Uncertainty measures on probability intervals from the imprecise Dirichlet model

Abstract

When we use a mathematical model to represent information, we can obtain a closed and convex set of probability distributions, also called a credal set. This type of representation involves two types of uncertainty called conflict (or randomness) and non-specificity, respectively. The imprecise Dirichlet model (IDM) allows us to carry out inference about the probability distribution of a categorical variable obtaining a set of a special type of credal set (probability intervals). In this paper, we shall present tools for obtaining the uncertainty functions on probability intervals obtained with the IDM, which can enable these functions in any application of this model to be calculated.

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

“…For the previously defined intervals and for a value of s = 1, Abellán's procedure [21] has a low computation cost. This value is the one recommended by Walley [12] and the one used to build decision trees in Abellán and Moral [10].…”

“…Let |B| be the cardinal of the set B. If we use p to denote the distribution where the maximum of entropy will be reached, the procedure of Abellán [21] for s = 1 can be expressed in the following way: In this procedure, we split the value of s as uniformly as it is possible between the states of C with lower frequency. For values s > 1, the algorithm to find the maximum entropy value has more computational cost.…”

“…The procedure of selecting attributes with the IIG method was implemented using data structures of Weka and the IDM parameter was set to s = 1, i.e., the value used in the original methods of [18]. The reasons to use this value were: first, it was the value recommended by Walley [12]; and, second, the procedure to obtain the maximum entropy value reaches its lowest computational cost for this value [21].…”

A New Robust Classifier on Noise Domains: Bagging of Credal C4.5 Trees

“…That representation gives rise to a specific kind of credal set on the variable , D ( ) [28]. This set is defined as follows:…”

“…It was introduced to build the so-called Credal Decision Tree (CDT) model. Probability intervals are obtained from the data set using Walley's Imprecise Dirichlet Model (IDM) [14] (a special type of credal sets [28]). The mathematical basis applied is described below.…”