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We present an efficient parameter-free approach for statistical learning from corrupted training sets. We identify corrupted and non-corrupted samples using latent Bernoulli variables, and therefore formulate the robust learning problem as maximization of the likelihood where latent variables are marginalized out. The resulting optimization problem is solved via variational inference using an efficient Expectation-Maximization based method. The proposed approach improves over the state-of-the-art by automatically inferring the corruption level and identifying outliers, while adding minimal computational overhead. We demonstrate our robust learning method on a wide variety of machine learning tasks including online learning and deep learning where it exhibits ability to adapt to different levels of noise and attain high prediction accuracy., Comment: 14 pages, 14 figures, and 2 tables