The decision about the solution of the problem is usually taken by the maximum of pignistic probability function **BetP**{.}. Let's remark the close ressemblance of the two pignistic transformations (18) and (19).

It can be shown that (19) reduces to (18) when the hyper-power set DΘ reduces to classical power set 2 Θ if we adopt Shafer's model. But (19) is a generalization of (18) since it can be used for computing pignistic probabilities for any models (including Shafer's model). It has been proved in [31] (Chap. 7) that **BetP**{.}
defined in (19) is indeed a probability distribution. In the following section, we introduce a new alternative to BetP which is presented in details in [36].

## 5 The Dsmp Transformation

In the theories of belief functions, the mapping from the belief to the probability domain is a controversial issue.

The original purpose of such mappings was to make (hard) decision, but contrariwise to erroneous widespread idea/claim, this is not the only interest for using such mappings nowadays. Actually the probabilistic transformations of belief mass assignments (as the pignistic transformation mentioned previously) are for example very useful in modern multitarget multisensor tracking systems (or in any other systems) where one deals with soft decisions (i.e. where all possible solutions are kept for state estimation with their likelihoods). For example, in a Multiple Hypotheses Tracker using both kinematical and attribute data, one needs to compute all probabilities values for deriving the likelihoods of data association hypotheses and then mixing them altogether to estimate states of targets. Therefore, it is very relevant to use a mapping which provides a high probabilistic information content (PIC) for expecting better performances.

In this section, we briefly recall a new probabilistic transformation, denoted **DSmP** and introduced in [10]
which will be explained in details in [36]. **DSmP** is straight and different from other transformations. The basic idea of **DSmP** consists in a new way of proportionalizations of the mass of each partial ignorance such as A1∪A2 or A1∪(A2∩A3) or (A1∩A2)∪(A3∩A4), etc. and the mass of the total ignorance A1∪A2∪**. . .**∪An, to the elements involved in the ignorances. This new transformation takes into account both the values of the masses and the cardinality of elements in the proportional redistribution process. We first remind what PIC criteria is and then shortly present the general formula for DSmP transformation with few numerical examples.

More examples and comparisons with respect to other transformations are given in [36].

## 5.1 The Probabilistic Information Content (Pic)

Following Sudano's approach [41, 42, 44], 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.

## - 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 by10 [25]:

$$H(P)\triangleq-\sum_{i=1}^{n}P\{\theta_{i}\}\log_{2}(P\{\theta_{i}\})\tag{20}$$
$P(\theta,\lambda,\dots,1,\ell,\sigma)$
27
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{.}.

10with common convention 0 log2 0 = 0.