*3.1. The Ming Vase*

A person is shown a Chinese vase and is asked whether the vase is a real vase of the Ming dynasty or a counterfeit. Sets *A* and *B* are assigned to the two possibilities, as shown in Table 1.

**Table 1.** The Chinese vase and the two considered sets.


Of course, looking at the vase, there could be different situations that also depend on the interviewed person, i.e., whether the person is an expert or not:

	- a. Substantial evidence on both sides.
	- b. Little evidence on both sides.

Let us now consider how these different situations can be handled with the probability and the belief functions.

In the first two cases, the same numerical values are given to both the probability and the belief functions (as shown in the first two cases of Table 2) since probably the interviewed person is an expert, and hence can recognize whether the vase is true or false.


**Table 2.** Assignments given to probability and belief functions in the considered cases.

On the other hand, the other two situations are treated in a different way by the probability and the belief functions since probability functions must obey the additivity rule, while belief functions need not.

Therefore, when cases 3A and 3B are considered, probability functions can take the values, for instance, given in Table 2, but no lower values can be assigned, even if little evidence is present on both *A* and *B*. On the other hand, when belief functions are considered, the person can indicate two numbers, which more precisely represent his/her idea about *A* and *B*.

In case 3A, it may happen that the same numbers are assigned to probability and belief functions (according to the degree of belief about *A* and *B*), but it may also happen that different numbers are assigned since, for belief functions, it is not necessary to satisfy the additivity rule (see Table 2). Furthermore, in case 3B, where there is little evidence on both sides, it is not possible to assign a small number to both *A* and *B* with probability functions, while this can be done with belief functions (see Table 2).

The different behavior of the probability and the belief functions is even more emphasized when Case 4 is considered, where the person is not an expert and therefore declares his/her ignorance about the vase. This is the classical situation, called, by Shafer, *total ignorance*, in which a zero value is assigned to all possible sets (and a unitary value is assigned only to the entire universal set, which include all possibilities). Therefore, as shown in Table 2, *Bel*(*A*) = 0 and *Bel*(*B*) = 0 in the case of total ignorance (Case 4). The probability functions, on the other side, must always obey the additivity rule, and therefore, even in the case of total ignorance (as in the case of equal evidence on both *A* and *B*) *Pro*(*A*) = 0.5 and *Pro*(*B*) = 0.5 are assigned, not to give preference to either *A* or *B*.

Total ignorance is, therefore, treated in a completely different way by the probability and the belief functions; an interesting question is determining which method is the better one. It seems that the belief functions are more suitable to represent total ignorance at least for two reasons. First, with probability functions, it is not possible to distinguish the two different cases where there is an equal degree of belief on both cases *A* and *B*, and there is no evidence about either *A* or *B*. In fact, in both these cases, *Pro*(*A*) = 0.5 and *Pro*(*B*) = 0.5 must be assigned. Second, probability functions may lead to incongruent results when more than two sets are considered, as in the following example of the Sirius star [9].
