We provide an in-depth analysis of the Bayes risk of clustering and the Bayes risk of classification in the context of Hidden Markov and i.i.d. models. In both settings, we identify the situations where the two risks are comparable or not and those where the associated minimizers are related or not, as well as the key quantity measuring the difficulty of both tasks. Then, leveraging the nonparametric identifiability of HMMs, we control the excess risk of a plug-in clustering procedure. Simulations illustrate our findings.

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