I am interested in supervised pattern recognition problems where the the label associated with each pattern gives the probability of membership for each of the $c$ classes, rather than assigning each pattern unequivocally to a single class. This can arise because the (possibly human) oracle providing the labels is unable to decide which class the pattern truly belongs to, or the labels are provided by another probabilistic classifier, and we don't want to throw away information by assigning each pattern to its most probable class. Note this is not a label noise or semi-supervised learning problem; the probabilistic labels accurately represent the oracles' state of knowledge.
I've been experimenting with this type of model for sometime, but I can't find any papers where this type of model s explicitly discussed, so I expect I am not searching with the right terms. Can anybody suggest any suitable books/papers?