Disaggregated total uncertainty measure for credal sets

Abstract

We present a new approach to measure uncertainty/information applicable to theories based on convex sets of probability distributions, also called credal sets. A definition of a total disaggregated uncertainty measure on credal sets is proposed in this paper motivated by recent outcomes. This definition is based on the upper and lower values of Shannon's entropy for a credal set. We justify the use of the proposed total uncertainty measure and the parts into which it is divided: the maximum difference of entropies, which can be used as a non-specificity measure (imprecision), and the minimum of entropy, which represents a measure of conflict (contradiction).

Improving the Naive Bayes Classifier via a Quick Variable Selection Method Using Maximum of Entropy

“…In Abellán, Klir and Moral [6], we can see that the function of the maximum entropy includes two kinds of uncertainty: conflict and non-specificity. The first one is somewhat similar to the one in PT, well quantified by the Shannon's entropy.…”

“…The problem lies in separating this function into others which really do measure the parts of conflict and non-specificity, respectively, and this entails the use of a credal set to represent the information. More recently, Abellán, Klir and Moral [6] presented a separation of the maximum of entropy into functions that are capable of coherently measuring the conflict and non-specificity of a credal set K on a finite variable X, as well as algorithms for facilitating its calculation in capacities of order 2 [6,18], and this may be expressed in the following way:…”

“…The maximum of entropy is the only measure that verifies a similar set of important properties on credal sets rather than the classic entropy in PT (see Abellán et al [6]). The use of the maximum entropy function on this type of sets is also justified with the following sentences of Jaynes [22] about the sense of this measure:…”

“…where S * (K) coherently quantifies the conflict part of the credal set K and (S * − S * )(K) represents the non-specificity part of K [6].…”

“…So far, the most successful one has been the maximum entropy measure (see Abellán et al [6], Abellán and Masegosa [7], and Abellán and Bossé [8]), which verifies a large set of important properties. This set of properties is similar to the one verified by entropy in PT.…”