

![0_image_0.png](0_image_0.png)

Fig. 3: Venn diagram of a DSm hybrid model for a 3D frame.

The *generalized belief and plausibility functions* are defined in almost the same manner as within DST, i.e.

$$\text{Bel}(A)=\sum_{\begin{subarray}{c}B\subseteq A\\ B\in G^{\text{th}}\end{subarray}}m(B)\qquad\qquad\text{Pl}(A)=\sum_{\begin{subarray}{c}B\cap A\neq\emptyset\\ B\in G^{\text{th}}\end{subarray}}m(B)\tag{3}$$

We recall that GΘ is the generic notation for the set on which the gbba is defined (GΘ can be 2 Θ, DΘ
or even S
Θ depending on the model chosen for Θ). These definitions are compatible with the definitions of the classical belief functions in DST framework when GΘ = 2Θ for fusion problems where Shafer's model M0(Θ) holds. We still have ∀A ∈ GΘ, Bel(A) ≤ Pl(A). Note that when working with the free DSm model Mf(Θ), one has always Pl(A) = 1 ∀A 6= ∅ ∈ (GΘ = DΘ) which is normal. Example: Let's consider the simple frame Θ = {**A, B**}, then depending on the model we choose for GΘ, one will consider either:
- GΘ as the power set 2 Θ and therefore:

$$m(A)+m(B)+m(A\cup B)=1$$
$G^\Theta$ as the h. 
- GΘ as the hyper-power set DΘ and therefore:

$$m(A)+m(B)+m(A\cup B)+m(A\cap B)=1$$

- GΘ as the super-power set S
Θ and therefore:

$m(A)+m(B)+m(A\cup B)+m(A\cap B)+m(c(A))+m(c(B))+m(c(A)\cup c(B))=1$.  

## 2.4 The Classic Dsm Rule Of Combination

When the free DSm model Mf(Θ) holds for the fusion problem under consideration, the classic DSm rule of combination mMf (Θ) ≡ m(.) , [m1 ⊕ m2](.) of two independent5sources of evidences B1 and B2 over the same frame Θ with belief functions Bel1(.) and Bel2(.) associated with gbba m1(.) and m2(.) corresponds to the conjunctive consensus of the sources. It is given by [31]:

$$\forall C\in D^{\Theta},\qquad m_{\mathcal{M}^{I}(\Theta)}(C)\equiv m(C)=\sum_{\begin{subarray}{c}A:B\in D^{\Theta}\\ A\cap B=C\end{subarray}}m_{1}(A)m_{2}(B)\tag{4}$$

Since DΘ is closed under ∪ and ∩ set operators, this new rule of combination guarantees that m(.) is a proper generalized belief assignment, i.e. m(.) : DΘ → [0, 1]. This rule of combination is commutative and associative and can always be used for the fusion of sources involving fuzzy concepts when free DSm model holds for the problem under consideration. This rule has been extended for s > 2 sources in [31].

5While independence is a difficult concept to define in all theories managing epistemic uncertainty, we follow here the interpretation of Smets in [37] and [38], p. 285 and consider that two sources of evidence are independent (i.e distinct and noninteracting) if each leaves one totally ignorant about the particular value the other will take.

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