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I understand that an Latent Dirichlet allocation models each document as a mixture of topics where a topic is a distribution over words.

What is not clear to me whether I need to manually specify the topics or whether this is something the LDA is doing for me, similarly to a clustering algorithm?

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Given a collection of documents there exist algorithms to infer (an approximation to) the most likely parameters according to the LDA model, including the topics. So you can use LDA to infer the topics across a set of documents.

The part you do need to specify is the number of topics since this is a hyperparameter of the model.

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