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This work introduces a novel classifier for a P300-based speller, which, contrary to common methods, can be trained entirely unsupervisedly using an Expectation Maximization approach, eliminating the need for costly dataset collection or tedious calibration sessions. We use publicly available datasets for validation of our method and show that our unsupervised classifier performs competitively with supervised state-of-the-art spellers. Finally, we demonstrate the added value of our method in different experimental settings which reflect realistic usage situations of increasing difficulty and which would be difficult or impossible to tackle with existing supervised or adaptive methods.