Combining nonspecificity measures in Dempster–Shafer theory of evidence

Abstract

In Dempster–Shafer theory (DST) of evidence coexist two types of uncertainty: conflict and nonspecificity. The maximum of entropy is a total uncertainty measure that verifies an important set of properties in this theory. We prove that this function has a fault of sensibility to changes in evidence in situations where only the part of nonspecificity is presented in the uncertainty (uncertainty without conflict), producing no-logical values. We analyse two measures of nonspecificity presented in DST, which have been used as part of total uncertainty measures. We compare their behaviour, focusing on situations where only the part of nonspecificity is presented in the uncertainty. We will see that it makes sense to combine them and use this combination as part of a total uncertainty measure verifying an important set of properties and behaviours. This new total measure has not the mentioned problem of lack of sensitivity to changes in evidence.

Keywords:

• imprecise probabilities
• theory of evidence
• uncertainty-based information
• nonspecificity
• total uncertainty

Acknowledgements

This work was supported by the Spanish ‘Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía’ and ‘Ministerio de Educación y Ciencia’, under Projects TIC-06016 and TIN2007-67418-C03-03.

Notes

1. log and are used indifferently in the literature for this aim. In this paper, we will use log.

2. We will use the notation or indifferently.

3. An alternative complementary interpretation is that there is no conflict when all the focal sets of a b.p.a. share an element (Abellán et al. 2006), which has more sense for extensions of this type of uncertainty on more general theories.

4. Considering the credal set associated with each b.p.a., a curious property is verified for this function. In our case, , and the credal set is the set associated with the b.p.a.: and . It is verified that . This property is always verified when the intersection of the credal sets associated with two b.p.a.s is associated with another b.p.a., but this is scope of another study.

5. Directly and via its parts: (nonspecificity) and (conflict), as is suggested by Abellán and Masegosa (2008).

6. We note that a general disaggregation as , , is not considered here.