where ǫ ≥ 0 is a tuning parameter and GΘ corresponds to the hyper-power set including eventually all the integrity constraints (if any) of the model M; C(X ∩Y ) and C(Y ) denote the DSm cardinals8 of the sets X ∩ Y and Y respectively. ǫ allows to reach the maximum PIC value of the approximation of m(.) into a subjective probability measure. The smaller ǫ, the better/bigger PIC value. In some particular degenerate cases however, the **DSmP**ǫ=0 values cannot be derived, but the **DSmP**ǫ>0 values can however always be derived by choosing ǫ as a very small positive number, say ǫ = 1/1000 for example in order to be as close as we want to the maximum of the PIC (see next sections for details and examples). When ǫ = 1 and when the masses of all elements Z having C(Z) = 1 are zero, (11) reduces to (3), i.e. DSmPǫ=1 = **BetP**. The passage from a free DSm model to a Shafer's model involves the passage from a structure to another one, and the cardinals change as well in the formula (11).

## B. Advantages Of Dsmp

DSmP works for all models (free, hybrid and Shafer's). In order to apply classical BetP, **CuzzP** or Sudano's mappings, we need at first to refine the frame (on the cases when it is possible!) in order to work with Shafer's model, and then apply their formulas. In the case where refinement makes sense, then one can apply the other subjective probabilities on the refined frame. **DSmP** works on the refined frame as well and gives the same result as it does on the nonrefined frame. Thus **DSmP** with ǫ > 0 works on any models and so is very general and appealing. It is a combination of P rBel and BetP. **P rBel** performs a redistribution of an ignorance mass to the singletons involved in that ignorance proportionally with respect to the singleton masses. While BetP also does a redistribution of an ignorance mass to the singletons involved in that ignorance but proportionally with respect to the singleton cardinals. **P rBel** does not work when the masses of all singletons involved in an ignorance are null since it gives the indetermination 0/0; and in the case when at least one singleton mass involved in an ignorance is zero, that singleton does not receive any mass from the distribution even if it was involved in an ignorance, which is not fair/good. So, DSmP solves the **P rBel** problem by doing a redistribution of the ignorance mass with respect to both the singleton masses and the singletons' cardinals in the same time. Now, if all masses of singletons involved in all ignorances are different from zero, then we can take ǫ = 0, and **DSmP** coincides with **P rBel** and both of them give the best result, i.e. the best PIC value. **P rNPl** is not satisfactory since it yields to an abnormal behavior. Indeed, in any model, when a bba m(.) is transformed into a probability, normally (we mean it is logically that) the masses of ignorances are transferred to the masses of elements of cardinal 1 (in Shafer's model these elements are singletons). Thus, the resulting probability of an element whose cardinal is 1 should be greater than or equal to the mass of that element. I. e. if A in GΘ and C(A) = 1, then P(A) ≥ m(A) for any probability transformation P(.).

This legitimate property is not satisfied by **P rNPl**, since for example if we consider Θ = {**A, B, C**} and m(A) = 0.2, m(B) = m(C) = 0 and m(B ∪ C) = 0.8, one obtains P rNPl(A) = 0.1112 < m(A) = 0.2. So it is abnormal that singleton A looses mass when m(.) is transformed into a subjective probability.

In summary, **DSmP** does an 'improvement' of all Sudano, Cuzzolin, and BetP formulas, in the sense that **DSmP**
mathematically makes a more accurate redistribution of the ignorance masses to the singletons involved in ignorances.

DSmP and **BetP** work in both theories: DST (= Shafer's model) and DSmT (= free or hybrid models) as well. In order to use Sudano's and Cuzzolin's in DSmT models, we have to refine the frame (see Example 5).

## Vi. The Probabilistic Information Content (Pic)

Following Sudano's approach [12], [13], [15], we adopt the Probabilistic Information Content (PIC) criterion as a metric depicting the strength of a critical decision by a specific probability distribution. It is an essential measure in any threshold-driven automated decision system. The PIC is the dual of the normalized Shannon entropy. A PIC value of one indicates the total knowledge to make a correct decision (one hypothesis has a probability value of one and the rest of zero).

A PIC value of zero indicates that the knowledge to make a correct decision does not exist (all the hypotheses have an equal probability value), i.e. one has the maximal entropy. The PIC is used in our analysis to sort the performances of the different pignistic transformations through several numerical examples. We first recall what Shannon entropy and PIC measure are and their tight relationship.

## A. Shannon Entropy

Shannon entropy, usually expressed in bits (binary digits),
of a probability measure P{.} over a discrete finite set Θ =
{θ1**, . . . , θ**n} is defined by9[5]:

$$H(P)\stackrel{{\Delta}}{{=}}-\sum_{i=1}^{n}P\{\theta_{i}\}\log_{2}(P\{\theta_{i}\})\tag{12}$$

H(P) is maximal for the uniform probability distribution over Θ, i.e. when P{θi} = 1/n for i = 1, 2**, . . . , n**. In that case, one gets H(P) = Hmax = −Pn i=1 1 n log2( 1 n
) = log2(n).

H(P) is minimal for a totally *deterministic* probability, i.e.

for any P{.} such that P{θi} = 1 for some i ∈ {1, 2**, . . . , n**} and P{θj} = 0 for j 6= i. H(P) measures the randomness carried by any discrete probability P{.}.

## B. The Pic Metric

The Probabilistic Information Content (PIC) of a probability measure P{.} associated with a probabilistic source over a discrete finite set Θ = {θ1**, . . . , θ**n} is defined by [13]:

$$P I C(P)=1+{\frac{1}{H_{\operatorname*{max}}}}\cdot\sum_{i=1}^{n}P\{\theta_{i}\}\log_{2}(P\{\theta_{i}\})\qquad{\mathrm{(13)}}$$

8We have omitted the index of the model M for notation convenience.

$^9$with common convention $0\log_20=0$. 