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For a linguistics course, we implemented Part of Speech (POS) tagging using a hidden Markov model, where the hidden variables were the parts of speech. We trained the system on some tagged data, and then tested it and compared our results with the gold data.

Would it have been possible to train the HMM without the tagged training set?

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Part of Speech Tagging (POS) is a process of tagging sentences with part of speech such as nouns, verbs, adjectives, and adverbs, etc. The Pomegranate library and various other libraries are used to build a hidden Markov model for part of speech tagging. However, having applied HMMs to part of speech, the error you get with the standard form will not be so satisfying. It is a form of expectation-maximization which only meets to local maxima. Rule-based approaches beat HMMs hands down, IIRC. 

Practically it is not possible to train the HMM without the tagged training set. But theoretically, it is possible. In that case, you would use the Baum-Welch-Algorithm. It is described very well in the following article: https://www.cs.ubc.ca/~murphyk/Bayes/rabiner.pdf

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