Skeptical inferences in multi-label ranking with sets of probabilities

Such MLC problems arise in a number of problems including text categorization [1, 2], music categorization [3], semantic scene classification [4], or protein function classification [5]. We refer to [6] and [7] for comprehensive survey articles on this topic. It is quite common in applications for the multi-label learner to output a ranking on each query instance, that is, a ranking of labels from most likely relevant to most likely irrelevant. A prediction of that kind is commonly evaluated in terms of the rank loss which is the fraction of incorrectly ordered label pairs, where a relevant and a irrelevant label are incorrectly ordered if the former does not precede the latter [8, 9, 10]. The problem of making skeptical inferences for MLC under the presence of uncertainty has been studied in the literature [11, 12, 13].