The propagation of probabilities in credal networks when probabilities are estimated with a global imprecise Dirichlet model is an important open problem. Only Zaffalon [21] has proposed an algorithm for the Naive classifier. The main difficulty is that, in general, computing upper and lower probability intervals implies the resolution of an optimization of a fraction of two polynomials. In the case of the Naive credal classifier, Zaffalon has shown that the function is a convex function of only one parameter, but there is not a similar result for general credal sets. In this paper, we propose the use of an imprecise global model, but we restrict the distributions to only the most extreme ones. The result is a model giving rise that in the case of estimating a conditional probability under independence relationships, it can produce smaller intervals than the global general model. Its main advantage is that the optimization problem is simpler, and available procedures can be directly applied, as the ones proposed in [7].

Bayesian network are powerful probabilistic graphical models for modelling uncertainty. Among others, classification represents an important application: some of the most used classifiers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided for quantification. This requirement is sometimes too narrow. Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions. This leads to a novel class of imprecise probabilistic graphical models, called credal networks. In particular, classifiers based on Bayesian networks are generalized to so-called credal classifiers. Unlike Bayesian classifiers, which always detect a single class as the one maximizing the posterior class probability, a credal classifier may eventually be unable to discriminate a single class. In other words, if the available information is not sufficient, credal classifiers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian classifiers.

This paper introduces a new approximate inference algorithm for credal networks. The algorithm consists of two major steps. It starts by representing the credal network as a compiled logical theory. The resulting graphical structure is the basis on which the subsequent steepest-ascent hill-climbing algorithm operates. The output of the algorithm is an inner approximation of the exact lower and upper posterior probabilities.