
## E. Example 5 (Free Dsm Model)

Let's assume the free DSm model (i.e. A ∩ B 6= ∅) and the generalized mass given in Table V. In the case of free-DSm (or hybrid DSm) models, the pignistic probability and the DSmP
can be derived directly from m(.) without the refinement of the frame Θ whereas Sudano's and Cuzzolin's probabilities cannot be derived directly from their formulas (4)-(9) for such models. However, they can be obtained indirectly after a refinement of the frame Θ into Θref which satisfies Shafer's model. More precisely, instead of working directly on the 2D frame Θ =
{**A, B**} with m(.) given in Table V, we need to work on the 3D frame Θref = {A′ , A\ {A∩B}, B′ , B \ {A∩B}, C′ ,
A∩B} satisfying Shafer's model with the equivalent bba m(.)
defined as in Table VI. The results are then given in Table VII. One sees that P IC(**DSmP**ǫ→0) is the maximum value.

P rBel does not work correctly because it cannot be directly evaluated for A and B since the underlying **P rBel**(A′) and P rBel(B′) are mathematically undefined in such case. If one works on the *refined frame* Θref and one applies the **DSmP**
mapping of the bba m(.) defined in Table VI, one obtains naturally the same results for **DSmP** as those given in table VII. Of course the results of **BetP** in Table VII are the same using directly the formula (3) as those using (1) on Θref. The verification is left to the reader.

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

C′ A′ ∪ C′ B′ ∪ C′ A′ ∪ B′ ∪ C′

![0_image_1.png](0_image_1.png)

![0_image_2.png](0_image_2.png)

![0_image_3.png](0_image_3.png)

m(.) 0.4 0.2 0.1 0.3 TABLE VI
QUANTITATIVE INPUTS ON THE REFINED FRAME ΘREF

A B A ∩ B **P IC**(.)

P rNP l(.) 0.7895 0.7368 0.5263 0.0741 CuzzP(.) 0.8400 0.8000 0.6400 0.1801 BetP(.) 0.8500 0.8000 0.6500 0.1931 P raP l(.) 0.8736 0.8421 0.7157 0.2789

P rP l(.) 0.9083 0.8544 0.7627 0.3570 P rHyb(.) 0.9471 0.9165 0.8636 0.5544

DSmPǫ=0.001(.) 0.9990 0.9988 0.9978 0.9842

P rBel(.) NaN NaN 1 1

DSmPǫ=0(.) 1 1 1 1

TABLE VII

RESULTS FOR EXAMPLE 5.

VIII. EXAMPLES ON A 3D FRAME

We work hereafter on the 3D frame Θ = {**A, B, C**}.

A. Example 6 (Shafer's model and a non-Bayesian mass)

This example is drawn from [15]. Let's assume Shafer's

model and the non-Bayesian belief mass given by m(A) =

0.35, m(B) = 0.25, m(C) = 0.02, m(A ∪ B) = 0.20, m(A∪C) = 0.07, m(B∪C) = 0.05 and m(A∪B∪C) = 0.06.

The results of the mappings are given in Table VIII. One sees

that **DSmP**ǫ→0 provides the same result as **P rBel** which

corresponds here to the best result in term of PIC metric.

A B C **P IC**(.)

P rNP l(.) 0.4722 0.3889 0.1389 0.0936

CuzzP(.) 0.5029 0.3937 0.1034 0.1377 BetP(.) 0.5050 0.3950 0.1000 0.1424 P raP l(.) 0.5294 0.3978 0.0728 0.1861 P rP l(.) 0.5421 0.4005 0.0574 0.2149 P rHyb(.) 0.5575 0.4019 0.0406 0.2517

DSmPǫ=0.001(.) 0.5665 0.4037 0.0298 0.2783

P rBel(.) 0.5668 0.4038 0.0294 0.2793

DSmPǫ=0(.) 0.5668 0.4038 0.0294 0.2793

TABLE VIII

RESULTS FOR EXAMPLE 6.

B. Example 7 (Shafer's model and a non-Bayesian mass)
Let's assume Shafer's model and change a bit the nonBayesian input mass by taking m(A) = 0.10, m(B) = 0, m(C) = 0.20, m(A ∪ B) = 0.30, m(A ∪ C) = 0.10, m(B ∪ C) = 0 and m(A ∪ B ∪ C) = 0.30. The results of the mappings are given in Table IX. One sees that **DSmP**ǫ→0 provides the best PIC value than all other mappings since P rBel is mathematically undefined. If one takes artificially P rBel(B) = 0, one gets the same result as with **DSmP**ǫ→0.

A B C **P IC**(.)

P rBel(.) 0.5333 NaN 0.4667 NaN P rNP l(.) 0.4000 0.3000 0.3000 0.0088 CuzzP(.) 0.3880 0.2470 0.3650 0.0163 BetP(.) 0.4000 0.2500 0.3500 0.0164

P raP l(.) 0.3800 0.2100 0.4100 0.0342 P rP l(.) 0.4486 0.2186 0.3328 0.0368 P rHyb(.) 0.4553 0.1698 0.3749 0.0650

DSmPǫ=0.001(.) 0.5305 0.0039 0.4656 0.3500

TABLE IX

RESULTS FOR EXAMPLE 7.

## C. Example 8 (Hybrid Dsm Model)

We consider the hybrid DSm model in which all intersections of elements of Θ are empty, but A∩B. In this case, GΘ
reduces to 9 elements {∅, A ∩ B, A, B, C, A ∪ B, A ∪ **C, B** ∪
C, A∪B∪C}. The input masses of focal elements are given by m(A∩B) = 0.20, m(A) = 0.10, m(C) = 0.20, m(A∪B) = 0.30, m(A ∪ C) = 0.10, and m(A ∪ B ∪ C) = 0.10. In order to apply Sudano's and Cuzzolin's mappings, we need to work on the refined frame Θref with Shafer's model as depicted on Figure 1 and masses given in the Table X.

D′ A′ ∪ D′ C′
m(.) 0.2 0.1 0.2 A′ ∪ B′ ∪ D′ A′ ∪ C′ ∪ D′ A′ ∪ B′ ∪ C′ ∪ D′
m(.) 0.3 0.1 0.1 TABLE X
QUANTITATIVE INPUTS ON THE REFINED FRAME FOR EXAMPLE 8 One sees from the Table XI that **DSmP**ǫ→0 provides the best results in term of PIC metric. The refined frame has been defined as: Θref = {A′ , A\(A∩B), B′ , B\(A∩B), C′ , C, D′ , A ∩ B} according to Figure 1.