A Class of DSm Conditioning Rules1
Florentin Smarandache, Mark Alford
Air Force Research Laboratory, RIEA,
525 Brooks Rd., Rome, NY 13441-4505, USA
Abstract:
In this paper we introduce two new DSm fusion conditioning rules with example, and as
a generalization of them a class of DSm fusion conditioning rules, and then extend them
to a class of DSm conditioning rules.
Keywords: conditional fusion rules, Dempster’s conditioning rule, Dezert-Smarandache
Theory, DSm conditioning rules
0. Introduction
In order to understand the material in this paper, it is first necessary to define the terms that we’ll
be using:
•
•
•
•
•
•
Frame of discernment = the set of all hypotheses.
Ignorance is the mass (belief) assigned to a union of hypotheses.
Conflicting mass is the mass resulted from the combination of many sources of
information of the hypotheses whose intersection is empty.
Fusion space = is the space obtained by combining these hypotheses using union,
intersection, or complement – depending on each fusion theory.
Dempster-Shafer Theory is a fusion theory, i.e. method of examination of hypotheses
based on measures and combinations of beliefs and plausibility in each hypothesis,
beliefs provided by many sources of information such as sensors, humans, etc.
Transferable Belief Model is also a fusion theory, an alternative of DST, whose method is
of transferring the conflicting mass to the empty set.
1
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
•
•
•
•
•
Dezert-Smarandache Theory is a fusion theory, which is a natural extension of DST and
works for high conflicting sources of information, and overcomes the cases where DST
doesn’t work.
Power set = is the fusion space of Dempster-Shafer Theory (DST) and Transferable
Belief Model (TBM) theory; the power set is the set of all subsets of the frame of
discernment, i.e. all hypotheses and all possible unions of hypotheses. {In the fusion
theory union of hypotheses means uncertainty about these hypotheses.}
Hyper-power set = the fusion set of Dezert-Smarandache Theory (DSmT); the hyperpower set is the set formed by all unions and intersections of hypotheses. {By
intersection of two or more hypotheses we understand the common part of these
hypotheses – if any. In the case when their intersection is empty, we consider these
hypotheses disjoint.}
Super-power set = the fusion space for the Unification of Fusion Theories and rules; the
super-power set is the set formed by all unions, intersection, and complements of the
hypotheses. {By a complement of a hypothesis we understand the opposite of that
hypothesis.}
Basic belief assignment (bba), also called mass and noted by m(.), is a subjective
probability (belief) function that a source assigns to some hypotheses or their
combinations. This function is defined on the fusion space and whose values are in the
interval [0, 1].
In the first section, we consider a frame of discernment and then we present the three known
fusion spaces. The first fusion space, the power set, is used by Dempster-Shafer Theory (DST)
and the Transferable Believe Model (TBM). The second fusion space, which is larger, the hyperpower set, is used by Dezert-Smarandache Theory (DSmT), while the third fusion space, the
super-power set, is the most general one, and it is used in the Unification of Fusion Theories and
Rules.
In the second section we present Dempster’s conditioning rule and the Bel(.) and Pl(.) functions.
In order to overcome some difficult corner cases where Dempster’s Conditioning Rule doesn’t
work, we design the first simple DSm conditioning rule and the second simple DSm conditioning
rule in section 3. These rules are referring to the fact that: if a source provides us some evidence
(i.e. a basic belief assignment), but later we find out that the true hypothesis is in a subset A of
the fusion space, then we need to compute the conditional belief m(.|A).
In section 4 we give a Class of DSm Conditioning Rules that generalizes two simple DSm
conditioning rules cover.
In section 5 we present two examples in military about target attribute identification.
2
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
1. Mathematical Preliminaries.
Let Θ = {θ1, θ2, …, θn}, with n ≥ 2, be a frame of discernment.
As fusion space, Shafer uses the power set 2 Θ , which means Θ closed under union of sets,
( Θ , ∪ ), and it is a lattice. In Dempster-Shafer Theory (DST) all hypotheses θi are considered
mutually exclusive, i.e. θi ∩ θj = ϕ for any i ≠ j, and exhaustive.
Dezert extended the power set to a hyper-power set D Θ in Dezert-Smarandache Theory (DSmT),
which means Θ closed under union and intersection of sets ( Θ , ∪ , ∩ ) and it is a distribute
lattice; in this case the hypotheses are not necessarily exclusive, so there could be two or more
hypotheses whose intersections are non-empty. Each model in DSmT is characterized by empty
and non-empty intersections. If all intersections are empty, we get Shafer’s model used in DST;
if some intersection are empty and others are not, we have a hybrid model; and if all intersection
are non-empty we have a free model.
Further on Smarandache [3] extended the hyper-power to a super-power set S Θ , as in UFT
(Unification of Fusion Theories), which means Θ closed under union, intersection, and
complement of sets ( Θ , ∪ , ∩ , C), that is a Boolean algebra.
We note by G any of these three fusion spaces, power set, hyper-power set, or super-power set.
2. Dempster’s Conditioning Rule (DCR).
Let’s have a bba (basic believe assignment, also called mass):
m1: G Θ Æ [0, 1], where ∑
= 1.
In the main time we find out that the truth is in B GΘ. We therefore need to adjust our bba
according to the new evidence, so we need to compute the conditional bba m1(X|B) for all
X GΘ.
Dempster’s conditioning rule means to simply fuse the mass m1(.) with m2(B) = 1 using
Dempster’s classical fusion rule.
A similar procedure can be done in DSmT, TBM, etc. by combining m1(.) with m2(B) = 1 using
other fusion rule.
In his book Shafer gave the conditional formulas for believe and plausible functions Bel(.) and
respectively Pl(.) only, not for the mass m(.).
In general we know that:
3
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
Bel(A) =
and
Pl(A) =
∑ m (X )
1
X ⊆A
∑
X ∩ A ≠φ
m1( X ) .
Let m1(.) and m2(.) be two bba’s defined on GΘ. The conjunctive rule for combining these bba’s
is the following:
(m1+ m2)(A) = ∑
, Є
In order to compute in DST the subjective conditional probability of B given A, i.e. m(A|B),
Shafer combines the masses m1(.) and m2(B)=1 using Dempster’s rule (pp. 71-72 in [2]) and he
gets:
∑ m ( X )m (Y )
m(A|B) =
1 − ∑ m ( X )m (Y )
φ
X ∩Y = A
1
2
1
X ∩Y =
∑ m ( X )m ( B)
=
1 − ∑ m ( X )m ( B)
φ
X ∩B= A
1
2
1
X ∩B=
(which is Dempster’s rule)
2
2
(since only m2(B) ≠ 0, all other values of m2(Y) = 0 for Y ≠ B)
∑ m (X ) ∑ m (X )
=
which is exactly what Milan Daniel got in [1], but with
=
1− ∑ m ( X )
m (X )
∑
φ
φ
X ∩B= A
X ∩B=
1
1
X ∩B= A
1
X ∩B≠
1
different notations.
Therefore, Dempster’s Conditioning Rule (DCR) referred to masses {not to Bel(.) or to Pl(.)
functions as designed by Shafer} is the following:
∀ A ∈ 2 Θ \ φ we have mDCR(A|B) =
∑ m (X )
.
∑φ m (X )
X ∩B= A
1
X ∩B≠
1
With M. Daniel’s notations, Dempster’s Conditioning Rule becomes:
4
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
∀ X ∈ D Θ \ φ we have mDCR(X|A) =
∑ m (Y )
.
∑ φ m (Y )
Y ∩ A= X
1
Y ∩ A≠
1
DCR doesn’t work when Pl(A) = 0 since its denominator becomes null.
3. Two DSm Conditioning Rules.
We can overcome this undefined division by constructing a DSm first simple conditioning rule
in the super-power set:
∀ X ∈ S Θ \ φ we have mDSmT1(X|A) =
∑
(Y ∩ A = X ) or (Y ∩ A=φ andX = A )
m(Y )
which works in any case.
In the corner case when Pl(A) = 0, we get mDSmT1(A|A) =1 and all other mDSmT1(X|A) = 0 for X
≠ A.
The DSm first simple conditioning rule transfers the masses which are outside of A (i.e. the
masses m(Y) with Y ∩ A = φ ) to A in order to keep the normalization of m(.), in order to avoid
doing normalization by division as DCR does.
Another way will be to uniformly split the total mass which is outside of A:
Kcond =
∑
Y ∩ A =φ
m(Y )
to the non-empty sets of P(A), i.e. sets whose mass is non-zero, where P(A) is the set of all parts
of A.
So, a DSm second simple conditioning rule is:
mDSmT 2( X | A) =
∑
Y ∩ A= X
m(Y ) +
1
CP ( A)
⋅
∑
Y ∩ A=φ
m(Y )
where CP ( A) is the cardinal of the set of elements from P(A) whose masses are not zero, i.e.
CP ( A) = Card{Z | Z ∈ S Θ , Z ⊆ A,
5
∑
Y ∩ A= Z
m(Y ) ≠ 0 }.
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
In the corner edge when CP ( A) = 0, we replace it with the number of singletons included in A if
any, the number of unions of singletons included in A if any, and A itself.
4. A Class of DSm Conditioning Rules.
In this way we can design a class of DSm conditioning rules taking into consideration not only
masses, but also other parameters that might influence the decision-maker in calculating the
subjective conditioning probability, and which is a generalization of Dempster’s conditioning
rule:
∑
α (Y )
β (Y )
mDSmTclass ( X | A) = Y ∩ A= X
α (Y )
∑
Y ∩ A=φ β (Y )
with α (Y) = α1(Y) ⋅ α2(Y) ⋅ … ⋅ αp(Y) , where all αi(Y), 1 ≤ i ≤ p, are parameters that Y is
directly proportional to;
and β(Y) = β 1(Y) ⋅ β 2(Y) ⋅ … ⋅ β r(Y), where all β j(Y), 1 ≤ j ≤ r, are parameters that Y is
inversely proportional to.
5. Examples of Conditioning Rules.
Example 5.1.
Let m1(.) be defined on the frame {F = friend, E = enemy, N = neutral}, where the hypotheses F,
E, N are mutually exclusive, in the following way (see the second row):
m1
m2
m1+ m2
mDCR(X|F ∪ E)
mTBM(X|F ∪ E)
mDSmT1(X|F ∪ E)
mDSmT2(X|F ∪ E)
6
ϕ
0
0
0
0
0.3
0
0
F
0.2
0
0.2
2/7
0.2
0.2
0.3
E
0.1
0
0.1
1/7
0.1
0.1
0.2
Table 1
N
0.3
0
0
0
0
0
0
F∪E
0.1
1
0.4
4/7
0.4
0.7
0.5
F∪E∪N
0.3
0
0
0
0
0
0
N ∩ (F ∪ E)
0
0
0.3
0
0
0
0
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
Suppose the truth is in the set F ∪ E. First we combine m1(.) with m2(E) = 1 using the
conjunctive rule, and its result m1+ m2 is in the fourth row in Table 1. All below conditioning
rules are referred to the result of this conjunctive rule, and they differ through the way the
conflicting mass, i.e. mass of empty intersections, is transferred to the other elements.
In DCR, since N ∩ (F ∪ E) = ϕ the conflicting mass m1(N)· m2 (F ∪ E) = 0.3·1 = 0.3, is
transferred to the non-empty sets F, E, and F ∪ E proportionally with respect to their masses
acquired after applying the conjunctive rule (m1+ m2), i.e. with respect to 0.2, 0.1, and
respectively 0.4. Thus, we get mDST(X|F ∪ E) as in the fifth row of Table 1, where X ∈ { ϕ, F, E,
N, F ∪ E, F ∪ E ∪ N, N ∩ (F ∪ E)}.
In Smets’ TBM (Transferable Believe Model), the conflicting mass, 0.3, is transferred to the
empty set, since TBM considers an open world (non-exhaustive hypotheses). See row # 6.
With DSm first conditioning rule (row # 7) the conflicting mass 0.3 is transferred to the whole
set that the truth belongs to, F ∪ E. So, mDSmT1(F ∪ E |F ∪ E) = (m1+ m2)( F ∪ E) + 0.3 = 0.4+0.3
= 0.7.
In DSm second conditioning rule (row # 8) the conflicting mass 0.3 is uniformly transferred to
the non-empty sets F, E, and F ∪ E, therefore each such set receives 0.3/3 = 0.1.
Example 5.2.
Let m1(.) be defined on the frame {A = Airplane, T = tank, S = ship, M = submarine}, where the
hypotheses A, T, S, M are mutually exclusive, in the following way (see the second row):
ϕ
A
T
S
M
A∪S
T∪M
m1
0
0.4
0
0.5
0
0.1
0
m2
m1+ m2
mDCR(X|T ∪ M)
mTBM(X|T ∪ M)
mDSmT1(X|T ∪ M)
mDSmT2(X|T ∪ M)
0
0
0
1
0
0
0
0
N/A
0
0
0
0
0
N/A
0
0
1/3
0
0
N/A
0
0
0
1
0
N/A
0
1
1/3
0
0
0
0
N/A N/A
0
0
0
0
1/3
0
A ∩ (T ∪ M)
S ∩ (T ∪ M)
(A ∪ S) ∩
(T ∪ M)
0.4
0.5
0.1
0
0
0
0
0
0
0
0
0
Table 2
Suppose the truth is in T ∪ M. Since the sets A ∩ (T ∪ M), S ∩ (T ∪ M), and (A ∪ S) ∩ (T ∪ M)
are empty, their masses 0.4, 0.5, and respectively 0.1 have to be transferred to non-empty sets
belonging to P(T ∪ M), where P(T ∪ M) means the set of all subsets of T ∪ M.
7
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
In this case, DCR does not work since it gets an undefined division 0/0.
In Smets’ TBM (Transferable Believe Model), the total conflicting mass, 0.4 + 0.5 + 0.1 = 1, is
transferred to the empty set, since TBM considers an open world (non-exhaustive hypotheses).
See row # 6.
With DSm first conditioning rule (row # 7) the total conflicting mass, 1, is transferred to the
whole set that the truth belongs to, T ∪ D. So, mDSmT1(T ∪ D |T ∪ D) = (m1+ m2)( T ∪ D) + 1 = 1.
In DSm second conditioning rule (row # 8) the total conflicting mass is 1. Since C(B ∪ D) = 0, the
total conflicting mass 1 is uniformly transferred to the sets T, D, and T ∪ D {i.e. the singletons
and unions of singletons included in T ∪ D}, therefore each such set receives 1/3.
Conclusion.
We have examined Dempster’s Conditioning Rule in terms of bba. We saw that in the second
military example, using DCR for target identification, the procedure failed mathematically.
That’s why we designed two DSm simple conditioning rules and could complete the procedure
of target identification. We have compared these approaching of target identification using DCR,
TBM conditioning, and the two DSm conditioning rules that got better results than DCR and
TBM. We also observed from these examples that the two DSm simple conditioning rules give
almost similar results.
References:
[1] Milan Daniel, Analysis of DSm belief conditioning rules and extension of their applicability,
Chapter 10 in [5], 2009.
[2] Glenn Shafer, A Mathematical Theory of Evidence, Princeton Univ. Press, Princeton, New
Jersey, 1976.
[3] F. Smarandache, Unification of Fusion Theories (UFT), International Journal of Applied
Mathematics & Statistics, Vol. 2, 1-14, 2004; Presented at NATO Advanced Study Institute,
Albena, Bulgaria, 16-27 May 2005.
[4] F. Smarandache, J. Dezert (Editors), Advances and Applications of DSmT for
Information Fusion, Volume 2, American Research Press, Rehoboth, 2006.
[5] F. Smarandache & J. Dezert, “Advances and Applications of DSmT for Information Fusion”,
Vol. 3, Arp, 2009.
8
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
[6] F. Smarandache, J. Dezert, Belief Conditioning Rules, Chapter 9 in [4], pp.
237–268, 2006.
[7] F. Smarandache, J. Dezert, Qualitative Belief Conditioning Rules (QBCR),
Fusion 2007 Int. Conf. on Information Fusion, Québec City, Canada, July 2007.
9
This research has been supported by Air Force Research Laboratory, Rome, NY, USA, in June
and July 2009.
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