We study probability intervals as an interesting tool to represent uncertain information. A number of basic operations necessary to develop a calculus with probability intervals, such as combination, marginalization, conditioning and integration are studied in detail. Moreover, probability intervals are compared with other uncertainty theories, such as lower and upper probabilities, Choquet capacities of order two and belief and plausibility functions. The advantages of probability intervals with respect to these formalisms in computational efficiency are also highlighted.

No abstract

No abstract

We present an application of the measure of total uncertainty on convex sets of probability distributions, also called credal sets, to the construction of classification trees. In these classification trees the probabilities of the classes in each one of its leaves is estimated by using the imprecise Dirichlet model. In this way, smaller samples give rise to wider probability intervals. Branching a classification tree can decrease the entropy associated with the classes but, at the same time, as the sample is divided among the branches the nonspecificity increases. We use a total uncertainty measure (entropy ϩ nonspecificity) as branching criterion. The stopping rule is not to increase the total uncertainty. The good behavior of this procedure for the standard classification problems is shown. It is important to remark that it does not experience of overfitting, with similar results in the training and test samples.

In this paper a new Monte-Carlo algorithm for the propagation of probabilities in Bayesian networks is proposed. This algorithm has two stages: in the ÿrst one an approximate propagation is carried out by means of a deletion sequence of the variables. In the second stage a sample is obtained using as sampling distribution the calculations of the ÿrst step. The di erent conÿgurations of the sample are weighted according to the importance sampling technique. We show how the use of probability trees to store and to approximate probability potentials, and a careful selection of the deletion sequence, make this algorithm able to propagate over large networks with extreme probabilities.