We study hierarchical classification in the general case when an instance could belong to more than one class node in the underlying taxonomy. Experiments done in previous work showed that a simple hierarchy of Support Vectors Machines (SVM) with a top-down evaluation scheme has a surprisingly good performance on this kind of task. In this paper, we introduce a refined evaluation scheme which turns the hierarchical SVM classifier into an approximator of the Bayes optimal classifier with respect to a simple stochastic model for the labels. Experiments on synthetic datasets, generated according to this stochastic model, show that our refined algorithm outperforms the simple hierarchical SVM. On real-world data, however, the advantage brought by our approach is a bit less clear. We conjecture this is due to a higher noise rate for the training labels in the low levels of the taxonomy.