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Typically you train a multi-class classifier (or multiple one-vs-rest or one-vs-one classifiers). The classifier then predicts the correct class acoording to the training BoW-features. You can get a way of "similarity" measure if you either choose a classifier which gives you probabilities right away or you compute probabilities from the classifier outputs, e.g. if you used SVMs you could get probabilities from the decision function via Platt calibration.
