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Due to the incompleteness of the external dictionaries and/or knowledge bases, such distantly annotated training data usually suffer from a high false negative rate. To this end, we formulate the Distantly Supervised NER (DS-NER) problem via Multi-class Positive and Unlabeled (MPU) learning and propose a theoretically and practically novel CONFidence-based MPU (Conf-MPU) approach. To handle the incomplete annotations, Conf-MPU consists of two steps. First, a confidence score is estimated for each token of being an entity token. Then, the proposed Conf-MPU risk estimation is applied to train a multi-class classifier for the NER task. 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