We consider the problem of estimating the class prior in an unlabeleddataset. Under the assumption that an additional labeled dataset is available,the class prior can be estimated by fitting a mixture of class-wise datadistributions to the unlabeled data distribution. However, in practice, such anadditional labeled dataset is often not available. In this paper, we show that,with additional samples coming only from the positive class, the class prior ofthe unlabeled dataset can be estimated correctly. Our key idea is to useproperly penalized divergences for model fitting to cancel the error caused bythe absence of negative samples. We further show that the use of the penalized$L_1$-distance gives a computationally efficient algorithm with an analyticsolution. The consistency, stability, and estimation error are theoreticallyanalyzed. Finally, we experimentally demonstrate the usefulness of the proposedmethod.
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