17
$$\begin{array}{c}{{2(\theta_{1}\cup\theta_{2})]}}\\ {{=-(\theta_{1})+\cdots+(\theta_{n}+\theta_{n})}}\end{array}$$
mDP (θ1 ∪ θ2) = [m1(θ1 ∪ θ2)m2(θ1 ∪ θ2)]
$$=[m_{1}(\theta_{1}\cup\theta_{2})m_{2}(\theta_{1}\cup\theta_{2})]$$ $$\quad+[m_{1}(\theta_{1}\cup\theta_{2})m_{2}(\theta_{3})+m_{2}(\theta_{1}\cup\theta_{2})m_{1}(\theta_{3})]$$ $$\quad+[m_{1}(\theta_{1})m_{2}(\theta_{2})+m_{2}(\theta_{1})m_{1}(\theta_{2})]$$ $$=[0.30.1]+[0.3\cdot0.3+0.1\cdot0.2]+[0.1\cdot0.1+0.5\cdot0.4]$$ $$=[0.03]+[0.09+0.02]+[0.01+0.20]$$ $$=0.03+0.11+0.21=0.35$$
Now if one adds up the masses, one gets 0 + 0.34 + 0.25 + 0.35 = 0.94 which is less than 1. Therefore Dubois & Prade's rule of combination does not work when a singleton, or an union of singletons, becomes empty (in a dynamic fusion problem). The products of such empty-element columns of the mass matrix M are lost; this problem is fixed in DSmT by the sum S2(.) in (5) which transfers these products to the total or partial ignorances.

- Finally, if one applies **DSmH rule**, one gets because θ3 M= ∅ :

$$\mathrm{(by~definition~of~DSmH)}$$
m**DSmH**(∅) = 0 (by definition of DSmH)
mDSmH(θ1) = 0.34 (same as mDP (θ1)) mDSmH(θ2) = 0.25 (same as mDP (θ2)) mDSmH(θ1 ∪ θ2) = [m1(θ1 ∪ θ2)m2(θ1 ∪ θ2)] + [m1(θ1 ∪ θ2)m2(θ3) + m2(θ1 ∪ θ2)m1(θ3)] + [m1(θ1)m2(θ2) + m2(θ1)m1(θ2)] + [m1(θ3)m2(θ3)] = 0.03 + 0.11 + 0.21 + 0.06 = 0.35 + 0.06 = 0.41 6= mDP (θ1 ∪ θ2)
We can easily verify that m**DSmH**(θ1) + m**DSmH**(θ2) + m**DSmH**(θ1 ∪ θ2) = 1. In this example, using the hybrid DSm rule, one transfers the product of the empty-element θ3 column, m1(θ3)m2(θ3) = 0.2 ·
0.3 = 0.06, to m**DSmH**(θ1 ∪ θ2), which becomes equal to 0.35 + 0.06 = 0.41. Clearly, DSmH rule doesn't provide the same result as Dubois and Prade's rule, but only when working on static frames of discernment (restricted cases).

## 2.7 Fusion Of Imprecise Beliefs

In many fusion problems, it seems very difficult (if not impossible) to have precise sources of evidence generating precise basic belief assignments (especially when belief functions are provided by human experts), and a more flexible plausible and paradoxical theory supporting imprecise information becomes necessary. In the previous sections, we presented the fusion of *precise* uncertain and conflicting/paradoxical generalized basic belief assignments (gbba) in DSmT framework. We mean here by precise gbba, basic belief functions/masses m(.) defined precisely on the hyper-power set DΘ where each mass m(X), where X belongs to DΘ, is represented by only one real number belonging to [0, 1] such that PX∈DΘ m(X) = 1. In this section, we present the DSm fusion rule for dealing with *admissible imprecise generalized basic belief assignments* mI(.) defined as real subunitary intervals of [0, 1], or even more general as real subunitary sets [i.e. sets, not necessarily intervals].

An imprecise belief assignment mI(.) over DΘ is said *admissible* if and only if there exists for every X ∈ DΘ at least one real number m(X) ∈ mI(X) such that PX∈DΘ m(X) = 1. The idea to work with imprecise belief structures represented by real subset intervals of [0, 1] is not new and has been investigated in [5, 6, 16] and references therein. The proposed works available in the literature, upon our knowledge were limited only to sub-unitary interval combination in the framework of Transferable Belief Model (TBM) developed by Smets [39, 40]. We extend the approach of Lamata & Moral and Denœux based on subunitary interval-valued masses to subunitary set-valued masses; therefore the closed intervals used by Denœux to denote imprecise masses are generalized to any sets included in [0,1], i.e. in our case these sets can be unions