>
,::•
.•
{';•·:;'"}
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Dempster, A.P. (1967) Upper and lower probabilities induced by a multivalued map,:1'�'1��
t
.·� t·'J
t�
r··�;-.
-:-.' ·,
:-�·-�
'',:.:. ,"!
·
·�-
j
•
'•i)
0,
..--�'
Garvey, T., and Lowrance, J. (1983) Evidenti �l reasoning: implementation for mull
< � ' . �}
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3
Gordon, J. and Shortliffe,
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5
Shafer, G. (1976) A Mathematical Theory of Evidence. Princeton University Press,
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6
Zadeh, L.A. (1979a) Fuzzy sets and information granularity.
Theory and Applications.
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<
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:,.,,.
r:-
In Advances in Fuzzy Set
(Gupta, M.M., Ragade, R. and Yager, R., eds.).
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7
Zadeh, L.A. (1979b) On the validity of Dempster's rule of combination of evidence.
ERL Memorandum M79!24, University of California, Berkeley, CA.
8
Zadeh, L.A. (1986) A simple view of the Dempster-Shafer theory of evidence and its
implication for the rule of combination. Institute of Cognitive Studies Report 33, 1985,
University of California, Berkeley, CA 94720. To appear in the Summer 1986 issue of
the AI Magazine.
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