Requirements for total uncertainty measures in Dempster–Shafer theory of evidence

Abstract

Recently, an alternative measure of total uncertainty in Dempster–Shafer theory of evidence (DST) has been proposed in place of the maximum entropy measure. It is based on the pignistic probability of a basic probability assignment and it is proved that this measure verifies a set of needed properties for such a type of measure. The proposed measure is motivated by the problems that maximum (upper) entropy has. In this paper, we analyse the requirements, presented in the literature, for total uncertainty measures in DST and the shortcomings found on them. We extend the set of requirements, which we consider as a set of requirements of properties, and we use the set of shortcomings found on them to define a set of requirements of the behaviour for total uncertainty measures in DST. We present the differences of the principal total uncertainty measures presented in DST taking into account their properties and behaviour.

Also, an experimental comparative study of the performance of total uncertainty measures in DST on a special type of belief decision trees is presented.

Keywords:

• imprecise probabilities
• theory of evidence
• uncertainty based information
• total uncertainty
• conflict
• non-specificity

Acknowledgement

This work has been supported by the Spanish Ministry of Science and Technology under the Projects TIN2007-67418-C03-03 and TIN2005-02516. A shortened version of this paper was presented at NAFIPS'08.

Notes

^1. log and log₂ are used indifferently in the literature for this aim. In this paper, we will use log.

^2. We will use the notation S(p) or S(p)(x ₁),p(x ₂),…) indifferently.

^3. It can be proved that if a b.p.a. m is contained by monotonicity axiom into another b.p.a. m′, we can define a set of b.p.a.s {m _i |i = 1,…, k} such that m ₁ = m and m _k = m′ and m _i is contained by monotone dispensability into m _i+1.

^4. If we want to quantify two types of uncertainty (one more than in the probability theory), perhaps the range requirement should be extended as suggested in existing literature on total uncertainty measures in DST. This is a question that requires further reflection. Function satisfies P1 − P6 requirements.

^5. This reasoning is compatible with the principle of maximum uncertainty [see Klir (2006)].

^6. Within these theories algorithms do exist for obtaining S^* and S_* , as we can see in Abellán and Moral 2005b, 2005c).

^7. We have repeated the experiments with this discretisation method and similar results have been obtained; only NB obtains slightly better results.