
## 2.3 Supporting Hypotheses For Decision

In the VBIED problem the main question (Q1) is related with the security of people in the building B**. The**
potential danger related with this building is of course θ8 = (A, V, B) i.e. the presence of A **in his/her car** V
near the building B**. This is however and unfortunately** not the only origin of the danger since the threat can also come from the possible presence of V **(possible A's**
improvised explosive vehicle) parked near the building B even if A **has left his/her car and is not himself/herself near the building. This second origin of danger**
is represented by θ7 = (A, V, B ¯ **). There exists also a** third origin of the danger represented by θ6 = (A, **V , B** ¯ )
which reflects the possibility to have A near the building without V car. θ6 **is also dangerous for the building**
B since A **can try to commit a suicidal terrorism attack** as human bomb against the building. Therefore based on these three sources of potential danger, the most reasonable/prudent supporting hypothesis for decisionmaking is consider

## Θ6 ∪ Θ7 ∪ Θ8 = (A, V , B ¯ ) ∪ (A, V, B ¯ ) ∪ (**A, V, B**)

If we assume that the danger is mostly due to presence of A**'s vehicle containing possibly a high charge of**
explosive near the building B **rather than the human** bomb attack, then one can prefer to consider only the following hypothesis for decision-making support evaluation

## Θ7 ∪ Θ8 = (A, V, B ¯ ) ∪ (**A, V, B**)

Finally if we are more optimistic**, we can consider**
that the real danger occurs if and only if A **drives** V near the building B **and therefore one could consider** only the supporting hypothesis θ8 = (A, V, B**) for the** danger in the decision-making support evaluation.

In the sequel, we adopt the worst scenario (we take the most prudent choice) and we consider all three origins of potential danger. Thus we will take θ6 ∪ θ7 ∪ θ8 **as cautious/prudent supporting hypothesis** for decision-making.

Thepropositions θ6 ∪ θ8 = (A, V , B ¯ ) ∪ (A, V, B**) and**
θ6∪θ7 = (A, V , B ¯ )∪(A, V, B ¯ **) represent also a potential**
danger and could serve as decision-support hypotheses also, and their imprecise probabilities can be evaluate easily following analysis presented in the sequel. They have not been reported in this paper to keep it at a reasonable size.

## 2.4 Choice Of Bba'S Of Sources

Let's define first the bba of each source without regard to what could be their reliability and importance in the fusion process. Reliability and importance will be examined in details in next section.

- Bba related with source 0 **(prior information):**
The prior information states that the suspect A
drives a white Toyota, and nothing is state about the prior information with respect to his location, so that we must consider the bba's representing the prior information as

$m_{0}(\theta_{4}\cup\theta_{8})=m_{0}((A,V,\bar{B})\cup(A,V,B))$  $=m_{0}((A,V,B\cup\bar{B}))$  $=1$
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- Bba related with source 1 **(Analyst 1 with 10**
years experience): The source 1 reports that the suspect A **is probably now near the building** B.

This source however doesn't report explicitly that the suspect A **is still with its white Toyota car or** not. So the fair way to model this report when working on Θ is to commit a high mass of belief to the element θ6 ∪ θ8**, that is**

$m_{1}(\theta_{6}\cup\theta_{8})=m_{1}((A,\bar{V},B)\cup(A,V,B))$  $=m_{1}((A,V\cup\bar{V},B))$  $=0.75$
and to commit the uncommitted mass to It **based**
on the principle of minimum of specificity, so that m1(θ6 ∪ θ8) = 0.75 and m1(It**) = 0**.25
- Bba related with source 2 **(ANPR system):**
The source 3 reports 30% probability that the vehicle is individual A**'s wite Toyota. Nothing is reported on the position information. The information provided by this source corresponds actually**
to incomplete probabilistic information. Indeed, when working on Θ1 × Θ2**, what we only know**
is that P{(A, V )} = 0.3 and P{(**A, V** ¯ ) ∪ (A, V¯ ) ∪
(A, ¯ V¯ )} = 0.7 (from additivity axiom of probability theory) and thus the bba m2(.**) we must choose**
on Θ1 × Θ2 × Θ3 **has to be compatible with this**
incomplete probabilistic information, i.e. the projection m′2
(.) , m
↓Θ1×Θ2 2(.) of m2(.**) on Θ**1 × Θ2 must satisfy the following constraints on belief and plausibility functions

$$B e l^{\prime}((A,V))=0.3$$
$$B e l^{\prime}((\bar{A},V)\cup(A,\bar{V})\cup(\bar{A},\bar{V}))=0.7$$
and also
$$P l^{\prime}((A,V))=0.3$$
$$P l^{\prime}((\bar{A},V)\cup(A,\bar{V})\cup(\bar{A},\bar{V}))=0.7$$

because belief and plausibility correspond to lower and upper bounds of probability measure [5]. So it is easy to verify that the following bba m′2
(.)