
## F. Derivation Of Pic From Qdsmp

We propose here the derivation of PIC from qualitative DSmP. Let's consider a finite space of discrete exclusive events Θ = {θ1, θ2**, . . . , θ**M} and a subjective qualitative alike probability measure qP(.) : Θ 7→ L = {L0, L1, . . . , Ln, Ln+1}.

Then one defines the entropy and PIC metrics from qP(.) as

$$H(qP)\triangleq-\sum_{i=1}^{M}qP\{\theta_{i}\}\log_{2}(qP\{\theta_{i}\})\tag{25}$$  $$PIC(qP)=1+\frac{1}{H_{\max}}\cdot\sum_{i=1}^{M}qP\{\theta_{i}\}\log_{2}(qP\{\theta_{i}\})\tag{26}$$  where $H_{\max}=\log_{2}(M)$ and in order to compute the log 
where Hmax = log2(M) and in order to compute the logarithms, one utilized the isomorphism Li = i/(n + 1).

X. EXAMPLE FOR QUALITATIVE DSMP
Let's consider the frame Θ = {**A, B, C**} with Shafer's model and the following set of linguistic labels L =
{L0, L1, L2, L3, L4, L5}, with L0 = Lmin and L5 = Lmax.

Let's consider the following qualitative belief assignment qm(A) = L1, qm(B ∪ C) = L4 and qm(X) = L0 for all X ∈ 2 Θ
P
\ {**A, B** ∪ C}. qm(.) is quasi-normalized since X∈2Θ qm(X) = L5 = Lmax. In this example, qm(B∪C) =
L4 is redistributed by **qDSmP**ǫ(.) to B and C only, since B
and C were involved in the ignorance, proportionally with respect to their cardinals (since their masses are L0 ≡ 0).

Applying **qDSmP**ǫ(.) formula (24), one gets for this example:

$$qDSmP_{\epsilon}(A)=L_{1}$$  $$qDSmP_{\epsilon}(B)=\frac{qm(B)+\epsilon\cdot\mathcal{C}(B)}{qm(B)+qm(C)+\epsilon\cdot\mathcal{C}(B\cup C)}qm(B\cup C)$$ $$=\frac{L_{0}+\epsilon\cdot1}{L_{0}+L_{0}+\epsilon\cdot2}\cdot L_{4}=\frac{L_{[0+(\epsilon\cdot1)\cdot5]}}{L_{[0+0+(\epsilon\cdot2)\cdot5]}}\cdot L_{4}$$ $$=\frac{L_{[\epsilon\cdot5]}}{L_{[\epsilon\cdot10]}}\cdot L_{4}=L_{[\frac{5\epsilon}{100}\cdot5]}\cdot L_{4}=L_{[2.5]}\cdot L_{4}$$ $$=L_{[2.5\cdot4/5]}=L_{[10/5]}=L_{2}$$  Similarly, one gets  $$qm(C)+\epsilon\cdot\mathcal{C}(C)$$
Similarly, the game  $qDSmP_\epsilon(C)=\dfrac{qm(C)+\epsilon\cdot\mathcal{C}(C)}{qm(B)+qm(C)+\epsilon\cdot\mathcal{C}(B\cup C)}qm(B\cup C)$ $\qquad\qquad=\dfrac{L_0+\epsilon\cdot1}{L_0+L_0+\epsilon\cdot2}L_4=L_2$  where the index in $[\cdot]$ has been computed at the very end for $[\cdot]$. 
where the index in [·] has been computed at the very end for the best accuracy. Thanks to the isomorphism between labels and numbers, all the properties of operations with numbers are transmitted to the operations with labels. **qDSmP**ǫ(.) is quasinormalized since qDSmPǫ(A)+qDSmPǫ(B)+**qDSmP**ǫ(C)
equals L1+L2+L2 = L5 = Lmax. Applying the PIC formula
(26), one obtains (here M =| Θ |= 3):

$$P I C(q D S m P_{\epsilon})=1+\frac{1}{\log_{2}3}(L_{1}\log_{2}(L_{1})$$ $$+L_{2}\log_{2}(L_{2})+L_{2}\log_{2}(L_{2}))\approx\frac{1}{5}L_{1}$$

where in order to compute the qualitative logarithms, one utilized the isomorphism Li = i n+1 .

## Xi. Conclusions

Motivated by the necessity to use a better (more informational) probabilistic approximation of belief assignment m(.)
for applications involving soft decisions, we have developed a new probabilistic transformation, called **DSmP**, for approximating m(.) into a subjective probability measure. **DSmP**
provides the maximum of the Probabilistic Information Content (PIC) of the source because it is based on proportional redistribution of partial and total uncertainty masses to elements of cardinal 1 with respect to their corresponding masses and cardinalities. **DSmP** works directly for any model (Shafer's, hybrid, or free DSm model) of the frame of the problem and the result can be obtained at any level of precision by a tuning positive parameter ǫ > 0. **DSmP**ǫ=0 coincides with Sudano's **P rBel** transformation for the cases when all masses of singletons involved in ignorances are nonzero. **P rBel**
formula is restricted to work on Shafer's model only while DSmPǫ>0 is always defined and for any model. We have clearly proved through simple examples that the classical BetP and Cuzzolin's transformations do not perform well in term of PIC criterion. It has been shown also how **DSmP**
can be extended to the qualitative domain to approximate qualitative belief assignments provided by human sources in natural language.

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